![OSCILLATIONS 15
that is proportional to the displacement. Then, when the particle is released, the
subsequent motion will be harmonic. A mass at the end of a coil spring (the other
end of the spring held fixed) is an example of such an oscillator. (It should be noted
though that in practice the condition that the restoring force be proportional to the
displacement is generally valid only for sufficiently small displacements.)
If the origin of the time scale is changed so that the displacement is zero at time
t = t1, we get
ξ(t) = A cos[ω(t − t1)] = A cos(ωt − φ) (2.6)
where φ = ωt1 is the phase angle or phase lag. Quantity A is the ampliutde and the
entire argument ωt − φ is sometimes called the ‘phase.’ In terms of the corresponding
motion along a circle, the representative point trails the point P, used earlier, by the
angle φ.
Example
The velocity that corresponds to the displacement in Eq. 2.6. is u = −Aω sin(ωt).
The speed is the absolute value |u| of the velocity. Thus, to get the average speed we
need consider only the average over the time during which u is positive, (i.e., in the
time interval from 0 to T/2), and we obtain
|u| = 2/T
T/2
0
Aω sin(ωt)dt = (2/π)umax, (2.7)
where umax = Aω is the maximum speed.
The mean square value of the velocity is the time average of the squared velocity
and the root mean square value, rms, is the the square root of the mean square value,
u2 = (1/T )
T
0 u2 dt = u2
max/2,
urms = umax/
√
2 (2.8)
where umax = Aω.
We shall take Eq. 2.6 to be the definition of harmonic motion. The velocity u is also
a harmonic function but we have to express it in terms of a cosine function to see what
the phase angle is. Thus, u = ω|ξ| sin(ωt) = ω|ξ| cos(ωt −π/2) is a harmonic motion
with the amplitude ω|ξ| and the phase angle (lag) π/2. Similarly, the acceleration is
a harmonic function a = −ω2|ξ| cos(ωt) = ω2|ξ| cos(ωt − π) with the amplitude
ω2|ξ| and the phase angle π.
One reason for the importance of the harmonic motion is that any periodic function,
period T and fundamental frequency 1/T , can be decomposed in a (Fourier) series of
harmonic functions with frequencies being multiples of the fundamental frequency,
as will be discussed shortly.
2.1.1 The Complex Amplitude
For a given angular velocity ω, a harmonic function ξ(t) = |ξ| cos(ωt −φ) is uniquely
defined by the amplitude |ξ| and the phase angle φ. Geometrically, it can be repre-
sented by a point in a plane at a distance |ξ| from the origin with the radius vector](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-38-320.jpg)
![16 ACOUSTICS
making an angle φ = 0 with the x-axis. This representation reminds us of a complex
number z = x+iy in the complex plane (see Appendix B), where x is the real part and
y the imaginary part. As we shall see, complex numbers and their algebra are indeed
ideally suited for representing and analyzing harmonic motions. This is due to the
remarkable Euler’s identity exp(iα) = cos α + i sin α, where i is the imaginary unit
number i =
√
−1 (i.e., i2 = −1). To prove this relation, expand exp(iα) in a power
series in α, making use of i2 = −1, and collect the real and imaginary parts; they are
indeed found to be the power series expansions of cos α and sin α, respectively. With
the proviso i2 = −1 (i3 = −i, etc.), the exponential exp(iα) is then treated in the
same way as the exponential for a real variable with all the associated algebraic rules.
It is sometimes useful to express cos α and sin α in terms of exp(iα); cos α =
(1/2)[exp(iα) + exp(−iα)] and sin α = (1/2i)[exp(iα) − exp(−iα)].
The complex number exp(iα) is represented in the complex plane by a point with
the real part cos α and the imaginary part sin α. The radius vector to the point makes
an angle α with the real axis. With standing for ‘the real part of’ and with α = ωt−φ,
the harmonic displacement ξ(t) = |ξ| cos(ωt − φ) can be expressed as
Definition of the complex amplitude
ξ(t) = |ξ| cos(ωt − φ) = {|ξ|e−i(ωt−φ)} = {|ξ|eiφ e−iωt } ≡ {ξ(ω)e−iωt }
ξ(ω) = |ξ|eiφ
.
(2.9)
At a given frequency, the complex amplitude ξ(ω) = |ξ| exp(iφ) uniquely defines
the motion.3 It is represented by a point in the complex plane (Fig. 2.1) a distance
|ξ| from the origin and with the line from the origin to the point making an angle φ
with the real axis.
The unit imaginary number can be written i = exp(iπ/2) (=cos(π/2) + i sin(π/2))
with the magnitude 1 and phase angle π/2; it is located at unit distance from the
origin on the imaginary axis. Multiplying the complex amplitude ξ̃(ω) = |ξ| exp(iφ)
by i = exp(iπ/2) increases the phase lag by π/2 and multiplication by −i reduces it
by the same amount.
Differentiation with respect to time in Eq. 2.9 brings down a factor (−iω) =
ω exp(−iπ/2) so that the complex amplitudes of the velocity ξ̇(t) and the acceleration
ξ̈(t) of the particle are (−iω)ξ(ω) and (−iω)2ξ(ω) = −ω2ξ(ω). The locations of
these complex amplitudes are indicated in Fig. 2.1 (with the tilde signatures omitted,
in accordance with the comment on notation given below); their phase lags are smaller
than that of the displacement by π/2 and π, respectively; this means that they are
running ahead of the displacement by these angles.
To visualize the time dependence of the corresponding real quantities, we can let
the complex amplitudes rotate with an angular velocity ω in the counter-clockwise
direction about the origin; the projections on the real axis then yield their time
dependence.
3We could equally well have used ξ(t) = {|ξ| exp[i(ωt − φ)]} = {[|ξ| exp(−iφ)] exp(iωt)} in the
definition of the complex amplitude. It merely involves replacing i by −i. This definition is sometimes used
in engineering where −i is denoted by j. Our choice will be used consistently in this book. One important
advantage becomes apparent in the description of a traveling wave in terms of a complex variable.](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-39-320.jpg)
![OSCILLATIONS 17
Figure 2.1: The complex plane showing the location of the complex amplitudes of displace-
ment ξ(ω), velocity ξ̇(ω) = −iωξ(ω) = ξ(ω) exp(−iπ/2), and acceleration ξ̈(ω) = −ω2ξ(ω).
All the terms in a differential equation for ξ(t) can be expressed in a similar manner
in terms of the complex amplitude ξ(ω). Thus, the differential equation is converted
into an algebraic equation for ξ(ω). Having obtained ξ(ω) by solving the equation,
we immediately get the amplitude |ξ| and the phase angle φ which then define the
harmonic motion ξ(t) = |ξ| cos(ωt − φ).
A Question of Notation
Sometimes the complex amplitude is given a ‘tilde’ symbol to indicate that the function
ξ̃(ω) is the complex amplitude of the displacement. In other words, the complex
amplitude is not obtained merely by replacing t by ω in the function ξ. However, for
convenience in writing and without much risk for confusion, we adopt from now on
the convention of dropping the tilde symbol, thus denoting the complex amplitude
merely by ξ(ω). Actually, as we get seriously involved in problem solving using
complex amplitudes, even the argument will be dropped and ξ alone will stand for
the complex amplitude; the context then will decide whether ξ(t) or ξ(ω) is meant.
Example
What is the complex amplitude of a displacement ξ(t) = |ξ| sin[ω(t − T/6)], where
T is the period of the motion.
The phase angle φ of the complex amplitude ξ(ω) is based on the displacement
being written as a cosine function, i.e., ξ(t) = |ξ| cos(ωt − φ). Thus, we have to
express the sine function in terms of a cosine function, i.e., sin α = cos(α − π/2).
Then, with ωT = 2π, we get sin[ω(t − T/6)] = cos(ωt − π/3 − π/2) = cos(ωt −
5π/6). Thus, the complex amplitude is ξ(ω) = |ξ| exp(i5π/6).
2.1.2 Problems
1. Harmonic motion, definitions
What is the angular frequency, frequency, period, phase angle (in radians), and ampli-
tude of a displacement ξ = 2 cos[100(t − 0.1)] cm, where t is the time in seconds?](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-40-320.jpg)
![OSCILLATIONS 19
Figure 2.2: The functions cos(ω1t), cos(
√
3 ω1t) and their sum (frequencies are incommen-
surable).
If the terms cos(nωt) are replaced by sin(nωt), the sum will still be periodic with
the period T , but it will be anti-symmetric (odd) in the sense that it changes sign
when t does.
If the terms are of the form
an cos(nωt − φn) = an[cos φn cos(nωt) + sin φn sin(nωt)],
where n is an integer, the sum contains a mixture of cosine and sine terms. The sum
will still be periodic with the period T , but the symmetry properties mentioned above
are no longer valid.
We leave it for the reader to experiment with and plot sums of this kind when the
frequencies of the individual terms are integer multiples of a fundamental frequency
or fractions thereof; we shall comment here on what happens when the fraction is an
irrational number.
Thus, consider the sum S(t) = 0.5 cos(ω1t) − 0.5 cos(
√
3 ω1t). The functions and
their sum are plotted in Fig. 2.2. The ratio of the two frequencies is
√
3, an irrational
number (the two frequencies are incommensurable), and no matter how long we wait,
the sum will not be periodic. In the present case, the sum starts out with the value 0
at t = 0 and then fluctuates in an irregular manner between −1 and +1.
On the other hand, if the ratio had been commensurable (i.e., a rational fraction)
the sum would have been periodic; for example, a ratio 2/3 results in a period 3T1.
The addition of two harmonic functions with slightly different frequencies leads
to the phenomenon of beats; it refers to a slow variation of the total amplitude of
oscillation. It is strictly a kinematic effect. It will be illustrated here by the sum of
two harmonic motions with the same amplitude but with different frequencies. The
mean value of the two frequencies is ω, and they are expressed as ω1 = ω − ω and
ω2 = ω+ω. Using the trigonometric identity cos(ωt ∓ωt) = cos(ωt) cos(ωt)±](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-42-320.jpg)
![20 ACOUSTICS
Figure 2.3: An example of beats produced by the sum of two harmonic motions with frequen-
cies 0.9ω and 1.1ω.
sin(ωt) sin(ωt), we find for the sum of the corresponding harmonic motions
ξ(t) = cos(ωt − ωt) + cos(ωt + ωt) = 2 cos(ωt) cos(ωt) (2.10)
which can be interpreted as a harmonic motion of frequency ω with a periodically
varying amplitude (“beats”) of frequency ω. The maximum value of the amplitude
is twice the amplitude of each of the components. An example is illustrated in Fig. 2.3.
In this case, with ω = 0.1ω, the period of the amplitude variation will be ≈10T ,
consistent with the result in the figure. Beats can be useful in experimental work
when it comes to an accurate comparison of the frequencies of two signals.
2.1.4 Heterodyning
The squared sum of two harmonic signals A1 cos(ω1t) and A2 cos(ω2t) produces
signals with the sum and difference frequencies ω1 + ω2 and ω1 − ω2, which can be
of considerable practical importance in signal analysis. The squared sum is
[A1cos(ω1t)+A2 cos(ω2t)]2
= A2
1 cos2
(ω1t)+A2
2 cos2
(ω2t)+2A1A2cos(ω1) cos(ω2t)
(2.11)
The time dependent part of each of the squared terms on the right-hand side is
harmonic with twice the frequency since cos2(ωt) = [1 + cos(2ωt)]/2. This is not
of any particular interest, however. The important part is the last term which can be
written
2A1A2 cos(ω1t) cos(ω2t) = A1A2[cos(ω1 + ω2)t + cos(ω1 − ω2)t]. (2.12)
It contains two harmonic components, one with the sum of the two primary fre-
quencies and one with the difference. This is what is meant by heterodyning, the
creation of sum and difference frequencies of the input signals. Normally, it is the
term with the difference frequency which is of interest.
There are several useful applications of heterodyning; we shall give but one example
here. A photo-cell or photo-multiplier is a device such that the output signal is
proportional to the square of the electric field in an incoming light wave. Thus, if the
light incident on the photo-cell is the sum of two laser signals, the output will contain
an electric current with the difference of the frequencies of the two signals.
Thus, consider a light beam which is split into two with one of the beams reflected
or scattered from a vibrating object, such as the thermal vibrations of the surface
of a liquid, where the reflected signal is shifted in frequency by an amount equal to](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-43-320.jpg)
![OSCILLATIONS 23
We start with the general expression for the harmonic displacement ξ = A cos
(ω0t − φ). It contains the two constants A and φ which are to be determined. Thus,
with t = 0, we obtain,
ξ(0) = A cos(φ)
u(0) = Aω0 sin(φ)
and
tan(φ) = u(0)/[ω0ξ(0)], A = ξ(0)/ cos(φ) = ξ(0)
1 + tan2(φ).
Inserting the numerical values we find
tan(φ) = 1/200 and A = 10
1 + (1/200)2 ≈ 10[1 + (1/2)(1/200)2
].
The subsequent displacement is ξ(t) = A cos(ω0t − φ).
Comment. With the particular initial values chosen in this problem the phase angle
is very small, and the amplitude of oscillation is almost equal to the initial displace-
ment. In other words, the oscillator is started out very nearly from the maximum value
of the displacement and the initial kinetic energy of the oscillator is much smaller
than the initial potential energy. How should the oscillator be started in order for the
subsequent motion to have the time dependence sin(ω0t)?
2.2.2 The ‘Real’ Spring. Compliance
The spring constant depends not only on the elastic properties of the material in
the spring but also on its length and shape. In an ordinary uniform coil spring, for
example, the pitch angle of the coil (helix) plays a role and another relevant factor
is the thickness of the material. The deformation of the coil spring is a complicated
combination of torsion and bending and the spring constant generally should be
regarded as an experimentally determined quantity; the calculation of it from first
principles is not simple.
The linear relation between force and deformation is valid only for sufficiently small
deformations. For example, for a very large elongation, the spring ultimately takes
the form of a straight wire or rod, and, conversely, a large compression will make it
into a tube-like configuration corresponding to a zero pitch angle of the coil. In both
these limits, the stiffness of the spring is much larger than for the relaxed spring.
It has been tactily assumed that the spring constant is determined from a static
deformation. Yet, this constant has been used for non-static (oscillatory) motion.
Although this is a good approximation in most cases, it is not always true. Materials
like rubber and plastics (and polymers in general) for which elastic constants depend
on the rate of strain, the spring constant is frequency dependent. For example,
there exist substances which are plastic for slow and elastic for rapid deformations
(remember ‘silly putty’?). This is related to the molecular structure of the material
and the effect is often strongly dependent on temperature. A cold tennis ball, for
example, does not bounce very well.
In a static deformation of the spring, the inertia of the spring does not enter. If
the motion is time dependent, this is no longer true, and another idealization is the](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-46-320.jpg)
![26 ACOUSTICS
and in the second
p2/2(M + m) − p2/2(M + 2m).
(a) Show that the two energy losses are the same if m/M = 1 +
√
2. Compare the two
energy losses as a function of m/M.
(b) Suppose that n shots are fired into the block under conditions of maximum amplitude
gain as explained in Example 8. What will be the amplitude of the oscillator after the
n:th shot?
2.3 Free Damped Motion of a Linear Oscillator
2.3.1 Energy Considerations
The mechanical energy in the harmonic motion of a mass-spring oscillator is the
sum of the kinetic energy Mu2/2 of the mass M and the potential energy V of the
spring. If the displacement from the equilibrium position is ξ, the force required for
this displacement is Kξ. The work done to reach this displacement is the potential
energy
V (ξ) =
ξ
0
Kξdξ = Kξ2
/2. (2.18)
In the harmonic motion, there is a periodic exchange between kinetic and potential
energy, each going from zero to a maximum value E, where E = Mu2/2 + Kξ2/2
is the total mechanical energy. In the absence of friction, this energy is a constant of
motion.
To see how this follows from the equation of motion, we write the harmonic oscil-
lator equation (2.13) in the form
Mu̇ + Kξ = 0, (2.19)
where u = ξ̇ is the velocity, and then multiply the equation by u. The first term in
the equation becomes Muu̇ = d/dt[Mu2/2]. In the second term, which becomes
Kuξ, we use u = ξ̇ so that it can be written Kξξ̇ = d/dt[Kξ2/2]. This means that
Eq. 2.19 takes the form
d/dt[Mu2
/2 + Kξ2
/2] = 0. (2.20)
The first term, Mu2/2, is the kinetic energy of the mass M, and the second term,
Kξ2/2, is the potential energy stored in the spring. Each is time dependent but the
sum, the total mechanical energy, remains constant throughout the motion. Although
no new physics is involved in this result (since it follows from Newton’s law), the
conservation of mechanical energy is a useful aid in problem solving.
In the harmonic motion, the velocity has a maximum when the potential energy is
zero, and vice versa, and the total mechanical energy can be expressed either as the
maximum kinetic energy or the maximum potential energy. The average kinetic and
potential energies (over one period) are the same.
When a friction force is present, the total mechanical energy of the oscillator is no
longer conserved. In fact, from the equation of motion Mu̇ + Kξ = −Ru it follows](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-49-320.jpg)
![OSCILLATIONS 27
by multiplication by u (see Eq. 2.20) that
d/dt[Mu2
/2 + Kξ2
/2] = −Ru2
. (2.21)
Thus, the friction drains the mechanical energy, at a rate −Ru2, and converts it
into heat.4
As a result, the amplitude of oscillation will decay with time and we can obtain
an approximate expression for the decay by assuming that the average potential and
kinetic energy (over one period) are the same, as is the case for the loss-free oscillator.
Thus, with the left-hand side of Eq. 2.21 replaced by d(Mu2]/dt, and the right-hand
side by Ru2, the time dependence of u2 will be
u(t)2
≈ u(0)2
e−(R/M)t
. (2.22)
The corresponding rms amplitude then will decay as exp[−(R/2M)t].
2.3.2 Oscillatory Decay
After having seen the effect of friction on the time dependence of the average energy,
let us pursue the effect of damping on free motion in more detail and determine the
actual decay of the amplitude and the possible effect of damping on the frequency of
oscillation.
The idealized oscillator considered so far had no other forces acting on the mass
than the spring force. In reality, there is also a friction force although in many cases
it may be small. We shall assume the friction force to be proportional to the velocity
of the oscillator. Such a friction force is often referred to as viscous or dynamic.
Normally, the contact friction with a table, for example, does not have such a simple
velocity dependence. Often, as a simplification, one distinguishes merely between a
‘static’ and a ‘dynamic’ contact friction, the magnitude of the latter often assumed to
be proportional to the magnitude of the velocity but with a direction opposite that of
the velocity. The ‘static’ friction force is proportional to the normal component of the
contact force and points in the direction opposite that of the horizontal component
of the applied force.
A friction force proportional to the velocity can be obtained by means of a dashpot
damper, as shown in Fig. 2.5. It is in parallel with the spring and is simply a ‘leaky’
piston which moves inside a cylinder. The piston is connected to the mass M of the
oscillator and the force required to move the piston is proportional to its velocity
relative to the cylinder (neglecting the mass of the piston). The cylinder is attached
to the same fixed support as the spring, as indicated in Fig. 2.5. The fluid in the
cylinder is then forced through a narrow channel (a ‘leak’) between the piston and
the cylinder and it is the viscous stresses in this flow which are responsible for the
friction force. Therefore, this type of damping is often referred to as viscous.
The friction on a body moving through air or some other fluid in free field will be
proportional to the velocity only for very low speeds and approaches an approximate
square law dependence at high speeds.
4When the concept of energy is extended to include other forms of energy other than mechanical, the
law of conservation of energy does bring something new, the first law of thermodynamics which can be
regarded as a postulate, the truth of which should be considered as an experimental fact.](https://image.slidesharecdn.com/1111702-250531205706-3599d212/85/Notes-On-Acoustics-1st-Edition-Uno-Ingard-50-320.jpg)
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- 8. Notes On ACOUSTICS By Uno Ingard INFINITY SCIENCE PRESS LLC Hingham, Massachusetts New Delhi, India
- 9. Copyright 2008 by INFINITY SCIENCE PRESS LLC All rights reserved. This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means or media, electronic or mechanical, including, but not limited to, photocopy, recording, Internet postings or scanning, without prior permission in writing from the publisher. INFINITY SCIENCE PRESS LLC 11 Leavitt Street Hingham, MA 02043 Tel. 877-266-5796 (toll free) Fax 781-740-1677 info@infinitysciencepress.com www.infinitysciencepress.com This book is printed on acid-free paper. Uno Ingard. Notes on Acoustics. ISBN: 978-1-934015-08-7 0419 The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Library of Congress Cataloging-in-Publication Data Ingard, K. Uno Acoustics / Uno Ingard. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-934015-08-7 (hardcover : alk. paper) 1. Sound. I. Title. QC225.15.I56 2008 534–dc22 2007019297 8 9 10 4 3 2 1 Our titles are available for adoption, license or bulk purchase by institutions, corpo- rations, etc. For additional information, please contact the Customer Service Dept. at 877-266-5796 (toll free).
- 10. Preface Having been involved periodically for many years in both teaching and research in acoustics has resulted in numerous sets of informal notes. The initial impetus for this book was a suggestion that these notes be put together into a book. However, new personal commitments of mine caused the project to be put on hold for several years and it was only after my retirement in 1991 that it was taken up seriously again for a couple of years. In order for the book to be useful as a general text, rather than a collection of research reports, new material had to be added including examples and problems, etc. The result is the present book, which, with appropriate choice of the material, can be used as a text in general acoustics. Taken as such, it is on the senior undergraduate or first year graduate level in a typical science or engineering curriculum. There should be enough material in the book to cover a two semester course. Much of the book includes notes and numerical results resulting to a large extent from my involvement in specific projects in areas which became of particular impor- tance at the early part of the jet aircraft era. In subsequent years, in the 1950’s and 1960’s, much of our work was sponsored by NACA and later by NASA. After several chapters dealing with basic concepts and phenomena follow discus- sions of specific topics such as flow-induced sound and instabilities, flow effects and nonlinear acoustics, room and duct acoustics, sound propagation in the atmosphere, and sound generation by fans. These chapters contain hitherto unpublished material. The introductory material in Chapter 2 on the oscillator is fundamental, but may appear too long as it contains summaries of well known results from spectrum analysis which is used throughout the book. As examples in this chapter can be mentioned an analysis of an oscillator, subject to both ‘dynamic’ and ‘dry’ friction, and an analysis of the frequency response of a model of the eardrum. In hindsight, I believe that parts of the book, particularly the chapters on sound generation by fans probably will be regarded by many as too detailed for an introduc- tory course and it should be apparent that in teaching a course based on this book, appropriate filtering of the material by the instructor is called for. As some liberties have been taken in regard to choice of material, organization, notation, and references (or lack thereof) it is perhaps a fair assessment to say that the ‘Notes’ in the title should be taken to imply that the book in some respects is less formal than many texts. In any event, the aim of the book is to provide a thorough understanding of the fundamentals of acoustics and a foundation for problem solving on a level compatible v
- 11. vi with the mathematics (including the use of complex variables) that is required in a typical science-engineering undergraduate curriculum. Each chapter contains exam- ples and problems and the entire chapter 11 is devoted to examples with solutions and discussions. Although great emphasis is placed on a descriptive presentation in hope of pro- viding ‘physical insight’ it is not at the expense of mathematical analysis. Admittedly, inclusion of all algebraic steps in many derivations can easily interrupt the train of thought, and in the chapter of sound radiation by fans, much of this algebra has been omitted, hopefully without affecting the presentation of the basic ideas involved. Appendix A contains supplementary notes and Appendix B a brief review of the algebra of complex numbers. Acknowledgment. I wish to thank colleagues and former students at M.I.T. as well as engineers and scientists in industry who have provided much of the stimulation and motivation for the preparation of this book. Special thanks go to several individuals who participated in some of the experiments described in the book, in particular to Stanley Oleson, David Pridmore Brown, George Maling, Daniel Galehouse, Lee W. Dean, J. A. Ross, Michael Mintz, Charles McMillan, and Vijay Singhal. At the time, they were all students at M.I.T. A grant from the Du Pont Company to the Massachusetts Institute of Technology for studies in acoustics is gratefully acknowledged. Uno Ingard, Professor Emeritus, M.I.T. Kittery Point, May, 2008
- 12. Contents 1 Introduction 1 1.1 Sound and Acoustics Defined . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Frequency Intervals. Musical Scale . . . . . . . . . . . . . . 3 1.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 An Overview of Some Specialties in Acoustics . . . . . . . . . . . . 3 1.2.1 Mathematical Acoustics . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Architectural Acoustics . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Sound Propagation in the Atmosphere . . . . . . . . . . . . 5 1.2.4 Underwater Sound, Geo-acoustics, and Seismology . . . . . 6 1.2.5 Infrasound. Explosions and Shock Waves . . . . . . . . . . . 7 1.2.6 Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.7 Aero-acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.8 Ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.9 Non-linear Acoustics . . . . . . . . . . . . . . . . . . . . . . 9 1.2.10 Acoustic Instrumentation . . . . . . . . . . . . . . . . . . . 9 1.2.11 Speech and Hearing . . . . . . . . . . . . . . . . . . . . . . 10 1.2.12 Musical Acoustics . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.13 Phonons and Laser Light Spectroscopy . . . . . . . . . . . . 10 1.2.14 Flow-induced Instabilities . . . . . . . . . . . . . . . . . . . 11 1.2.15 Aero-thermo Acoustics. Combustion Instability . . . . . . . 11 1.2.16 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Oscillations 13 2.1 Harmonic Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 The Complex Amplitude . . . . . . . . . . . . . . . . . . . . 15 2.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Sums of Harmonic Functions. Beats . . . . . . . . . . . . . 18 2.1.4 Heterodyning . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 The Linear Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 The ‘Real’ Spring. Compliance . . . . . . . . . . . . . . . . 23 2.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Free Damped Motion of a Linear Oscillator . . . . . . . . . . . . . 26 2.3.1 Energy Considerations . . . . . . . . . . . . . . . . . . . . . 26 vii
- 13. viii 2.3.2 Oscillatory Decay . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3 Use of Complex Variables. Complex Frequency . . . . . . . 28 2.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Forced Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Without Complex Amplitudes . . . . . . . . . . . . . . . . . 31 2.4.2 With Complex Amplitudes . . . . . . . . . . . . . . . . . . . 32 2.4.3 Impedance and Admittance . . . . . . . . . . . . . . . . . . 33 2.4.4 Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.5 Acoustic Cavity Resonator (Helmholtz Resonator) . . . . . . 35 2.4.6 Torsion Oscillator . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.7 Electro-mechanical Analogs . . . . . . . . . . . . . . . . . . 37 2.4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Impulse Response and Applications . . . . . . . . . . . . . . . . . . 38 2.5.1 General Forced Motion of an Oscillator . . . . . . . . . . . 39 2.5.2 Transition to Steady State . . . . . . . . . . . . . . . . . . . 39 2.5.3 Secular Growth . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5.4 Beats Between Steady State and Transient Motions . . . . . 40 2.5.5 Pulse Excitation of an Acoustic Resonator . . . . . . . . . . 41 2.5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Fourier Series and Fourier Transform . . . . . . . . . . . . . . . . . 42 2.6.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 44 2.6.3 Spectrum Densities; Two-sided and One-sided . . . . . . . . 45 2.6.4 Random Function. Energy Spectra and Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.6.5 Random Excitation of the Linear Oscillator . . . . . . . . . 49 2.6.6 Impulse and Frequency Response Functions; Generalization and Summary . . . . . . . . . . . . . . . . . 50 2.6.7 Cross Correlation, Cross Spectrum Density, and Coherence Function . . . . . . . . . . . . . . . . . . . 52 2.6.8 Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . 53 2.6.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 The Potential Well and Nonlinear Oscillators . . . . . . . . . . . . . 55 2.7.1 Period of Oscillation, Large Amplitudes . . . . . . . . . . . 57 2.7.2 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.7.3 Oscillator with ‘Static’ and ‘Dynamic’ Contact Friction . . . 58 2.7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 Sound Waves 63 3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.2 The Complex Wave Amplitude . . . . . . . . . . . . . . . . 65 3.1.3 Standing Wave . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . 66 3.1.5 Wave Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.6 The Doppler Effect . . . . . . . . . . . . . . . . . . . . . . 67
- 14. ix 3.1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2 Sound Wave in a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.1 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.2 Piston Source of Sound . . . . . . . . . . . . . . . . . . . . 73 3.2.3 Sound Speed and Wave Impedance . . . . . . . . . . . . . . 74 3.2.4 Acoustic Levels. Loudness . . . . . . . . . . . . . . . . . . 79 3.2.5 Hearing Sensitivity and Ear Drum Response . . . . . . . . . 82 3.2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Waves on Bars, Springs, and Strings . . . . . . . . . . . . . . . . . . 85 3.3.1 Longitudinal Wave on a Bar or Spring . . . . . . . . . . . . 85 3.3.2 Torsional Waves . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3.3 Transverse Wave on a String. Polarization . . . . . . . . . . 87 3.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Normal Modes and Resonances . . . . . . . . . . . . . . . . . . . . 90 3.4.1 Normal Modes and Fourier Series . . . . . . . . . . . . . . 90 3.4.2 The ‘Real’ Mass-Spring Oscillator . . . . . . . . . . . . . . . 92 3.4.3 Effect of Source Impedance . . . . . . . . . . . . . . . . . . 94 3.4.4 Free Motion of a String. Normal Modes . . . . . . . . . . . 95 3.4.5 Forced Harmonic Motion of a String . . . . . . . . . . . . . 95 3.4.6 Rectangular Membrane . . . . . . . . . . . . . . . . . . . . 97 3.4.7 Rectangular Cavity . . . . . . . . . . . . . . . . . . . . . . . 98 3.4.8 Modal Densities . . . . . . . . . . . . . . . . . . . . . . . . 98 3.4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.5 The Flow Strength of a Sound Source . . . . . . . . . . . . . . . . . 100 3.5.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 Sound on the Molecular Level . . . . . . . . . . . . . . . . . . . . . 102 4 Sound Reflection, Absorption, and Transmission 105 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.1 Reflection, an Elastic Particle Collision Analogy . . . . . . . 105 4.1.2 Gaseous Interface . . . . . . . . . . . . . . . . . . . . . . . 106 4.1.3 Reflection from an Area Discontinuity in a Duct . . . . . . . 107 4.1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Sound Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2.1 Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2.2 The Viscous Boundary Layer . . . . . . . . . . . . . . . . . 111 4.2.3 The Thermal Boundary Layer . . . . . . . . . . . . . . . . . 114 4.2.4 Power Dissipation in the Acoustic Boundary Layer . . . . . 118 4.2.5 Resonator Absorber . . . . . . . . . . . . . . . . . . . . . . 119 4.2.6 Generalization; Impedance Boundary Condition . . . . . . . 120 4.2.7 Measurement of Normal Incidence Impedance and Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . 125 4.2.8 Uniform Porous Absorber . . . . . . . . . . . . . . . . . . . 126 4.2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.3 Sound Transmission Through a Wall . . . . . . . . . . . . . . . . . . 132 4.3.1 Limp Wall Approximation . . . . . . . . . . . . . . . . . . . 132
- 15. x 4.3.2 Effect of Bending Stiffness . . . . . . . . . . . . . . . . . . 135 4.3.3 Measurement of Transmission Loss . . . . . . . . . . . . . . 139 4.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Transmission Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4.1 The Acoustic ‘Barrier’ . . . . . . . . . . . . . . . . . . . . . 141 4.4.2 Acoustic Impedance . . . . . . . . . . . . . . . . . . . . . . 143 4.4.3 Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . 143 4.4.4 Absorption Coefficient . . . . . . . . . . . . . . . . . . . . . 144 4.4.5 Examples of Matrices . . . . . . . . . . . . . . . . . . . . . 144 4.4.6 Choice of Variables and the Matrix Determinant . . . . . . . 146 4.4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5 The Wave Equation 149 5.1 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.1.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 149 5.1.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . 151 5.1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2 Pulsating Sphere as a Sound Source . . . . . . . . . . . . . . . . . . 154 5.2.1 The Point Source. Monopole . . . . . . . . . . . . . . . . . 156 5.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.3 Source and Force Distributions . . . . . . . . . . . . . . . . . . . . 158 5.3.1 Point Force (Dipole) . . . . . . . . . . . . . . . . . . . . . . 159 5.3.2 The Oscillating Compact Sphere . . . . . . . . . . . . . . . 160 5.3.3 Realization of Source and Force Distributions . . . . . . . . 161 5.3.4 Quadrupole and Higher Multipoles . . . . . . . . . . . . . . 162 5.3.5 Circular Piston in an Infinite Baffle . . . . . . . . . . . . . . 163 5.3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.4 Random Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.1 Two Point Sources . . . . . . . . . . . . . . . . . . . . . . . 166 5.4.2 Finite Line Source . . . . . . . . . . . . . . . . . . . . . . . 167 5.4.3 Circular Source Distribution . . . . . . . . . . . . . . . . . 168 5.4.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.5 Superposition of Waves; Nonlinearity . . . . . . . . . . . . . . . . . 169 5.5.1 Array of Line Sources. Strip Source . . . . . . . . . . . . . 171 5.5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6 Room and Duct Acoustics 175 6.1 Diffuse Field Approximation . . . . . . . . . . . . . . . . . . . . . . 175 6.1.1 Reverberation Time . . . . . . . . . . . . . . . . . . . . . . 175 6.1.2 Measurement of Acoustic Power . . . . . . . . . . . . . . . 178 6.1.3 Measurement of the (Sabine) Absorption coefficient . . . . 178 6.1.4 Measurement of Transmission Loss of a Wall . . . . . . . . . 179 6.1.5 Wave Modes in Rooms . . . . . . . . . . . . . . . . . . . . . 180 6.1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.2 Waves in Ducts with Hard Walls . . . . . . . . . . . . . . . . . . . . 181 6.2.1 Wave Modes. Cut-off Frequency and Evanescence . . . . . 182
- 16. xi 6.2.2 Simple Experiment. Discussion . . . . . . . . . . . . . . . . 188 6.2.3 Sound Radiation into a Duct from a Piston . . . . . . . . . . 190 6.3 Lined Ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.3.1 Attenuation Spectra . . . . . . . . . . . . . . . . . . . . . . 195 6.3.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7 Flow-induced Sound and Instabilities 201 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2 Fluid-Solid Body Interaction . . . . . . . . . . . . . . . . . . . . . . 202 7.2.1 Boundary Layers and Drag . . . . . . . . . . . . . . . . . . 202 7.2.2 Model of a Porous Material; Lattice of Spheres . . . . . . . 206 7.3 Flow Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.3.1 Sound from Flow-Solid Body Interaction . . . . . . . . . . . 206 7.3.2 Noise from Turbulence . . . . . . . . . . . . . . . . . . . . 208 7.3.3 Jet Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.4 ‘Spontaneous’ Instabilities . . . . . . . . . . . . . . . . . . . . . . . 213 7.4.1 Single Shear Layer . . . . . . . . . . . . . . . . . . . . . . . 213 7.4.2 Parallel Shear Layers. Kármán Vortex Street . . . . . . . . . 213 7.4.3 Flow Damping . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.5 ‘Stimulated’ Flow Instabilities, a Classification . . . . . . . . . . . . 216 7.6 Flutter; Mechanically Stimulated Flow Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.6.1 Kármán Vortex Street . . . . . . . . . . . . . . . . . . . . . 216 7.6.2 Instability of a Cylinder in Nonuniform Flow . . . . . . . . 217 7.7 Flute Instabilities; Acoustically Stimulated Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.7.1 Cylinder in a Flow Duct. Heat Exchangers . . . . . . . . . . 218 7.7.2 Pipe and Orifice Tones . . . . . . . . . . . . . . . . . . . . . 221 7.7.3 Flow Excitation of a Resonator in Free Field . . . . . . . . . 224 7.7.4 Flow Excitation of a Side-Branch Resonator in a Duct . . . . 226 7.8 Valve Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.8.1 Axial Valve Instability . . . . . . . . . . . . . . . . . . . . . 229 7.8.2 Lateral Valve Instability . . . . . . . . . . . . . . . . . . . . 235 7.8.3 Labyrinth Seal Instability . . . . . . . . . . . . . . . . . . . 239 7.9 Heat Driven Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 241 7.9.1 The Rijke Tube . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.9.2 Combustion Instabilities . . . . . . . . . . . . . . . . . . . . 242 8 Sound Generation by Fans 245 8.1 Axial Fan in Free Field . . . . . . . . . . . . . . . . . . . . . . . . . 245 8.1.1 Sound Generation; Qualitative Observations . . . . . . . . . 247 8.1.2 Point Dipole Simulation . . . . . . . . . . . . . . . . . . . . 251 8.1.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 255 8.1.4 Simulation with Span-wise Distributions of Dipoles . . . . . 257 8.1.5 Effect of Nonuniform Inflow . . . . . . . . . . . . . . . . . 262
- 17. xii 8.2 Fan in a Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.2.1 Modal Cut-off Condition and Exponential Decay . . . . . . 264 8.2.2 Effect of a Nonuniform Flow . . . . . . . . . . . . . . . . . 265 8.2.3 Rotor-Stator Interaction . . . . . . . . . . . . . . . . . . . . 266 8.3 Centrifugal Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.3.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 9 Atmospheric Acoustics 271 9.1 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2 The Earth’s Surface Boundary Layer . . . . . . . . . . . . . . . . . 275 9.2.1 The Stratification of the Atmosphere . . . . . . . . . . . . . 275 9.2.2 Wind Profile . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2.3 Wind Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 276 9.2.4 The Temperature Field . . . . . . . . . . . . . . . . . . . . 277 9.3 Sound Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.3.1 Visco-Thermal Absorption . . . . . . . . . . . . . . . . . . . 278 9.3.2 ‘Molecular’ Absorption . . . . . . . . . . . . . . . . . . . . . 280 9.3.3 Proposed Explanation of Tyndall’s Paradox . . . . . . . . . . 283 9.3.4 Effect of Turbulence . . . . . . . . . . . . . . . . . . . . . . 284 9.3.5 Effect of Rain, Fog, and Snow . . . . . . . . . . . . . . . . . 287 9.3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 9.4 The Effect of Ground Reflection . . . . . . . . . . . . . . . . . . . 288 9.4.1 Pure Tone . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 9.4.2 Random Noise . . . . . . . . . . . . . . . . . . . . . . . . . 291 9.4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.5 Refraction Due to Temperature and Wind Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.5.2 Law of Refraction . . . . . . . . . . . . . . . . . . . . . . . 293 9.5.3 Acoustic ‘Shadow’ Zone . . . . . . . . . . . . . . . . . . . . 296 9.5.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 9.6 Propagation from a High Altitude Source . . . . . . . . . . . . . . . 303 9.6.1 The ‘Real’ Atmosphere . . . . . . . . . . . . . . . . . . . . 304 9.6.2 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 9.6.3 Attenuation Due to Absorption (Vibrational Relaxation) . . . . . . . . . . . . . . . . . . . . 309 9.6.4 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 312 10 Mean-flow Effects and Nonlinear Acoustics 315 10.1 Review of Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . 315 10.1.1 Sound Propagation in a Duct . . . . . . . . . . . . . . . . . 317 10.2 Conservation of Acoustic Energy; Energy Density and Intensity . . . . . . . . . . . . . . . . . . . . . 319 10.2.1 Effect of Mean Flow . . . . . . . . . . . . . . . . . . . . . . 321 10.2.2 Radiation into a Duct with Flow . . . . . . . . . . . . . . . 323 10.2.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
- 18. xiii 10.3 Flow-Induced Acoustic Energy Loss . . . . . . . . . . . . . . . . . 324 10.3.1 Orifice and Pipe Flow . . . . . . . . . . . . . . . . . . . . . 324 10.3.2 Flow-Induced Damping of a Mass-Spring Oscillator . . . . . 327 10.3.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 10.4 The Mass Flux Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 328 10.4.1 Resolution of the Paradox . . . . . . . . . . . . . . . . . . . 328 10.5 Mean Pressure in a Standing Wave . . . . . . . . . . . . . . . . . . 330 10.5.1 Fountain Effect and Mode Visualization . . . . . . . . . . . 330 10.5.2 Acoustic Levitation . . . . . . . . . . . . . . . . . . . . . . . 333 10.5.3 Other Demonstrations . . . . . . . . . . . . . . . . . . . . . 333 10.5.4 Acoustic Radiation Pressure . . . . . . . . . . . . . . . . . . 335 10.5.5 Acoustic ‘Propulsion’ . . . . . . . . . . . . . . . . . . . . . . 335 10.6 Vorticity and Flow Separation in a Sound Field . . . . . . . . . . . . 336 10.6.1 Nonlinear Orifice Resistance . . . . . . . . . . . . . . . . . 338 10.6.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 10.7 Acoustically Driven Mean Flow of Heat . . . . . . . . . . . . . . . . 340 10.8 Formation of a Periodic Shock Wave (‘Saw-Tooth’ Wave) . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.8.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 10.9 Nonlinear Reflection from a Flexible Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 10.9.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.9.2 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . 346 11 Examples 349 A Supplementary Notes 389 A.1 Fourier Series and Spectra . . . . . . . . . . . . . . . . . . . . . . . 389 A.1.1 Fourier Transform. Spectrum of Finite Harmonic Wave Train . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 A.1.2 Fourier Transform and Energy Spectrum . . . . . . . . . . 389 A.1.3 Measurement of Intensity by Means of a Probe . . . . . . . 391 A.2 Radiation from a Circular Piston in an Infinite Wall . . . . . . . . . 391 A.2.1 The Far Field. Radiated Power . . . . . . . . . . . . . . . . 392 A.2.2 Near Field and the Radiation Impedance . . . . . . . . . . 393 A.3 Radiation from Pistons into Ducts . . . . . . . . . . . . . . . . . . . 394 A.3.1 Rectangular Piston in a Rectangular Duct . . . . . . . . . . 394 A.3.2 Circular Piston in a Circular Tube . . . . . . . . . . . . . . . 397 A.4 One-Dimensional Green’s Functions . . . . . . . . . . . . . . . . . 399 A.4.1 Free Field or Infinite Duct and no Mean Flow . . . . . . . . 399 A.4.2 Finite Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 A.4.3 Effects of Mean Flow . . . . . . . . . . . . . . . . . . . . . 403 A.4.4 Radiation from a Piston in the Side Wall of a Duct . . . . . . 405 A.5 Sound from an Axial Fan in Free Field . . . . . . . . . . . . . . . . 409 A.5.1 Point Force Simulation of Axial Fan in Free Field . . . . . . 409 A.5.2 Fan Simulation by Swirling Line Forces . . . . . . . . . . . 412
- 19. xiv A.5.3 Nonuniform Flow . . . . . . . . . . . . . . . . . . . . . . . 413 B Complex Amplitudes 417 B.1 Brief Review of Complex Numbers . . . . . . . . . . . . . . . . . . 417 B.1.1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 417 B.1.2 Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . 418 B.1.3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 418 B.1.4 Euler’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . 419 B.1.5 The Complex Amplitude of a Harmonic Function . . . . . . 420 B.1.6 Discussion. Sign Convention . . . . . . . . . . . . . . . . . 420 B.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 C References 425 C.1 Fourier series in chapter 2 . . . . . . . . . . . . . . . . . . . . . . . 425 C.2 Loudness. Figure 3.4 in chapter 3 . . . . . . . . . . . . . . . . . . . 425 C.3 ‘Molecular’ sound absorption in chapter 9 . . . . . . . . . . . . . . . 425 C.4 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
- 20. List of Figures 2.1 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Sum of Harmonic Motions. Frequencies Incommensurable . . . . 19 2.3 Example of Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Mass-spring Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Oscillator with Dash-pot Damper . . . . . . . . . . . . . . . . . . 28 2.6 Acoustic Cavity Resonator . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Random Function with a Sample of Length . . . . . . . . . . . 47 2.8 Potential Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.9 Decay Curves of Oscillator with Dry Contact Friction . . . . . . . 60 3.1 Doppler Effect. Moving Source, Stationary Observer . . . . . . . 68 3.2 Supersonic Sound Source . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Generation of Sound by a Piston in a Tube . . . . . . . . . . . . . 73 3.4 Equal Loudness Contours . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Normalized Input Impedance of the Eardrum . . . . . . . . . . . 82 3.6 Frequency Response of the Velocity Amplitude of the Eardrum . . 84 3.7 Wave on a String . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.8 Generation of String Wave by Oscillator . . . . . . . . . . . . . . . 89 3.9 Schematic of an Electrodynamic Loudspeaker . . . . . . . . . . . 95 3.10 One-Dimensional Acoustic Source . . . . . . . . . . . . . . . . . 101 4.1 Area Discontinuity in a Duct . . . . . . . . . . . . . . . . . . . . . 107 4.2 Reflection Coefficient and end Correction at the Open End of a Pipe108 4.3 Obliquely Incident Wave on a Boundary . . . . . . . . . . . . . . 121 4.4 Diffuse Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.5 Single Porous Sheet-cavity Absorber . . . . . . . . . . . . . . . . 124 4.6 Absorption Spectra of Sheet Absorber . . . . . . . . . . . . . . . . 125 4.7 Absorption Spectra of a Locally Reacting Rigid Porous Layer . . . 130 4.8 TL of Limp Panel; Angular Dependence and Diffuse Field Average 134 4.9 Plane Wave Incident on a Panel . . . . . . . . . . . . . . . . . . . 136 4.10 TL of a Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.11 Acoustic ‘Barrier’ . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.12 Four-pole Network . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.1 The Normalized Radiation Impedance of a Pulsating Sphere . . . 155 5.2 Piston Radiator in an Infinite Baffle . . . . . . . . . . . . . . . . . 163 xv
- 21. xvi LIST OF FIGURES 5.3 Sound Pressure Level Contours About Two Random Noise Sources 166 5.4 Sound Pressure Level Contours of a Random Noise Line Source . 168 5.5 Sound Radiation from a Moving, Corrugated Board . . . . . . . . 172 6.1 Illustration of Phase Velocity . . . . . . . . . . . . . . . . . . . . . 183 6.2 Circular Tube Coordinates . . . . . . . . . . . . . . . . . . . . . . 185 6.3 Annular Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.4 Generation of a Higher Acoustic Mode in a Duct . . . . . . . . . . 188 6.5 Mass End Correction of a Piston in a Duct . . . . . . . . . . . . . 193 6.6 Attenuation Spectra of a Rectangular Duct. Local Reaction . . . . 196 6.7 Attenuation Spectra of a Circular Duct with a Locally Reacting Porous Liner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.8 Acoustically Equivalent Duct Configurations for the Fundamental Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.1 Flow Interaction with a Sphere . . . . . . . . . . . . . . . . . . . 205 7.2 Drag Coefficient of a Sphere . . . . . . . . . . . . . . . . . . . . . 205 7.3 Power Spectrum of a Subsonic Jet . . . . . . . . . . . . . . . . . . 211 7.4 Kármán Vortex Street . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.5 Transverse Force on Cylinder from Vortex Shedding . . . . . . . . 214 7.6 The Degree of Correlation of Vortex Shedding Along a Cylinder . 215 7.7 Flow-Induced Instabilities, a Classification. . . . . . . . . . . . . . 216 7.8 Cylinder in Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . 217 7.9 Acoustically Stimulated Kármán Street Through Feedback . . . . 219 7.10 Orifice Whistle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.11 Tone Generation in Industrial Driers . . . . . . . . . . . . . . . . 223 7.12 Stability Diagram of a Flow Excited Resonator . . . . . . . . . . . 225 7.13 Flow Excited Resonances of a Side-branch Cavity in a Duct . . . . 226 7.14 Mode Coupling in Flow Excited Resonators . . . . . . . . . . . . 227 7.15 Flow Excitation of a Slanted Resonator in a Duct . . . . . . . . . . 228 7.16 Control Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.17 Stability Contour for Axial Control Valve . . . . . . . . . . . . . . 234 7.18 Lateral Valve Instability . . . . . . . . . . . . . . . . . . . . . . . 235 7.19 Stability Diagram for Flow-Driven Lateral Oscillations of a Valve . 238 7.20 Influence of Flow Direction on a Lateral Valve Instability . . . . . 238 7.21 Concerning Seal Instability . . . . . . . . . . . . . . . . . . . . . . 239 7.22 Seal Instability, Acoustic and Structural Modes . . . . . . . . . . . 240 8.1 Fan Curve and Load Line . . . . . . . . . . . . . . . . . . . . . . 246 8.2 Moving Corrugated Board and Linear Blade Cascade . . . . . . . 248 8.3 Pressure Perturbations Caused by Nonuniform Flow into a Fan . . 250 8.4 Re Point Force Simulation of a Fan (Propeller) in Free Field . . . 251 8.5 Sound Pressure Distribution in the Far Field of a Propeller . . . . 253 8.6 SPL Versus Polar Angle From a Fan in Free Field . . . . . . . . . 254 8.7 SPL spectrum From Fan in Free Field . . . . . . . . . . . . . . . 256 8.8 Radiation Efficiency of a Fan in Free Field . . . . . . . . . . . . . 257
- 22. LIST OF FIGURES xvii 8.9 Directivity Pattern of SPL of Fan in Free Field . . . . . . . . . . . 258 8.10 SPL Spectra from Fan Simulated by Swirling Line Sources . . . . 259 8.11 Sound Pressure Versus Time in the Far Field of a Fan . . . . . . . 260 8.12 Radiated Power Versus Tip Mach Number of Fan . . . . . . . . . 261 8.13 Power Level Spectrum of Fan . . . . . . . . . . . . . . . . . . . . 261 8.14 Effect of Nonuniform Flow on Sound Radiation From a Fan . . . 263 8.15 Circumferential Variation of SPL, Fan in Nonuniform Flow . . . . 263 8.16 Whirling Tube Model of a Centrifugal Fan . . . . . . . . . . . . . 267 9.1 Steam-Driven Siren . . . . . . . . . . . . . . . . . . . . . . . . . 272 9.2 Sound Attenuation in the Atmosphere . . . . . . . . . . . . . . . . 282 9.3 Point Source Above a Reflecting Plane . . . . . . . . . . . . . . . 289 9.4 Field Distribution from a Point Source Above a Boundary . . . . . 291 9.5 Refraction of Sound to a Temperature Gradient . . . . . . . . . . 293 9.6 Law of Refraction in a Moving Fluid . . . . . . . . . . . . . . . . 294 9.7 Refraction of Sound in the Atmosphere . . . . . . . . . . . . . . . 295 9.8 Acoustic Shadow Formation Due to Refraction . . . . . . . . . . . 296 9.9 Directional Dependence of Acoustic Shadow Zone Distance . . . 297 9.10 Effect of Ground Surface on Shadow Formation . . . . . . . . . . 298 9.11 Experimental Data on Sound in a Shadow Zone . . . . . . . . . . 299 9.12 Attenuation Caused by Acoustic Shadow Formation . . . . . . . . 300 9.13 SPL Distribution as Influenced by Shadow Formation . . . . . . . 302 9.14 Temperature, Pressure, Wind, and Humidity Distribution in the atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.15 Sound Emission from an Aircraft in an Inhomogeneous Atmosphere 306 9.16 Emission Angle and Viewing Angle . . . . . . . . . . . . . . . . . 308 9.17 Altitude Dependence of Attenuation Per Unit Length . . . . . . . 310 9.18 Total Attenuation versus Emission Angle from Aircraft in Flight . . 311 9.19 Record of Measured SPL from Over-flight of Air Craft . . . . . . . 313 10.1 Geometrical Interpretation of Dispersion Relation in a Duct with Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 10.2 Sound Beam in the Atmosphere and its Convection by Wind . . . 322 10.3 Experimental Data; Upstream Versus Downstream Radiation . . . 323 10.4 Effect of Flow on Pipe Resonances . . . . . . . . . . . . . . . . . 326 10.5 Flow Damping of Mass-spring Oscillator . . . . . . . . . . . . . . 327 10.6 The Time Average of Eulerian Velocity in a Traveling Harmonic Wave is Negative . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 10.7 Demonstration of Mean Pressure in a Standing Wave . . . . . . . 331 10.8 Contact Print of Liquid Surface Deformation by Acoustic Mode . 332 10.9 Demonstration of Mean Pressure in a Sound Wave . . . . . . . . . 333 10.10 Demonstration of Mean Pressure Distribution in an Acoustic Mode 334 10.11 Liquid Sheet Formation in a Cylindrical Cavity . . . . . . . . . . . 335 10.12 Nonlinear Liquid Sheet Formation in a Standing Wave . . . . . . 335 10.13 Acoustic ‘Propulsion’ . . . . . . . . . . . . . . . . . . . . . . . . . 336 10.14 Acoustically Driven Steady Circulations . . . . . . . . . . . . . . . 337
- 23. xviii LIST OF FIGURES 10.15 Amplitude Dependence of Sound Speed . . . . . . . . . . . . . . 342 10.16 ‘Saw-tooth’ Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 10.17 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.18 Shock Wave Reflection Coefficients . . . . . . . . . . . . . . . . . 346 10.19 Shock Wave Reflections from Flexible Porous Layers . . . . . . . 347 10.20 Compression of Flexible Layer by a Shock Wave . . . . . . . . . . 348 11.1 The Transverse Displacement of the String . . . . . . . . . . . . . 361 11.2 Wave Lines from a Point Source Moving Source . . . . . . . . . . 363 11.3 Pulse Reflection and Transmission on a String . . . . . . . . . . . 370 A.1 Spectrum of a Finite Harmonic Function . . . . . . . . . . . . . . 390 A.2 Radiation from a Circular Piston Source in an Infinite Baffle . . . 393 A.3 Rectangular Piston in a Rectangular Tube . . . . . . . . . . . . . . 395 A.4 Concerning the One-dimensional Green’s Function . . . . . . . . 399 A.5 Concerning the One-dimensional Green’s Function of Finite Duct 402 A.6 Piston Source in the Side-wall of a Duct . . . . . . . . . . . . . . . 406 A.7 Point Force Simulation of a Fan (Propeller) in Free Field . . . . . 409 B.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 419
- 24. Chapter 1 Introduction 1.1 Sound and Acoustics Defined In everyday conversational language, ‘acoustics’ is a term that refers to the quality of enclosed spaces such as lecture and concert halls in regard to their effect on the perception of speech and music. It is supposed to be used with a verb in its plural form. The term applies also to outdoor theaters and ‘bowls.’ From the standpoint of the physical sciences and engineering, acoustics has a much broader meaning and it is usually defined as the science of waves and vibrations in matter. On the microscopic level, sound is an intermolecular collision process, and, unlike an electromagnetic wave, a material medium is required to carry a sound wave.1 On the macroscopic level, acoustics deals with time dependent variations in pressure or stress, often cyclic, with the number of cycles per second, cps or Hz, being the frequency. The frequency range extends from zero to an upper limit which, in a gas, is of the order of the intermolecular collision frequency; in normal air it is ≈ 109 Hz and the upper vibration frequency in a solid is ≈ 1013 Hz. Thus, acoustics deals with problems ranging from earthquakes (and the vibrations induced by them) at the low-frequency end to thermal vibrations in matter on the high. A small portion of the acoustic spectrum, ≈ 20 to ≈ 20, 000 Hz, falls in the audible range and ‘sound’ is often used to designate waves and vibrations in this range. In this book, ‘sound’ and ‘acoustic vibrations and waves’ are synonymous and signify mechanical vibrations in matter regardless of whether they are audible or not. In the audible range, the term ‘noise’ is used to designate ‘undesireable’ and dis- turbing sound. This, of course, is a highly subjective matter. The control of noise has become an important engineering field, as indicated in Section 1.2.6. The term noise is used also in signal analysis to designate a random function, as discussed in Ch. 2. Below and above that audible range, sound is usually referred to as infrasound and ultrasound, respectively. 1Since molecular interactions are electrical in nature, also the acoustic wave can be considered elec- tromagnetic in origin. 1
- 25. 2 ACOUSTICS There is an analogous terminology for electromagnetic waves, where the visible portion of the electromagnetic spectrum is referred to as ‘light’ and the prefixes ‘infra’ and ‘ultra’ are used also here to signify spectral regions below and above the visible range. To return to the microscopic level, a naive one-dimensional model of sound trans- mission depicts the molecules as identical billiard balls arranged along a straight line. We assume that these balls are at rest when undisturbed. If the ball at one end of the line is given an impulse in the direction of the line, the first ball will collide with the second, the second with the third, and so on, so that a wave disturbance will travel along the line. The speed of propagation of the wave will increase with the strength of the impulse. This, however, is not in agreement with the normal behavior of sound for which the speed of propagation is essentially the same, independent of the strength. Thus, our model is not very good in this respect. Another flaw of the model is that if the ball at the end of the line is given an impulse in the opposite direction, there will be no collisions and no wave motion. A gas, however, can support both compression and rarefaction waves. Thus, the model has to be modified to be consistent with these experimental facts. The modification involved is the introduction of the thermal random motion of the molecules in the gas. Through this motion, the molecules collide with each other even when the gas is undisturbed (thermal equilibrium). If the thermal speed of the molecules is much greater than the additional speed acquired through an external impulse, the time between collisions and hence the time of communication between them will be almost independent of the impulse strength under normal conditions. Through collision with its neighbor to the left and then with the neighbor to the right, a molecule can probe the state of motion to the left and then ‘report’ it to the right, thus producing a wave that travels to the right. The speed of propagation of this wave, a sound wave, for all practical purposes will be the thermal molecular speed since the perturbation in molecular velocity typically is only one millionth of the thermal speed. Only for unusually large amplitudes, sometimes encountered in explosive events, will there be a significant amplitude dependence of the wave speed. The curious reader may wish to check to see if our definitions of sound and acoustics are consistent with the dictionary versions. The American Heritage Dictionary tells us that (a): “Sound is a vibratory disturbance in the pressure and density of a fluid or in the elastic strain in a solid, with frequencies in the approximate range between 20 and 20,000 cycles per second, and capable of being detected by the organs of hearing,” and (b): “Loosely, such a disturbance at any frequency.” In the same dictionary, Acoustics is defined as 1. “The scientific study of sound, specially of its generation, propagation, perception, and interaction with materials and with other forms of radiation. Used with a singular verb.” 2. “The total effect of sound, especially as produced in an enclosed space. Used with a plural verb.”
- 26. INTRODUCTION 3 1.1.1 Frequency Intervals. Musical Scale The lowest frequency on a normal piano keyboard is 27.5 Hz and the highest, 4186 Hz. Doubling the frequency represents an interval of one octave. Starting with the lowest C (32.7 Hz), the keyboard covers 7 octaves. The frequency of the A-note in the fourth octave has been chosen to be 440 Hz (international standard). On the equally tempered chromatic scale, an octave has 12 notes which are equally spaced on a logarithmic frequency scale. A frequency interval f2 − f1 represents log2(f2/f1) octaves (logarithm, base 2) and the number of decades is log10(f2/f1). A frequency interval covering one nth of an octave is such that log2(f2/f1) = 1/n, i.e., f2/f1 = 21/n. The center frequency of an interval on the logarithmic scale is the (geometrical) mean value, fm = √ f1f2. Thus, the ratio of the frequencies of two adjacent notes on the equally tempered chromatic scale (separation of 1/12th of an octave) is 21/12 ≈ 1.059 which defines a semitone interval, half a tone. The intervals in the major scale with the notes C, D, E, F, G, A, and B, are 1 tone, 1 tone, 1/2 tone, 1 tone, 1 tone, 1 tone, and 1/2 tone. Other measures of frequency intervals are cent and savart. One cent is 0.01 semi- tones and one savart is 0.001 decades. 1.1.2 Problems 1. Frequencies of the normal piano keyboard The frequency of the A note in the fourth octave on the piano is 440 Hz. List the frequencies of all the other notes on the piano keyboard. 2. Pitch discrimination of the human ear Pitch is the subjective quantity that is used in ordering sounds of different frequencies. To make a variation f in frequency perceived as a variation in pitch, f/f must exceed a minimum value, the difference limen for pitch, that depends on f . However, in the approximate range from 400 to 4000 Hz this ratio is found to be constant, ≈ 0.003 for sound pressures in the normal range of speech. In this range, what is the smallest detectable frequency variation f/f in (a) octaves, (b) cents, (c) savarts? 3. Tone intervals The ‘perfect fifth,’ ‘perfect fourth,’ and ‘major third’ refer to tone intervals for which the frequency ratios are 3/2, 4/3, and 5/4. Give examples of pairs of notes on the piano keyboard for which the ratios are close to these values. 4. Engineering acoustics and frequency bands (a) Octave band spectra in noise control engineering have the standardized center fre- quencies 31.5, 63, 125, 250, 500, 1000, 2000, 4000, and 8000 Hz. What is the bandwidth in Hz of the octave band centered at 1000 Hz. (b) One-third octave bands are also frequently used. What is the relative bandwidth f/fm of a 1/3 octave where f = f2 − f1 and fm is the center frequency? 1.2 An Overview of Some Specialties in Acoustics An undergraduate degree in acoustics is generally not awarded in colleges in the U.S.A., although general acoustics courses are offered and may be part of a de- partmental requirement for a degree. On the graduate level, advanced and more
- 27. 4 ACOUSTICS specialized courses are normally available, and students who wish to pursue a career in acoustics usually do research in the field for an advanced degree in whatever de- partment they belong to. Actually, the borderlines between the various disciplines in science and engineering are no longer very well defined and students often take courses in departments different from their own. Even a thesis advisor can be from a different department although the supervisor usually is from the home department. This flexibility is rather typical for acoustics since it tends to be interdisciplinary to a greater extent than many other fields. Actually, to be proficient in many areas of acoustics, it is almost necessary to have a working knowledge in other fields such as dynamics of fluids and structures and in signal processing. In this section we present some observations about acoustics to give an idea of some of the areas and applications that a student or a professional in acoustics might get involved with. There is no particular logical order or organization in our list of examples, and the lengths of their description are not representative of their relative significance. A detailed classification of acoustic disciplines can be found in most journals of acoustics. For example, the Journal of the Acoustical Society of America contains about 20 main categories ranging from Speech production to Quantum acoustics, each with several subsections. There are numerous other journals such as Sound and Vibration and Applied Acoustics in the U.K. and Acustica in Germany. 1.2.1 Mathematical Acoustics We start with the topic which is necessary for a quantitative understanding of acous- tics, the physics and mathematics of waves and oscillations. It is not surprising that many acousticians have entered the field from a background of waves acquired in electromagnetic theory or quantum mechanics. The transition to linear acoustics is then not much of a problem; one has to get used to new concepts and solve a number of problems to get a physical feel for the subject. To become well-rounded in aero- acoustics and modern problems in acoustics, a good knowledge of aerodynamics and structures has to be acquired. Many workers in the field often spend several years and often a professional career working on various mathematical wave problems, propagation, diffraction, radiation, interaction of sound with structures, etc., sometimes utilizing numerical techniques. These problems frequently arise in mathematical modeling of practical problems and their solution can yield valuable information, insights, and guidelines for design. 1.2.2 Architectural Acoustics Returning to the two definitions of acoustics above, one definition refers to the per- ception of speech and music in rooms and concert halls. In that case, as mentioned, the plural form of the associated verb is used. Around the beginning of the 20th century, the interest in the acoustical character- istics of rooms and concert halls played an important role in the development of the field of acoustics as a discipline of applied physics and engineering. To a large extent, this was due to the contributions by Dr. Wallace C. Sabine, then a physics professor
- 28. INTRODUCTION 5 at Harvard University, with X-rays as his specialty. His acoustic diversions were moti- vated initially by his desire to try to improve the speech intelligibility in an incredibly bad lecture hall at Harvard. He used organ pipes as sound sources, his own hearing for sound detection, and a large number of seat cushions (borrowed from a nearby theater) as sound absorbers. To eliminate his own absorption, he placed himself in a wooden box with only his head exposed. With these simple means, he established the relation between reverberation time and absorption in a room, a relation which now bears his name. The interest was further stimulated by his involvement with the acoustics of the Boston Symphony Hall. These efforts grew into extensive sys- tematic studies of the acoustics of rooms, which formed the foundation for further developments by other investigators for many years to come. ItisafarcryfromSabine’ssimpleexperimentstomodernresearchinroomacoustics with sophisticated computers and software, but the necessary conditions for ‘good’ acoustics established by Sabine are still used. They are not sufficient, however. The difficulty in predicting the response of a room to music and establishing subjective measures of evaluation are considerable, and it appears that even today, concert hall designers arerelyingheavilyon empiricismandtheirknowledgeofexisting‘good’halls as guides. With the aid of modern signal analysis and data processing, considerable research is still being done to develop a deeper understanding of this complex subject. Architectural acoustics deals not only with room acoustics, i.e., the acoustic re- sponse of an enclosed space, but also with factors that influence the background noise level in a room such as sound transmission through walls and conduits from external sources and air handling systems. 1.2.3 Sound Propagation in the Atmosphere Many other areas of acoustics have emerged from specific practical problems. A typical example is atmospheric acoustics. For the past 100 years the activity in this field has been inspired by a variety of societal needs. Actually, interest in the field goes back more than 100 years. The penetrating crack of a bolt of lightning and the rolling of thunder always have aroused both fear and curiosity. It is not until rather recently that a quantitative understanding of these effects is emerging. The early systematic studies of atmospheric acoustics, about a century ago, were not motivated by thunder, however, but rather by the need to improve fog horn signaling to reduce the hazards and the number of ship wrecks that were caused by fog in coastal areas. Many prominent scientists were involved such as Tyndall and Lord Rayleigh in England and Henry in the United States. Through their efforts, many important results were obtained and interesting questions were raised which stimulated further studies in this field. Later, a surge of interest in sound propagation in the atmosphere was generated by the use of sound ranging for locating sound sources such as enemy weapons. In this country and abroad, several projects on sound propagation in the atmosphere were undertaken and many theoretical physicists were used in these studies. In Russia, one of their most prominent quantum theorists, Blokhintzev, produced a unique document on sound propagation in moving, inhomogeneous media, which later was translated by NACA (now NASA).
- 29. 6 ACOUSTICS The basic problem of atmospheric acoustics concerns sound propagation over a sound absorptive ground in an inhomogeneous turbulent atmosphere with tempera- ture and wind gradients. The presence of wind makes the atmosphere acoustically anisotropic and the combination of these gradients and the effect of the ground gives rise to the formation of shadow zones. The theoretical analysis of sound propagation under these conditions is complicated and it is usually supplemented by experimental studies. The advent of the commercial jet aircraft created community noise problems and again sound propagation in the atmosphere became an important topic. Numerous extensive studies, both theoretical and experimental, were undertaken. The aircraft community noise problem in the US led to federal legislation (in 1969) for the noise certification of aircraft, and this created a need for measurement of the acoustic power output of aircraft engines. It was soon realized that atmospheric and ground conditions significantly affected the results and again detailed studies of sound propagation were undertaken. The use of sound as a diagnostic tool (SODAR, SOund Detection And Ranging) for exploration of the conditions of the lower atmosphere also should be mentioned as having motivated propagation studies. An interesting application concerns the possibility of using sound scattering for monitoring the vortices created by a large aircraft at airports. These vortices can remain in the atmosphere after the landing of the aircraft, and they have been found to be hazardous for small airplanes coming in for landing in the wake of a large plane. 1.2.4 Underwater Sound, Geo-acoustics, and Seismology The discussion of atmospheric acoustics above illustrates how a particular research activity often is stimulated and supported from time to time by many different societal needs and interests. Atmospheric acoustics has its counterparts in the sea and in the ground, sometimes referred to as ocean and geo-acoustics, respectively. From studies of the sound transmission characteristics, it is possible to get information about the sound speed profiles which in turn contain information about the structure and composition of the medium. Geo-acoustics and seismology deal with this problem for exploring the structure of the Earth, where, for example, oil deposits are the target of obvious commercial interests. Sound scattering from objects in the ocean, be it fish, submarines, or sunken ships, can be used for the detection and imaging of these objects in much the same way as in medical acoustics in which the human body is the ‘medium’ and organs, tumors, and fetuses might be the targets. During World War II, an important battleground was underwater and problems of sound ranging in the ocean became vitally important. This technology developed rapidly and many acoustical laboratories were established to study this problem. It was in this context the acronym SONAR was coined. More recently, the late Professor Edgerton at M.I.T., the inventor of the modern stroboscope, developed underwater scanners for the exploration of the ocean floor and for the detection of sunken ships and other objects. He used them extensively in
- 30. INTRODUCTION 7 collaboration with his good friend, the late Jacques Cousteau, on many oceanographic explorations. 1.2.5 Infrasound. Explosions and Shock Waves Geo-acoustics, mentioned in the previous section, deals also with earthquakes in which most of the energy is carried by low frequencies below the audible range (i.e., in the infrasound regime). These are rather infrequent events, however, and the interest in infrasound, as far as the interaction with humans and structures is concerned, is usually focused on various industrial sources such as high power jet engines and gas turbine power plants for which the spectrum of significant energy typically goes down to about 4 Hz. The resonance frequency of walls in buildings often lie in the infrasonic range and infrasound is known to have caused unacceptable building vibration and even structural damage. Shock waves, generated by explosions or supersonic air craft (sonic boom), for example, also contain energy in the infrasonic range and can have damaging effects on structures. For example, the spectrum often contain substantial energy in a frequency range close to the resonance frequency of windows which often break as a result of the ‘push-pull’ effect caused by such waves. The break can occur on the pull half-cycle, leaving the fragments of the window on the outside. 1.2.6 Noise Control In atmospheric acoustics research, noise reduction was one of the motivating societal needs but not necessarily the dominant one. In many other areas of acoustics, how- ever, the growing concern about noise has been instrumental in promoting research and establishing new laboratories. Historically, this concern for noise and its effect on people has not always been apparent. During the Industrial Revolution, 100 to 150 years ago, we do not find much to say about efforts to control noise. Rather, part of the reason was probably that noise, at least industrial noise, was regarded as a sign of progress and even as an indicator of culture. Only when it came to problems that involved acoustic privacy in dwellings was the attitude somewhat different. Actually, building constructions incorporating design principles for high sound insulation in multi-family houses can be traced back as far as to the 17th century, and they have been described in the literature for more than 100 years. As a historical aside, we note that in 1784 none less than Michael Faraday was hired by the Commissioner of Jails in England to carry out experiments on sound transmission of walls in an effort to arrive at a wall construction that would prevent communication between prisoners in adjacent cells. This was in accord with the then prevailing attitude in penology that such an isolation would be beneficial in as much as it would protect the meek from the savage and provide quiet for contemplation. More recently, the need for sound insulation in apartment buildings became par- ticularly acute when, some 50 to 60 years ago, the building industry more and more turned to lightweight constructions. It quickly became apparent that the building in- dustry had to start to consider seriously the acoustical characteristics of materials and
- 31. 8 ACOUSTICS building constructions. As a result, several acoustical laboratories were established with facilities for measurement of the transmission loss of walls and floors as well as the sound absorptive characteristics of acoustical materials. At the same time, major advances were made in acoustical instrumentation which made possible detailed experimental studies of basic mechanisms and understanding of sound transmission and absorption. Eventually, the results thus obtained were made the basis for standardized testing procedures and codes within the building industry. Noise control in other areas developed quickly after 1940. Studies of noise from ships and submarines became of high priority during the second World War and spe- cialized laboratories were established. Many mathematicians and physical scientists were brought into the field of acoustics. Of more general interest, noise in transportation, both ground based and air borne, has rapidly become an important problem which has led to considerable investment on the part of manufacturers on noise reduction technology. Related to it is the shielding of traffic noise by means of barriers along highways which has become an industry all of its own. Aircraft noise has received perhaps even more attention and is an important part of the ongoing work on the control of traffic noise and its societal impact. 1.2.7 Aero-acoustics The advent of commercial jet air craft in the 1950s started a new era in acoustics, or more specifically in aero-acoustics, with the noise generation by turbulent jets at the core. Extensive theoretical and experimental studies were undertaken to find means of reducing the noise, challenging acousticians, aerodynamicists, and mathematicians in universities, industrial, and governmental laboratories. Soon afterwards, by-pass engines were introduced, and it became apparent that the noise from the ducted fan in these engines represented a noise problem which could be even more important than the jet noise. In many respects, it is also more difficult than the jet noise to fully understand since it involves not only the generation of sound from the fan and guide vane assemblies but also the propagation of sound in and radiation from the fan duct. Extensive research in this field is ongoing. 1.2.8 Ultrasonics There are numerous other areas in acoustics ranging from basic physics to various industrial applications. One such area is ultrasonics which deals with high frequency sound waves beyond the audible range, as mentioned earlier. It contains many sub- divisions. Medical acoustics is one example, in which ultrasonic waves are used as a means for diagnostic imaging as a supplement to X-rays. Surgery by means of focused sound waves is also possible and ultrasonic microscopy is now a reality. Ultrasonic ‘drills,’ which in essence are high frequency chip hammers, can produce arbitrarily shaped holes, and ultrasonic cleaning has been known and used for a long time. Ultrasound is used also for the detection of flaws in solids (non-destructive testing) and ultrasonic transducers can be used for the detection of acoustic emission from
- 32. INTRODUCTION 9 stress-induced dislocations. This can be used for monitoring structures for failure risk. High-intensity sound can be used for emulsification of liquids and agglomeration of particles and is known to affect many processes, particularly in the chemical industry. Ultrasonic waves in piezo-electric semi-conductors, both in bulk and on the surface, can be amplified by means of a superimposed electric field. Many of these and related industrial applications are sometimes classified under the heading Sonics. 1.2.9 Non-linear Acoustics In linear acoustics, characterized by sound pressures much smaller than the static pressure, the time average value of the sound pressure or any other acoustic variable in a periodic signal is zero for most practical purposes. However, at sufficiently large sound pressures and corresponding fluid velocity amplitudes, the time average or mean values can be large enough to be significant. Thus, the static pressure variation in a standing sound wave in an enclosure can readily be demonstrated by trapping light objectsandmovingthembyalteringthestandingwavefieldwithoutanyothermaterial contact with the body than the air in the room. This is of particular importance in the gravity free environment in a laboratory of an orbiting satellite. Combination of viscosity and large amplitudes can also produce significant acous- tically induced mean flow (acoustic streaming) in a fluid and a corresponding particle transport. Similarly, the combination of heat conduction and large amplitudes can lead to a mean flow of heat and this effect has been used to achieve acoustically driven refrigeration using acoustic resonators driven at resonance to meet high amplitude requirements. Other interesting effects in nonlinear acoustics include interaction of a sound wave with itself which makes an initially plane harmonic wave steepen as it travels and ultimately develop into a saw tooth wave. This is analogous to the steepening of surface waves on water. Interaction of two sound waves of different frequencies leads to the generation of sum and difference frequencies so that a low-frequency wave can be generated from two high-frequency waves. 1.2.10 Acoustic Instrumentation Much of what we have been able to learn in acoustics (as in most other fields) has been due to the availability of electronic equipment both for the generation, detection, and analysis of sound. The rapid progress in the field beginning about 1930 was due to the advent of the radio tube and the equipment built around it. This first electronic ‘revolution,’ the electronic ‘analog’ era, was followed with a second with the advent of the transistor which led into the present ‘digital’ era. The related development of equipment for acoustic purposes, from Edison’s original devices to the present, is a fascinating story in which many areas of acoustics have been involved, including the electro-mechanics of transducers, sound radiation, room acoustics, and the perception of sound.
- 33. 10 ACOUSTICS 1.2.11 Speech and Hearing The physics, physiology, and psychology of hearing and speech occupies a substantial part of modern acoustics. The physics of speech involves modeling the vocal tract as a duct of variable area (both in time and space) driven at the vocal chords by a modulated air stream. A wave theoretical analysis of the response of the vocal tract leads to an understanding of the frequency spectrum of the vowels. In the analysis of the fricative sounds, such as s, sh, ch, and t, the generation of sound by turbulent flow has to be accounted for. On the basis of the understanding thus obtained, synthetic speech generators have been developed. Hearing represents a more complicated problem, even on the physics level, which deals with the acoustics of the ear canal, the middle ear, and, in particular, the fluid dynamics in the inner ear. In addition, there are the neurological aspects of the problem which are even more complex. From extensive measurements, however, much of the physics of hearing has been identified and understood, at least in part, such as the frequency dependence of the sensitivity of the human ear, for example. 1.2.12 Musical Acoustics The field of musical acoustics is intimately related to that of speech. The physics now involves an understanding of sound generation by various musical instruments rather than by the vocal tract. A thorough understanding of wind instruments requires an intimate knowledge of aero-acoustics. For string instruments, like the violin and the piano, the vibration and radiation characteristics of the sounding boards are essential, and numerous intricate experiments have been carried out in efforts to make the vibrations visible. 1.2.13 Phonons and Laser Light Spectroscopy The thermal vibrations in matter can be decomposed into (random) acoustic waves over a range of wavelengths down to the distance between molecules. The exper- imental study of such high-frequency waves (‘hypersonics’) requires a ‘probe’ with the same kind of resolution and the use of (Brillouin) scattering of laser light is the approach that has been used (photon-phonon interaction). By analysis of the light scattered by the waves in a transparent solid (heterodyne spectroscopy), it is possible to determine the speed of sound and the attenuation in this high-frequency regime. The scattered light is shifted in frequency by an amount equal to the frequency of the acoustic wave and this shift is measured. Furthermore, the line shape of the scattered light provides another piece of information so that both the sound speed and attenuation can be determined. A similar technique can be used also for the thermal fluctuations of a liquid surface which can be decomposed into random high-frequency surface waves. The upper frequency limit varies from one liquid to the next but the corresponding wavelength is of the order of the intermolecular distance. Again by using the technique of laser light heterodyne spectroscopy, both surface tension and viscosity can be determined. Actually, even for the interface between two liquids which do not mix, these quantities
- 34. INTRODUCTION 11 can be determined. The interfacial surface tension between water and oil, for exam- ple, is of considerable practical interest. 1.2.14 Flow-induced Instabilities The interaction of a structure with fluid flow can lead to vibrations which under certain conditions can be unstable through feedback. The feedback can be a result of the interactions between fluid flow, sound, and the structure. In some musical wind instruments, such as an organ pipe or a flute, the structure can be regarded as rigid as far as the mechanism of the instability is concerned, and it is produced as a result of the interaction of vorticity and sound. The sound produced by a vortex can react on the fluid flow to promote the growth of the vortex and hence give rise to a growing oscillation and sound that is sustained by the flow through this feedback. A similar instability, which is very important in some industrial facilities, is the ‘stimulated’ Kármán vortex behind a cylinder in a duct. The periodic vortex can be stimulated through feedback by an acoustic cross mode in the duct if its resonance frequency is equal to (or close to) the vortex frequency. This is a phenomenon which can occur in heat exchangers and the amplitude can be so large that it represents an environmental problem and structural failure can also result. A stimulation of the Kàrmàn vortex can result also if the cylinder is flexible and if the transverse resonance frequency of the cylinder is the same as the vortex frequency. Large vibrations of a chimney can occur in this manner and the structural failure of the Tacoma bridge is a classic example of the destructive effects that can result from this phenomenon. In a reed type musical instrument, or in an industrial control valve, the reed or the valve plug represents a flexible portion of the structure. In either of these cases, this flexible portion is coupled to the acoustic resonator which, in the case of the plug, is represented by the pipe or duct involved. If the resonance frequencies of the structure and the pipe are sufficiently close, the feedback can lead to instability and very large vibration amplitudes, known to have caused structural failures of valves. 1.2.15 Aero-thermo Acoustics. Combustion Instability This designation as a branch of acoustics is sometimes used when heat sources and heat conduction have a significant influence on the acoustics. For example, the sound generation in a combustor falls into this category as does the acoustic refrigeration mentioned earlier. The rate of heat release Q in a combustor acts like a source of sound if Q is time dependent with the acoustic source strength being proportional to dQ/dt. If Q is also pressure dependent, the sound pressure produced in the combustion chamber can feed back to the combustor and modulate the acoustic output. This can lead to an instability with high amplitude sound (and vibration) as a consequence. The vibrations can be so violent that structural failure can result when a facility, such as a gas turbine power plant, is operating above a certain power setting. The challenge, of course, is to limit the amplitude of vibrations or, even better, to eliminate the
- 35. 12 ACOUSTICS instability. An acoustic analysis can shed valuable light on this problem and can be most helpful in identifying its solution. 1.2.16 Miscellaneous As in most other fields of science and engineering, there are numerous activities dealing with regulations, codes, standards, and the like. They are of considerable importance in industry and in government agencies and there is great need for inputs from experts. Working in such a field, even for a short period, is apt to provide famil- iarity with various government agencies and international organizations and serve as an introduction to the art of politics.
- 36. Chapter 2 Oscillations As indicated in the Preface, it is assumed that the reader is familiar with the content of a typical introductory course in mechanics that includes a discussion of the basics of the harmonic oscillator. It is an essential element in acoustics and it will be reviewed and extended in this chapter. The extension involves mainly technical aspects which are convenient for problem solving. Thus, the use of complex variables, in particular the complex amplitude, is introduced as a convenemt and powerful way of dealing with oscillations and waves. With modern digital instrumentation, many aspects of signal processing are read- ily made available and to be able to fully appreciate them, it is essential to have some knowledge of the associated mathematics. Thus, Fourier series and Fourier transforms, correlation functions, spectra and spectrum analysis are discussed. As an example, the response of an oscillator to a completely random driving force is deter- mined. This material is discussed in Section 2.6. However, it can be skipped at a first reading without a lack of continuity. The material referred to above is all ‘standard’; it is important to realize, though, that it is generally assumed that the oscillators involved and the related equations of motion are linear. This is an idealization, and is valid, at best, for small amplitudes of oscillations. But even for small amplitudes, an oscillator can be non-linear, and we end this chapter with a simple example. It involves a damped mass-spring oscillator. Normally, the friction force is tactily assumed to be proportional to the velocity in which case the equation of motion becomes linear and a solution for the displacement is readily found. However, consider the very simple case of a mass sliding on a table and subject not only to a (‘dynamic’) friction force proportional to the velocity but also to a (‘static’) friction force proportional to the static friction coefficient. 2.1 Harmonic Motions A periodic motion is one that repeats itself after a constant time interval, the period, denoted T . The number of periods (cycles) per second, cps, is called the frequency f (i.e. f = 1/T cps or Hz).1 For example, a period of 0.5 seconds corresponds to a frequency of 2 Hz. 1The unit Hz after the German physicist Heinrich Hertz (1857−1894). 13
- 37. 14 ACOUSTICS The periodic motion plays an important role in nature and everyday life; the spin of the earth and the orbital motion (assumed uniform) of the earth and of the moon are obvious examples. The ordinary pendulum is familiar to all but note that the period of oscillation increases with the amplitude of oscillation. This effect, however, is insignificant at small amplitudes. To obtain periodicity to a very high degree of accuracy, one has to go down to the atomic level and consider the frequency of atomic ‘vibrations.’ Actually, this is the basis for the definition of the unit of time. A good atomic clock, a Cesium clock, loses or gains no more than one second in 300,000 years and the unit of time, one second, is defined as the interval for 9,191,631,770 periods of the Cesium atom.2 Harmonic motion is a particular periodic motion and can be described as follows. Consider a particle P which moves in a circular path of radius A with constant speed. The radius vector to the particle makes an angle with the x-axis which is proportional to time t, expressed as ωt, where ω is the angular velocity (for rectilinear motion, the position of the particle is x = vt, where v is the linear velocity). It is implied that the particle crosses the x-axis at t = 0. After one period T of this motion, the angle ωt has increased by 2π, i.e., ωT = 2π or ω = 2π/T = 2πf (2.1) where f = 1/T , is the frequency, introduced above. In general discussions, the term frequency, rather than angular frequency, is often used also for ω. Of course, in numerical work one has to watch out for what quantity is involved, ω or f . The time dependence of the x-coordinate of the particle P defines the harmonic motion ξ = A cos(ωt). (2.2) It is characteristic of harmonic motion that ω does not depend on time. But note that a motion can be periodic even if ω is time dependent. This is the case for the motion of a planet in an elliptical orbit, for example. The velocity in the harmonic motion is u = ξ̇ = −Aω sin(ωt) (2.3) and the acceleration a = ξ̈ = −Aω2 cos(ωt) = −ω2 ξ. (2.4) It follows that the harmonic motion satisfies the differential equation ξ̈ = −ω2 ξ. (2.5) Thus, if an equation of this form is encountered in the study of motion, we know that the harmonic motion is a solution. As we shall see, such is the case when a particle, displaced from its equilibrium position, is acted on by a restoring force 2The unit of length, one meter, is defined in such a way as to make the speed of light exactly 3×108 m/sec; thus, the unit of length, one meter, is the distance traveled by light in (1/3)10−8 sec. This unit is very close to the unit of length based on the standard meter (a bar of platinum-iridium alloy) kept at the International Bureau of Weight and Measures at Sèvres, France.
- 38. OSCILLATIONS 15 that is proportional to the displacement. Then, when the particle is released, the subsequent motion will be harmonic. A mass at the end of a coil spring (the other end of the spring held fixed) is an example of such an oscillator. (It should be noted though that in practice the condition that the restoring force be proportional to the displacement is generally valid only for sufficiently small displacements.) If the origin of the time scale is changed so that the displacement is zero at time t = t1, we get ξ(t) = A cos[ω(t − t1)] = A cos(ωt − φ) (2.6) where φ = ωt1 is the phase angle or phase lag. Quantity A is the ampliutde and the entire argument ωt − φ is sometimes called the ‘phase.’ In terms of the corresponding motion along a circle, the representative point trails the point P, used earlier, by the angle φ. Example The velocity that corresponds to the displacement in Eq. 2.6. is u = −Aω sin(ωt). The speed is the absolute value |u| of the velocity. Thus, to get the average speed we need consider only the average over the time during which u is positive, (i.e., in the time interval from 0 to T/2), and we obtain |u| = 2/T T/2 0 Aω sin(ωt)dt = (2/π)umax, (2.7) where umax = Aω is the maximum speed. The mean square value of the velocity is the time average of the squared velocity and the root mean square value, rms, is the the square root of the mean square value, u2 = (1/T ) T 0 u2 dt = u2 max/2, urms = umax/ √ 2 (2.8) where umax = Aω. We shall take Eq. 2.6 to be the definition of harmonic motion. The velocity u is also a harmonic function but we have to express it in terms of a cosine function to see what the phase angle is. Thus, u = ω|ξ| sin(ωt) = ω|ξ| cos(ωt −π/2) is a harmonic motion with the amplitude ω|ξ| and the phase angle (lag) π/2. Similarly, the acceleration is a harmonic function a = −ω2|ξ| cos(ωt) = ω2|ξ| cos(ωt − π) with the amplitude ω2|ξ| and the phase angle π. One reason for the importance of the harmonic motion is that any periodic function, period T and fundamental frequency 1/T , can be decomposed in a (Fourier) series of harmonic functions with frequencies being multiples of the fundamental frequency, as will be discussed shortly. 2.1.1 The Complex Amplitude For a given angular velocity ω, a harmonic function ξ(t) = |ξ| cos(ωt −φ) is uniquely defined by the amplitude |ξ| and the phase angle φ. Geometrically, it can be repre- sented by a point in a plane at a distance |ξ| from the origin with the radius vector
- 39. 16 ACOUSTICS making an angle φ = 0 with the x-axis. This representation reminds us of a complex number z = x+iy in the complex plane (see Appendix B), where x is the real part and y the imaginary part. As we shall see, complex numbers and their algebra are indeed ideally suited for representing and analyzing harmonic motions. This is due to the remarkable Euler’s identity exp(iα) = cos α + i sin α, where i is the imaginary unit number i = √ −1 (i.e., i2 = −1). To prove this relation, expand exp(iα) in a power series in α, making use of i2 = −1, and collect the real and imaginary parts; they are indeed found to be the power series expansions of cos α and sin α, respectively. With the proviso i2 = −1 (i3 = −i, etc.), the exponential exp(iα) is then treated in the same way as the exponential for a real variable with all the associated algebraic rules. It is sometimes useful to express cos α and sin α in terms of exp(iα); cos α = (1/2)[exp(iα) + exp(−iα)] and sin α = (1/2i)[exp(iα) − exp(−iα)]. The complex number exp(iα) is represented in the complex plane by a point with the real part cos α and the imaginary part sin α. The radius vector to the point makes an angle α with the real axis. With standing for ‘the real part of’ and with α = ωt−φ, the harmonic displacement ξ(t) = |ξ| cos(ωt − φ) can be expressed as Definition of the complex amplitude ξ(t) = |ξ| cos(ωt − φ) = {|ξ|e−i(ωt−φ)} = {|ξ|eiφ e−iωt } ≡ {ξ(ω)e−iωt } ξ(ω) = |ξ|eiφ . (2.9) At a given frequency, the complex amplitude ξ(ω) = |ξ| exp(iφ) uniquely defines the motion.3 It is represented by a point in the complex plane (Fig. 2.1) a distance |ξ| from the origin and with the line from the origin to the point making an angle φ with the real axis. The unit imaginary number can be written i = exp(iπ/2) (=cos(π/2) + i sin(π/2)) with the magnitude 1 and phase angle π/2; it is located at unit distance from the origin on the imaginary axis. Multiplying the complex amplitude ξ̃(ω) = |ξ| exp(iφ) by i = exp(iπ/2) increases the phase lag by π/2 and multiplication by −i reduces it by the same amount. Differentiation with respect to time in Eq. 2.9 brings down a factor (−iω) = ω exp(−iπ/2) so that the complex amplitudes of the velocity ξ̇(t) and the acceleration ξ̈(t) of the particle are (−iω)ξ(ω) and (−iω)2ξ(ω) = −ω2ξ(ω). The locations of these complex amplitudes are indicated in Fig. 2.1 (with the tilde signatures omitted, in accordance with the comment on notation given below); their phase lags are smaller than that of the displacement by π/2 and π, respectively; this means that they are running ahead of the displacement by these angles. To visualize the time dependence of the corresponding real quantities, we can let the complex amplitudes rotate with an angular velocity ω in the counter-clockwise direction about the origin; the projections on the real axis then yield their time dependence. 3We could equally well have used ξ(t) = {|ξ| exp[i(ωt − φ)]} = {[|ξ| exp(−iφ)] exp(iωt)} in the definition of the complex amplitude. It merely involves replacing i by −i. This definition is sometimes used in engineering where −i is denoted by j. Our choice will be used consistently in this book. One important advantage becomes apparent in the description of a traveling wave in terms of a complex variable.
- 40. OSCILLATIONS 17 Figure 2.1: The complex plane showing the location of the complex amplitudes of displace- ment ξ(ω), velocity ξ̇(ω) = −iωξ(ω) = ξ(ω) exp(−iπ/2), and acceleration ξ̈(ω) = −ω2ξ(ω). All the terms in a differential equation for ξ(t) can be expressed in a similar manner in terms of the complex amplitude ξ(ω). Thus, the differential equation is converted into an algebraic equation for ξ(ω). Having obtained ξ(ω) by solving the equation, we immediately get the amplitude |ξ| and the phase angle φ which then define the harmonic motion ξ(t) = |ξ| cos(ωt − φ). A Question of Notation Sometimes the complex amplitude is given a ‘tilde’ symbol to indicate that the function ξ̃(ω) is the complex amplitude of the displacement. In other words, the complex amplitude is not obtained merely by replacing t by ω in the function ξ. However, for convenience in writing and without much risk for confusion, we adopt from now on the convention of dropping the tilde symbol, thus denoting the complex amplitude merely by ξ(ω). Actually, as we get seriously involved in problem solving using complex amplitudes, even the argument will be dropped and ξ alone will stand for the complex amplitude; the context then will decide whether ξ(t) or ξ(ω) is meant. Example What is the complex amplitude of a displacement ξ(t) = |ξ| sin[ω(t − T/6)], where T is the period of the motion. The phase angle φ of the complex amplitude ξ(ω) is based on the displacement being written as a cosine function, i.e., ξ(t) = |ξ| cos(ωt − φ). Thus, we have to express the sine function in terms of a cosine function, i.e., sin α = cos(α − π/2). Then, with ωT = 2π, we get sin[ω(t − T/6)] = cos(ωt − π/3 − π/2) = cos(ωt − 5π/6). Thus, the complex amplitude is ξ(ω) = |ξ| exp(i5π/6). 2.1.2 Problems 1. Harmonic motion, definitions What is the angular frequency, frequency, period, phase angle (in radians), and ampli- tude of a displacement ξ = 2 cos[100(t − 0.1)] cm, where t is the time in seconds?
- 41. 18 ACOUSTICS 2. Harmonic motion. Phase angle The harmonic motion of two particles are A cos(ωt) and A cos(ωt − π/6). (a) The latter motion lags behind the former in time. Determine this time lag in terms of the period T . (b) At what times do the particles have their (positive) maxima of velocity and accelera- tion? (c) If the amplitude A is 1 cm, at what frequency (in Hz) will the acceleration equal g = 981 cm/sec2? 3. Complex amplitudes and more Consider again Problem 1. (a) What are the complex amplitudes of displacement, velocity, and acceleration? (b) Indicate their location in the complex plane. (c) What is the average speed in one period? (d) What is the rms value of the velocity? 4. Sand on a membrane A membrane is excited by an incoming sound wave at a frequency of 50 Hz. At a certain level of the sound, grains of sound on the membrane begin to bounce. What then, is the displacement amplitude of the membrane? (This method was used by Tyndall in 1874 in his experiments on sound propagation over ocean to determine the variation of the range of fog horn signals with weather and wind.) 2.1.3 Sums of Harmonic Functions. Beats Same Frequencies Thesum(superposition)oftwoharmonicmotionsξ1(t) = A1 cos(ωt−φ1)andξ2(t) = A2 cos(ω2t − φ2) with the same frequencies but with different amplitudes and phase angles is a harmonic function A cos(ωt − φ). To prove that, use the trigonometric identity cos(a−b) = cos(a) cos(b)+sin(a) sin(b) and collect the resulting terms with cos(ωt) and sin(ωt) and then compare the expression thus obtained for both the sum and for A cos(ωt −φ). It is left as a problem to carry out this calculation (Problem 1). If we use the complex number representation, we can express the two harmonic functions as B1 exp(−iωt) and B2 exp(−iωt) where B1 and B2 are complex, in this case B1 = A1 exp(iφ1) and B2 = A2 exp(iφ2). The sum is then (B1 +B2) exp(−iωt), with the new complex amplitude B = B1 + B2. The real and imaginary parts of B1 are A1 cos(φ1) and A1 sin(φ) with similar expressions for B2 and B. By equating the real and imaginary parts in B = B1 + B2, we readily find A and φ. The result applies to the sum of an arbitrary number of harmonic functions of the same frequency. Different Frequencies Consider the sum of two harmonic motions, C1 cos(ωt) and C2 cos(2ωt). The period of the first is T and of the second, T/2. The sum will be periodic with the period T since both functions repeat after this time. Furthermore, the sum will be symmetric (even) with respect to t; it is the same for positive and negative values of t since this is true for each of the components. The same holds true for the sum of any number of harmonic functions of the form An cos(nωt).
- 42. OSCILLATIONS 19 Figure 2.2: The functions cos(ω1t), cos( √ 3 ω1t) and their sum (frequencies are incommen- surable). If the terms cos(nωt) are replaced by sin(nωt), the sum will still be periodic with the period T , but it will be anti-symmetric (odd) in the sense that it changes sign when t does. If the terms are of the form an cos(nωt − φn) = an[cos φn cos(nωt) + sin φn sin(nωt)], where n is an integer, the sum contains a mixture of cosine and sine terms. The sum will still be periodic with the period T , but the symmetry properties mentioned above are no longer valid. We leave it for the reader to experiment with and plot sums of this kind when the frequencies of the individual terms are integer multiples of a fundamental frequency or fractions thereof; we shall comment here on what happens when the fraction is an irrational number. Thus, consider the sum S(t) = 0.5 cos(ω1t) − 0.5 cos( √ 3 ω1t). The functions and their sum are plotted in Fig. 2.2. The ratio of the two frequencies is √ 3, an irrational number (the two frequencies are incommensurable), and no matter how long we wait, the sum will not be periodic. In the present case, the sum starts out with the value 0 at t = 0 and then fluctuates in an irregular manner between −1 and +1. On the other hand, if the ratio had been commensurable (i.e., a rational fraction) the sum would have been periodic; for example, a ratio 2/3 results in a period 3T1. The addition of two harmonic functions with slightly different frequencies leads to the phenomenon of beats; it refers to a slow variation of the total amplitude of oscillation. It is strictly a kinematic effect. It will be illustrated here by the sum of two harmonic motions with the same amplitude but with different frequencies. The mean value of the two frequencies is ω, and they are expressed as ω1 = ω − ω and ω2 = ω+ω. Using the trigonometric identity cos(ωt ∓ωt) = cos(ωt) cos(ωt)±
- 43. 20 ACOUSTICS Figure 2.3: An example of beats produced by the sum of two harmonic motions with frequen- cies 0.9ω and 1.1ω. sin(ωt) sin(ωt), we find for the sum of the corresponding harmonic motions ξ(t) = cos(ωt − ωt) + cos(ωt + ωt) = 2 cos(ωt) cos(ωt) (2.10) which can be interpreted as a harmonic motion of frequency ω with a periodically varying amplitude (“beats”) of frequency ω. The maximum value of the amplitude is twice the amplitude of each of the components. An example is illustrated in Fig. 2.3. In this case, with ω = 0.1ω, the period of the amplitude variation will be ≈10T , consistent with the result in the figure. Beats can be useful in experimental work when it comes to an accurate comparison of the frequencies of two signals. 2.1.4 Heterodyning The squared sum of two harmonic signals A1 cos(ω1t) and A2 cos(ω2t) produces signals with the sum and difference frequencies ω1 + ω2 and ω1 − ω2, which can be of considerable practical importance in signal analysis. The squared sum is [A1cos(ω1t)+A2 cos(ω2t)]2 = A2 1 cos2 (ω1t)+A2 2 cos2 (ω2t)+2A1A2cos(ω1) cos(ω2t) (2.11) The time dependent part of each of the squared terms on the right-hand side is harmonic with twice the frequency since cos2(ωt) = [1 + cos(2ωt)]/2. This is not of any particular interest, however. The important part is the last term which can be written 2A1A2 cos(ω1t) cos(ω2t) = A1A2[cos(ω1 + ω2)t + cos(ω1 − ω2)t]. (2.12) It contains two harmonic components, one with the sum of the two primary fre- quencies and one with the difference. This is what is meant by heterodyning, the creation of sum and difference frequencies of the input signals. Normally, it is the term with the difference frequency which is of interest. There are several useful applications of heterodyning; we shall give but one example here. A photo-cell or photo-multiplier is a device such that the output signal is proportional to the square of the electric field in an incoming light wave. Thus, if the light incident on the photo-cell is the sum of two laser signals, the output will contain an electric current with the difference of the frequencies of the two signals. Thus, consider a light beam which is split into two with one of the beams reflected or scattered from a vibrating object, such as the thermal vibrations of the surface of a liquid, where the reflected signal is shifted in frequency by an amount equal to
- 44. OSCILLATIONS 21 the vibration frequency ω0. (Actually, the reflected light contains both an up-shifted and a down-shifted frequency, ± ω0, which can be thought of as being Doppler shifted by the vibrating surface.) Then, if both the direct and the reflected beams are incident on the photo-cell, the output signals will contain the frequency of vibration. This frequency might be of the order of ≈105 Hz whereas the incident light frequency typically would be ≈ 1015 Hz. In this case, the shift is very small, however, only 1 part in 1010, and conventional spectroscopic methods would not be able to resolve such a small shift. With the heterodyne technique, heterodyne spectroscopy, this problem of resolu- tion is solved. Since the output current contains the difference frequency ω0, the vibration frequency, which can be detected and analyzed with a conventional elec- tronic analyzer. 2.1.5 Problems 1. Sum of harmonic functions (a) With reference to the outline at the beginning of Section 2.1.3, show that A1 cos(ωt − φ1) + A1 cos(ωt − φ2) can be written as a new harmonic function A cos(ωt − φ) and determine A and φ in terms of A1, A2, φ1, and φ2. (b) Carry out the corresponding calculation using complex amplitude description of the harmonic functions as outlined in Section 2.1.3. 2. Heterodyning In heterodyning, the sum of two signals with the frequencies ω1 and ω2 are processed with a square law detector producing the output sum and differences of the input signal frequencies. What frequencies would be present in the output of a cube-law detector? 2.2 The Linear Oscillator 2.2.1 Equation of Motion So far, we have dealt only with the kinematics of harmonic motion without regard to the forces involved. The real ‘physics’ enters when we deal with the dynamics of the motion and it is now time to turn to it. One reason for the unique importance of the harmonic motion is that in many cases in nature and in applications, a small displacement of a particle from its equi- librium position generally results in a restoring (reaction) force proportional to the displacement. If the particle is released from the displaced position, the only force acting on it in the absence of friction will be the restoring force and, as we shall see, the subsequent motion of the particle will be harmonic. The classical example is the mass-spring oscillator illustrated in Fig. 2.4. A particle of mass M on a table, assumed friction-less, is attached to one end of a spring which has its opposite end clamped. The displacement of the particle is denoted ξ. Instead of sliding on the table, the particle can move up and down as it hangs from the free end of a vertical spring with the upper end of the spring held fixed, as shown. Itisfoundexperimentallythatforsufficientlysmalldisplacements, theforcerequired to change the length of the spring by an amount ξ is Kξ, where K is a constant. It
- 45. 22 ACOUSTICS Figure 2.4: Mass-spring oscillator. is normally called the spring constant and, with the force being a linear function of ξ, the oscillator is referred to as a linear oscillator. The reaction force on M is in the opposite direction to the displacement and is −Kξ. After releasing the particle, the equation of motion, Newton’s law, will be Mξ̈ = −Kξ, where we have used the ‘dot’- notation for the time derivative. Furthermore, with K/M denoted ω2 0, this equation can be written ξ̈ + ω2 0ξ = 0 ω2 0 = K/M. (2.13) This has the same form as Eq. 2.5 which we already know to be satisfied by a harmonic motion. Withreferencetostandardmathematicstexts, thegeneralsolutiontoasecondorder linear differential equation of this kind is a linear combination of two independent so- lutions, in this case cos(ω0t) and sin(ω0t). (A criterion for solutions to be independent is that the functions be orthogonal which means, in this context, that the integral of the product of the two functions over one period is zero.) The general solution is a linear combination of the two independent solutions, i.e., ξ(t) = C cos(ω0t) + S sin(ω0)t, where C and S are constants. The physical meaning of C is the displacement at t = 0, C = ξ(0), and Sω0 is the initial particle velocity, ξ̇(0). We can replace C and S by two other constants A and φ defined by C = A cos(φ) and S = A sin(φ), and the solution can then be written in the familiar form ξ(t) = A cos(ω0t − φ), (2.14) which is the harmonic motion discussed above, where A is the amplitude and φ the phase angle. The motion is uniquely specified by the initial displacement and the initial velocity in terms of which A and φ can be expressed, as indicated above. Actually, the displacement and velocity at any other time can also be used for the determination of A and φ. Example A harmonic motion has the angular frequency ω0 = 400 sec−1. At t = 0 the dis- placement is 10 cm and the velocity is 20 cm/sec. Determine the subsequent motion. What are the amplitudes of displacement, velocity, and acceleration? Denote the displacement and velocity of the oscillator at t = 0 by ξ(0) and u(0) (initial conditions) which in our case are 10 cm and 20 cm/sec.
- 46. OSCILLATIONS 23 We start with the general expression for the harmonic displacement ξ = A cos (ω0t − φ). It contains the two constants A and φ which are to be determined. Thus, with t = 0, we obtain, ξ(0) = A cos(φ) u(0) = Aω0 sin(φ) and tan(φ) = u(0)/[ω0ξ(0)], A = ξ(0)/ cos(φ) = ξ(0) 1 + tan2(φ). Inserting the numerical values we find tan(φ) = 1/200 and A = 10 1 + (1/200)2 ≈ 10[1 + (1/2)(1/200)2 ]. The subsequent displacement is ξ(t) = A cos(ω0t − φ). Comment. With the particular initial values chosen in this problem the phase angle is very small, and the amplitude of oscillation is almost equal to the initial displace- ment. In other words, the oscillator is started out very nearly from the maximum value of the displacement and the initial kinetic energy of the oscillator is much smaller than the initial potential energy. How should the oscillator be started in order for the subsequent motion to have the time dependence sin(ω0t)? 2.2.2 The ‘Real’ Spring. Compliance The spring constant depends not only on the elastic properties of the material in the spring but also on its length and shape. In an ordinary uniform coil spring, for example, the pitch angle of the coil (helix) plays a role and another relevant factor is the thickness of the material. The deformation of the coil spring is a complicated combination of torsion and bending and the spring constant generally should be regarded as an experimentally determined quantity; the calculation of it from first principles is not simple. The linear relation between force and deformation is valid only for sufficiently small deformations. For example, for a very large elongation, the spring ultimately takes the form of a straight wire or rod, and, conversely, a large compression will make it into a tube-like configuration corresponding to a zero pitch angle of the coil. In both these limits, the stiffness of the spring is much larger than for the relaxed spring. It has been tactily assumed that the spring constant is determined from a static deformation. Yet, this constant has been used for non-static (oscillatory) motion. Although this is a good approximation in most cases, it is not always true. Materials like rubber and plastics (and polymers in general) for which elastic constants depend on the rate of strain, the spring constant is frequency dependent. For example, there exist substances which are plastic for slow and elastic for rapid deformations (remember ‘silly putty’?). This is related to the molecular structure of the material and the effect is often strongly dependent on temperature. A cold tennis ball, for example, does not bounce very well. In a static deformation of the spring, the inertia of the spring does not enter. If the motion is time dependent, this is no longer true, and another idealization is the
- 47. 24 ACOUSTICS omission of the mass of the spring. This is justified if the mass attached to the spring is much larger than the spring mass. The effect of the spring mass will be discussed shortly. The inverse of the spring constant K is called the compliance C = 1/K. (2.15) It is proportional to the length of the spring. Later, in the study of wave motion on a spring, we shall introduce the compliance per unit length. Frequently, several springs are combined in order to obtain a desired resulting spring constant. If the springs are in ‘parallel,’ the deformations will be the same for all springs and the restoring forces will add. The resulting spring constant is then the sum of the individual spring constants; the resulting spring will be ‘harder.’ If the springs are in ‘series,’ the force in each spring will be the same and the deformations add. The resulting compliance is then the sum of the individual compliances; the resulting combined spring will be ‘softer’ than any of the individual springs. Effect of the Mass of the Spring As already indicated, the assumption of a mass-less spring in the discussion of the mass-spring oscillator is of course an idealization and is not a good assumption unless the spring mass m is much smaller than the mass M of the body attached to the spring. This shows up as a defect in Eq. 2.13 for the frequency of oscillation, ω0 = √ K/M. According to it, the frequency goes to infinity as M goes to zero. In reality, this cannot be correct since removal of M still yields a finite frequency of oscillation of the spring alone. This problem of the spring mass will be considered later in connection with wave propagation and it will be shown that for the lowest mode of oscillation, the effect of the spring mass m can be accounted for approximately, if m/M 1, by adding one-third of this mass to the mass M in Eq. 2.13. Thus, the corrected expression for the frequency of oscillation (lowest mode) is ω0 ≈ K M + m/3 = ω0 √ 1 + m/3M . (2.16) Air Spring For an isothermal change of state of a gas, the relation between pressure P and volume V is simply PV = constant (i.e., dP/P = −dV/V ). For an isentropic (adiabatic) change, this relation has to be replaced by dP/P = −γ (dV/V ), where γ is the specific heat ratio Cp/Cv, which for air is ≈1.4. Consider a vertical tube of length L and closed at the bottom and with a piston riding on the top of the air column in the tube. If the piston is displaced into the tube by a small among ξ, the volume of the air column is changed by dV = −Aξ, where A is the area of the tube. On the assumption that the compression is isentropic, the pressure change will be dP = γ (PA/V )ξ and the corresponding force on the piston
- 48. OSCILLATIONS 25 will oppose the displacement so that F = −γ (PA2/V )ξ. This means that the spring constant of the air column is K = γ (PA2 /V ) = γ (PA/L), (2.17) whereL = V/Aisthelengthofthetube. Thespringconstantisinverselyproportional to the length and, hence, the compliance C = 1/K is proportional to the length. After releasing the piston, it will oscillate in harmonic motion with the angular frequency √ K/M. As will be shown later, the adiabatic approximation in a situation like this is valid except at very low frequencies. By knowing the dimensions of the tube and the mass M, a measurement of the frequency can be used as a means of determining the specific heat ratio γ . A modified version of this experiment, often used in introductory physics laboratory, involves a flask or bottle with a long, narrow neck in which a steel ball is used as a piston. For a large volume change, the motion will not be harmonic since the relation between the displacement and the restoring force will not be linear. Thus, with the initial quantities denoted by a subscript 1, a general displacement ξ yields a new volume V2 = V1 − Aξ and the new pressure is obtained from P2V γ 2 = P1V γ 1 . The restoring force A(P2−P1) no longer will be proportional to ξ and we have a non-linear rather than a linear oscillator. 2.2.3 Problems 1. Static compression and resonance frequency A weight is placed on top of a vertical spring and the static compression of the spring is found to be ξst . Show that the frequency of oscillation of the mass-spring oscillator is determined solely by the static displacement and the acceleration of gravity g. 2. Frequency of oscillation A body of mass m on a horizontal friction-less plane is attached to two springs, one on each side of the body. The spring constants are K1 and K2. The relaxed lengths of each spring is L. The free ends of the springs are pulled apart and fastened to two fixed walls a distance 3L apart. (a) Determine the equilibrium position of the body. (b) What is the frequency of oscillation of the body about the equilibrium position? (c) Suppose that the supports are brought close together so that the their separation will be L/2. What, then, will be the equilibrium position of M and the frequency of oscillation? 3. Lateral oscillations on a spring (a) In Example 2, what will be the frequency of small amplitude oscillations of M in a direction perpendicular to the springs? (b) Suppose that the distance between the end supports of the spring equals the length of the spring so that the spring is slack. What will be the restoring force for a lateral displacement ξ of M? Will the oscillation be harmonic? 4. Initial value problem The collisions in Example 8 in Ch.11 are inelastic and mechanical energy will be lost in a collision. The mechanical energy loss in the first collision is p2/2m − p2/2(M + m) = (p2/2m)(M/(M + m))
- 49. 26 ACOUSTICS and in the second p2/2(M + m) − p2/2(M + 2m). (a) Show that the two energy losses are the same if m/M = 1 + √ 2. Compare the two energy losses as a function of m/M. (b) Suppose that n shots are fired into the block under conditions of maximum amplitude gain as explained in Example 8. What will be the amplitude of the oscillator after the n:th shot? 2.3 Free Damped Motion of a Linear Oscillator 2.3.1 Energy Considerations The mechanical energy in the harmonic motion of a mass-spring oscillator is the sum of the kinetic energy Mu2/2 of the mass M and the potential energy V of the spring. If the displacement from the equilibrium position is ξ, the force required for this displacement is Kξ. The work done to reach this displacement is the potential energy V (ξ) = ξ 0 Kξdξ = Kξ2 /2. (2.18) In the harmonic motion, there is a periodic exchange between kinetic and potential energy, each going from zero to a maximum value E, where E = Mu2/2 + Kξ2/2 is the total mechanical energy. In the absence of friction, this energy is a constant of motion. To see how this follows from the equation of motion, we write the harmonic oscil- lator equation (2.13) in the form Mu̇ + Kξ = 0, (2.19) where u = ξ̇ is the velocity, and then multiply the equation by u. The first term in the equation becomes Muu̇ = d/dt[Mu2/2]. In the second term, which becomes Kuξ, we use u = ξ̇ so that it can be written Kξξ̇ = d/dt[Kξ2/2]. This means that Eq. 2.19 takes the form d/dt[Mu2 /2 + Kξ2 /2] = 0. (2.20) The first term, Mu2/2, is the kinetic energy of the mass M, and the second term, Kξ2/2, is the potential energy stored in the spring. Each is time dependent but the sum, the total mechanical energy, remains constant throughout the motion. Although no new physics is involved in this result (since it follows from Newton’s law), the conservation of mechanical energy is a useful aid in problem solving. In the harmonic motion, the velocity has a maximum when the potential energy is zero, and vice versa, and the total mechanical energy can be expressed either as the maximum kinetic energy or the maximum potential energy. The average kinetic and potential energies (over one period) are the same. When a friction force is present, the total mechanical energy of the oscillator is no longer conserved. In fact, from the equation of motion Mu̇ + Kξ = −Ru it follows
- 50. OSCILLATIONS 27 by multiplication by u (see Eq. 2.20) that d/dt[Mu2 /2 + Kξ2 /2] = −Ru2 . (2.21) Thus, the friction drains the mechanical energy, at a rate −Ru2, and converts it into heat.4 As a result, the amplitude of oscillation will decay with time and we can obtain an approximate expression for the decay by assuming that the average potential and kinetic energy (over one period) are the same, as is the case for the loss-free oscillator. Thus, with the left-hand side of Eq. 2.21 replaced by d(Mu2]/dt, and the right-hand side by Ru2, the time dependence of u2 will be u(t)2 ≈ u(0)2 e−(R/M)t . (2.22) The corresponding rms amplitude then will decay as exp[−(R/2M)t]. 2.3.2 Oscillatory Decay After having seen the effect of friction on the time dependence of the average energy, let us pursue the effect of damping on free motion in more detail and determine the actual decay of the amplitude and the possible effect of damping on the frequency of oscillation. The idealized oscillator considered so far had no other forces acting on the mass than the spring force. In reality, there is also a friction force although in many cases it may be small. We shall assume the friction force to be proportional to the velocity of the oscillator. Such a friction force is often referred to as viscous or dynamic. Normally, the contact friction with a table, for example, does not have such a simple velocity dependence. Often, as a simplification, one distinguishes merely between a ‘static’ and a ‘dynamic’ contact friction, the magnitude of the latter often assumed to be proportional to the magnitude of the velocity but with a direction opposite that of the velocity. The ‘static’ friction force is proportional to the normal component of the contact force and points in the direction opposite that of the horizontal component of the applied force. A friction force proportional to the velocity can be obtained by means of a dashpot damper, as shown in Fig. 2.5. It is in parallel with the spring and is simply a ‘leaky’ piston which moves inside a cylinder. The piston is connected to the mass M of the oscillator and the force required to move the piston is proportional to its velocity relative to the cylinder (neglecting the mass of the piston). The cylinder is attached to the same fixed support as the spring, as indicated in Fig. 2.5. The fluid in the cylinder is then forced through a narrow channel (a ‘leak’) between the piston and the cylinder and it is the viscous stresses in this flow which are responsible for the friction force. Therefore, this type of damping is often referred to as viscous. The friction on a body moving through air or some other fluid in free field will be proportional to the velocity only for very low speeds and approaches an approximate square law dependence at high speeds. 4When the concept of energy is extended to include other forms of energy other than mechanical, the law of conservation of energy does bring something new, the first law of thermodynamics which can be regarded as a postulate, the truth of which should be considered as an experimental fact.
- 51. 28 ACOUSTICS Figure 2.5: Oscillator with dash-pot damper. With a friction force proportional to the velocity, the equation of motion for the oscillatorbecomeslinearsothatasolutioncanbeobtainedinasimplemanner. Fordry contact friction or any other type of friction, the equation becomes non-linear and the solution generally has to be found by numerical means, as will be demonstrated in Section 2.7.3. With dξ/dt ≡ ξ̇, we shall express the friction force as −Rξ̇ and the equation of motion for the mass element in an oscillator becomes Mξ̈ = −Kξ − Rξ̇ or, with K/M = ω2 0, Free oscillations, damped oscillator ξ̈ + (R/M)ξ̇ + ω2 0ξ = 0 ξ(t) = Ae−γ t cos(ω 0t − φ) γ = R/2M, ω 0 = ω2 0 − γ 2 . (2.23) The general procedure to solve a linear differential equation is aided considerably with the use of complex variables (Section 2.3.3). For the time being, however, we use a ‘patchwork’ approach to construct a solution, making use of the result obtained in the decay of the energy in Eq. 2.22 from which it is reasonable to assume that the solution ξ(t) will be of the form given in Eq. 2.23, where γ , and ω 0 are to be determined. Thus, we insert this expression for ξ(t) into the first equation in 2.23 and write the left-hand side as a sum of sin(ω 0t) and cos(ω 0t) functions. Requiring that each of the coefficients of these functions be zero to satisfy the equation at all times, we get the required values of γ and ω 0 in Eq. 2.23. Actually, the value of γ is the same as obtained in Eq. 2.22. The damping makes the ω 0 lower than ω0. When there is no friction, i.e., γ = 0, the solution reduces to the harmonic motion discussed earlier, where A is the amplitude and φ the phase angle. The damping produces an exponential decay of the amplitude and also causes a reduction of the frequency of oscillation. If the friction constant is large enough to that ω 0 = 0, the motion is non-oscillatory and the oscillator is then said to be critically damped. If γ ω0, the frequency ω 0 formally becomes imaginary and the solution has to be reexamined, as will be done shortly. As it turns out, the general solution then consists of a linear combination of two decaying exponential functions. 2.3.3 Use of Complex Variables. Complex Frequency With the use of complex variables in solving the damped oscillator equation, there is no need for the kind of patchwork that was used in Section 2.3.2. We merely let the mathematics do its job and present us with the solution.
- 52. OSCILLATIONS 29 It should be familiar by now, that the complex amplitude ξ(ω) of ξ(t) is defined by ξ(t) = {ξ(ω)e−iωt }. (2.24) The corresponding complex amplitudes of the velocity and the acceleration are then −iωξ(ω) and −ω2ξ(o) and if these expressions are used in Eq. 2.23 we obtain the following equation for ω ω2 + i2γ − ω2 0 = 0 (2.25) in which γ = R/2M. Formally, the solution to this equation yields complex frequencies ω = −iγ ± ω2 − γ 2. (2.26) The general solution is a linear combination of the solutions corresponding to the two solutions for ω, i.e., ξ(t) = e−γ t {A1eiω 0t + A2e−iω 0t }, (2.27) where ω 0 = ω2 − γ 2 and A1 and A2 are complex constants to be determined from initial conditions. We distinguish between the three types of solutions which correspond to γ ω0, γ ω0, and γ = ω0. Oscillatory decay, γ ω0. In this case, ω 0 is real, and the oscillator is sometimes referred to as underdamped; the general solution takes the form ξ(t) = A e−γ t cos(ω 0t − φ) (2.28) which is the same as in Eq. 2.23. The constants A and φ are determined by the initial conditions of the oscillator. Overdamped oscillator, γ ω0. The frequency ω 0 now is purely imaginary, ω 0 = i γ 2 − ω2 0, and the two solutions to the frequency equation (6.18) become ω+ = −i(γ − γ 2 − ω2 0) ≡ −iγ1 ω− = −i(γ + γ 2 − ω2 00 ≡ −iγ2. (2.29) The motion decays monotonically (without oscillations) and the corresponding gen- eral solution for the displacement is the sum of two exponential functions with the decay constants γ1 and γ2, ξ(t) = C1e−γ1t + C2e−γ2t , (2.30) where the two (real) constants are to be determined from the initial conditions. Critically damped oscillator, γ = ω0. A special mention should be made of the ‘degenerate’ case in which the two solutions to the frequency equation are the same, i.e., when γ1 = γ2 = ω0. To obtain the general solution for ξ in this case
- 53. 30 ACOUSTICS requires some thought since we are left with only one adjustable constant. The general solution must contain two constants so that the two conditions of initial displacement and velocity can be satisfied (formally, we know that the general solution to a second order differential equation has two constants of integration). To obtain the general solution we can proceed as follows. We start from the overdamped motion ξ = C1 exp(−γ1t)+C2 exp(−γ2t). Let γ2 = γ1+anddenotetemporarilyexp(−γ2t)byf (γ2, t). Expansionofthisfunctiontothe first order in yields f (γ2, t) = f (γ1, t)+(∂f/∂)0 = exp(−γ t)−t exp(−γ1t). The expression for the displacement then becomes ξ = (C1 + C2) exp(−γ1t) − t(C2) exp(−γ1t), or ξ = (C + Dt)e−ω0t , (2.31) where C = C1 +C2 and D = −C2, C2 being adjusted in such a way that D remains finite as → 0. Direct insertion into the differential equation ξ̈ + 2γ ξ̇ + ω2 0ξ = 0 (Eq. 2.23) shows that this indeed is a solution when γ = ω0. In summary, the use of complex amplitudes in solving the frequency equation (6.18) and accepting a complex frequency as a solution, we have seen that it indeed has a physical meaning; the real part being the quantity that determines the period of oscillation (for small damping) and the imaginary part, the damping. In this manner, the solution for the displacement emerged automatically from the equation of motion. 2.3.4 Problems 1. Oscillatory decay of damped oscillator The formal solution for the displacement of a damped oscillator in free motion is given in Eq. 2.27, in which A1 and A2 are two independent complex constants, each with a magnitude and phase angle. Show in algebraic detail that the general solution can be expressed as in Eq. 2.28, in which A and φ are real constants. 2. Critically damped oscillator. Impulse response In the degenerate case of a damped oscillator when γ = ω0 so that ω 0 = 0, the general solution for the displacement is ξ(t) = (A + Bt)e−ω0t , (2.32) where A and B are constants to be determined by the initial conditions. (a) Prove this by direct insertion into the equation of motion. (b) The oscillator, initially at rest, is given a unit impulse at t = 0. Determine the subsequently motion. 3. Paths in the complex plane It is instructive to convince oneself that as γ increases, the two solutions for the complex frequency in Example 9 in Ch.11 follow along circular paths in the complex plane when the motion is oscillatory. They meet on the negative imaginary axis when the damping is critical, i.e., γ = ω0, and then move apart in opposite direction along the imaginary axis. Sketch in some detail the paths and label the values of γ at critical points, as you go along. 4. Impulse response. Maximum excursion The oscillator in Example 9 in Ch.11 is started from rest by an impulse of 10 Ns. For the underdamped, critically damped, and overdamped conditions in (a) and (c),
- 54. OSCILLATIONS 31 (a) determine the maximum excursion of the mass element and the corresponding time of occurrence and (b) determine the amount of mechanical energy lost during this excursion. 5. Overdamped harmonic oscillator (a) With reference to the expressions for the two decay constants in Eq. 2.29 show that if γ ω0 we obtain γ1 ≈ K/R and γ2 ≈ R/M. (b) What is the motion of an oscillator, started from rest with an initial displacement ξ(0), in which R is so large that the effect of inertia can be neglected? (c) Do the same for an oscillator, started from ξ = 0, with an initial velocity u(0), in which the effect of the spring force can be neglected in comparison with the friction force. 2.4 Forced Harmonic Motion 2.4.1 Without Complex Amplitudes Toanalyzetheforcedharmonicmotionofthedampedoscillator, weaddadrivingforce F(t) = |F| cos(ωt) on the right-hand side of Eq. 2.23. The corresponding steady state expression for the displacement is assumed to be ξ = |ξ| cos(ωt − φ). Inserting this into the equation of motion, we get for the first term −Mω2|ξ| cos(ωt − φ), for the second, −Rω|ξ| sin(ωt − φ), and for the third, K|ξ| cos(ωt − φ). Next, we use the trigonometric identities cos(ωt − φ) = cos(ωt) cos φ + sin(ωt) sin φ and sin(ωt − φ) = sin(ωt) cos φ − cos(ωt) sin φ and express each of these three terms as a sum of cos(ωt)- and sin(ωt)-terms. Since we have only a cos(ωt)-term on the right-hand side, the sum of the sine terms on the left-hand side has to be zero in order to satisfy the equation at all times and the amplitude of the sum of the cosine terms must equal |F|. These conditions yield two equations from which |ξ| and φ can be determined. It is left as a problem to fill in the missing algebraic steps and show that |ξ| = |F|/ω R2 + (K/ω − ωM)2 tan φ = ωR/(K − ω2M). (2.33) At very low frequencies, the displacement approaches the static value |ξ| ≈ |F|/K and is in phase with the driving force. At resonance, |ξ| = |F|/(ωR) which means that the velocity amplitude is |u| = |F|/R, with the velocity in phase with the driving force. At very high frequencies where the inertia dominates, the phase angle becomes ≈π; the displacement is then opposite to the direction of the driving force. A good portion of the algebra has been skipped here, and what remains is a de- ceptively small amount. This should be kept in mind when it is compared with the complex amplitude approach used in Section 2.4.2. The driving force F(t) = |F| cos(ωt) and the ‘steady state’ motion it produces are idealizations since they have no beginning and no end. A realistic force would be one which is turned on at time t = 0, say, and then turned off at a later time. This introduces additional motions, so called transients, which have to be added to the steady state motion. An obvious indication of the shortcoming of the present analysis is that it leads to an infinite displacement at resonance if the damping is zero.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. Did Farringford call you his son? asked Mr. Gray, turning to me. Yes, sir, he did; but not while I held Lynch down. It was while we were in Plum Street, I replied. What trick were you engaged in? demanded Mr. Gray, rather sternly. Why did he call you his son? I am his son. He is my father, I answered. Farringford looked at me with an expression of disapproval, as if to reproach me for the falsehood he believed I had uttered.
- 57. Y CHAPTER XIV. IN WHICH PHIL RECOVERS HIS MONEY. ou don't mean to say that Farringford here, whom everybody in St. Louis knows, is your father—do you? continued Mr. Gray, apparently amazed at the absurdity of the proposition, while his friend and the sergeant laughed heartily. That is precisely what I mean to say, I replied, in the most determined tone. Farringford shook his head, and was apparently sorry that I had turned out to be such an abominable liar. What is your name? inquired the sergeant. Philip Farringford. I had taken especial pains not to give my full name to my father when he questioned me, and he doubtless supposed that I had invented the name for the occasion. He looked at me, and shook his head. Very likely, by this time, he was willing to believe I had deceived him, and that I had lost no money, for if I could lie about one thing I could about another. Do you justify this young man in calling you his father, Farringford? said Mr. Gray. I am sorry to say I cannot. Gentlemen, I have endeavored to act in good faith, replied my father. I have always found that the truth would serve me better than falsehood. Did you call him your son?
- 58. I did, but used the expression as a kind of harmless fib to carry my purpose with this Lynch, who had robbed the boy of nearly a hundred dollars. It is false! exclaimed Lynch. Keep cool, if you please, sir, interposed the sergeant. We have heard your story, and now we will hear the other side. Philip may have deceived me, but I believed that he had been robbed, and I did the best I could to get his money back, after he had pointed out to me the man who took it from him. Certainly he is not my son. I never saw him till yesterday; and I am sorry he has thought it necessary to repeat my fib, or falsehood, if you please, continued Farringford. Nevertheless, I hope I shall be able to prove in due time that he is my father, I added. But, my lad, everybody knows that Farringford has no children, said Mr. Gray. Never mind that now. I want to know whether any robbery has been committed, interposed the sergeant, impatiently. Let the boy tell his own story, replied Mr. Gray. Here is Lynch's purse, I began, handing it to the sergeant. Then you did take these things from him? I did; but he told me to put my hand in his pocket and take out the pocket-book and the purse. Very probable! sneered Lynch. It's all true, said Farringford. Well, go on, young man. I was coming down the Missouri River in the steamer Fawn—
- 59. She arrived last Tuesday morning, interposed Mr. Lamar, the gentleman with Mr. Gray. Yes, sir. I was with Mr. Gracewood and his family. What Gracewood? Henry. Is he a brother of Robert Gracewood of Glencoe? I don't know. He had a brother in St. Louis, said Mr. Lamar, who was an elderly gentleman, and appeared to know everybody and everything. He bought a place at Glencoe a year ago. His wife's brother was a Mr. Sparkley. It's the same man. But he separated from his wife years ago, cleared out, and has not been heard from since. I explained that the family had been reunited, and were on their way to St. Louis. I had endeavored to find Mr. Gracewood's brother, but without success, in order to inform him of what had occurred up the river. The fact that he had moved from the city explained why I had not found his name in the Directory. I continued my story, with frequent interruptions, much to the disgust of the sergeant, who was interested only in the criminal aspect of the case. I told how Lynch had robbed me at Leavenworth, how I had identified him in St. Louis, and followed him and Farringford from Forstellar's to Front Street. Every word of that story is true so far as it relates to me, said Farringford. I watched Lynch and Farringford, the former trying to get rid of the latter all the time, until at last he laid violent hands upon him, I continued. I couldn't stand it any longer; I went up behind Lynch, threw my hands around his neck, and stuck my knees into his back
- 60. till he went down. He begged me to let him up, and promised to restore my money if I would. Then, when I was not willing to let him up without some security, he told me to take his pocket-book and purse. That was just what was going on when these gentlemen came out of Plum Street. Then you did not knock him down till he laid hands upon Farringford? added the sergeant. No, sir; I did not till he took hold of my father. Your father! exclaimed Mr. Gray. The rest of your story is so straightforward that I hoped you would abandon that fiction. It is no fiction. It matters not to me whether it is fact or fiction, interposed the sergeant. I only wish to know whether or not a crime has been committed in St. Louis. If the boy knocked this Lynch down in order to save Farringford from injury, it is no crime, whether father or not. I cried, 'Police!' as loud as I could, as soon as we struck the ground, I added. Can you identify your money? asked the sergeant. Not every piece of it; but there was a five-dollar gold coin, with a hole through the middle, dated 1850. The clerk of the Fawn would not take it for my passage for five dollars. The officer poured the gold from the purse upon the table, and instantly picked out the coin I had described, which Lynch had perhaps found it as difficult to pass as I had. He looked at the date, and declared it was 1850. That is very good evidence, my boy, said the officer, bestowing a smile of approval upon me. Can you give me any more. If you can find Captain Davis, of the Fawn, he will say that I left the boat with Lynch.
- 61. Where is he? He has gone up to Alton with the Fawn. When Mr. Gracewood comes, he will tell you the same thing. Your witnesses are not at hand. In what boat did you come down the river. In the Fawn. And you, Mr. Lynch? In the Daylight. Where from? St. Joe. The sergeant continued to question and cross-question Lynch for half an hour. His statements were confused and contradictory, and being based upon falsehoods, they could not well be otherwise. It appeared that the Daylight, in which he had arrived, came down the river immediately after the Fawn, which made my story the more probable. I do not see that any crime has been committed in St. Louis, said the officer, after his long and patient investigation. Then you don't call it a crime to knock a man down, and take his purse and pocket-book from him? added Lynch, in deep disgust. I believe the young man's story, replied the officer. If your money had been taken from you by force, you would not have walked quietly through the streets with those who robbed you, passing an officer on your way without hinting at what had happened. The young man's story is straightforward and consistent, except as to his relations with Farringford, which is not material. I am of the opinion that you commenced the assault upon Farringford.
- 62. Not so. Both Farringford and the young man agree in all essential points. Lynch growled and protested, but finally declared that he was satisfied to let the matter drop where it was. He had recovered his money, and he could not complain. But I have not recovered mine, and I am not satisfied, I added, feeling that the discharge of Lynch was total defeat to me. You were robbed in the territory of Kansas, and not in the city of St. Louis, replied the officer. Must I lose my money for that reason? Certainly not; but the complaint against Lynch must be made at Leavenworth, and a requisition from the governor of the territory must be sent here. The case was full of difficulties, and Lynch, in charge of a policeman, was sent out of the room to enable us to consider the best means of proceeding. I could not go back to Leavenworth very conveniently, and it would cost me more than the amount of money I had lost. We decided to let the matter rest till the next day, and Lynch was called in again. I propose to detain you till to-morrow, when Farringford will complain of you for an assault, said the officer. I would rather give a hundred dollars than be detained, said Lynch. We don't settle cases in that way. Of course we intend to reach the robbery matter in some manner. I will give the boy the money he claims to have lost, added the culprit.
- 63. If you wish to restore the money, you can, replied the sergeant. I do not admit the truth of his story. Then you shall not give him any money. You shall not be swindled here. If I admit the— Don't commit yourself unless you choose to do so. Whatever you say may be used as evidence to convict you. You put me in a tight place, said Lynch. If I commit myself, you will prosecute me. If I don't commit myself, I cannot give the boy the money. I did not say I should prosecute you. The crime, if any, was committed beyond the limits of this state. I cannot enter a complaint. The young man may do so if he thinks best. Can I make Phil a present of a hundred dollars? demanded Lynch, desperately. You can do as you please with your own money, answered the officer. The robber counted a hundred dollars from his pocket-book, and handed it to Mr. Lamar, who declared that the amount was right, and the bills were good. It was passed to me; but I declined to receive any more than I had lost, and changing a bill, I returned two dollars and a half. I will make no complaint for assault now, said Farringford. Then I cannot detain him. If the young man chooses to complain of Lynch in Leavenworth, he is still liable to prosecution. I will risk that, said Lynch, more cheerfully. You can leave, added the officer.
- 64. The rascal promptly availed himself of this permission, and left the office. I am sorry to have a case settled in that manner. I know that man as a notorious blackleg, continued the officer. I don't see that it could be settled in any other way now, replied Mr. Gray. We have done nothing to prejudice the interests of justice. The young man can prosecute now. I can't afford to go to Kansas to do so, I replied. We will keep watch of him, said the sergeant. We all left the office together. The two gentlemen who had manifested so much interest in the affair were unwilling to part with Farringford and me. Mr. Gray asked me what had induced me to say that Farringford was my father. It's a long story, gentlemen; and I have to convince him as well as you of the truth of what I say. If you will go to my boarding- house I will do so. I told them where it was, and they consented to accompany me. When we reached the house, Mrs. Greenough was astonished at the number of my visitors, but I conducted them all to my chamber.
- 65. H CHAPTER XV. IN WHICH PHIL PRODUCES THE RELICS OF HIS CHILDHOOD. aving seated my party in my chamber, I told the last part of my story first. I began by saying that I had been brought up on the upper Missouri, by Matt Rockwood, relating all my experience down to the present moment, including the history of the Gracewoods. That's all very well, Phil; but where were you born? asked Mr. Gray. You left that part out, and told us everything except that which we wished to know. I don't know where I was born. You must ask my father? Do you still persist in saying that Farringford is your father? I still persist. But he has no children. I had one child, interposed Farringford, trembling with emotion, as well as from the effects of inebriation. I remember, said Mr. Lamar. You lost that child when the Farringford was burned. Yes, replied my father, with a shudder. Will you state precisely how that child was lost, sir? I continued. I would not ask you to do so if it were not necessary, for I know the narrative is painful. I suppose you claim to be this child, which, if I remember rightly, was a girl, added Mr. Lamar.
- 66. No; it was a boy, responded Mr. Farringford. Gentlemen, I shall leave you to draw your own conclusions, after you have heard the rest of the story. Can it be possible that you are my lost child, Philip? said my father. Let us see the evidence before we decide, I replied. Now, how was the child lost? My wife's brother, Lieutenant Collingsby, was stationed at a fort on the upper Missouri. My wife was anxious to see him, and we started in one of the steamers I owned then, with our little boy two years old, Mr. Farringford began. The boat had our family name, and was the finest one I owned. We enjoyed the trip very much. I didn't drink very hard at that time, gentlemen, though I occasionally took too much in the evening, or on a festive occasion. On the night the steamer was burned, we were within thirty miles of the fort to which we were going, and where we intended to remain till the Farringford returned from her trip to the mouth of the Yellowstone. I know my wife did not undress the child, because we hoped to reach the fort, and spend the night at the barracks. Expecting to part with the passengers that evening, we had a merry time; and I drank till I was, in a word, intoxicated. I supplied whiskey and champagne for everybody on board, not excepting the officers, crew, and firemen, who would drink them. Even the two or three ladies who were on board partook of the sparkling beverage. Wishing to reach the fort as early as possible, I told the firemen and engineers to hurry up when I gave them their whiskey. They obeyed me to the letter, and the furnaces were heated red hot. I do not know to this day how the boat took fire; but I do know that a barrel of camphene, belonging to some army stores on board, was stove, and its contents ran all over the forward deck. All hands worked hard to save the boat; but they worked in vain. The pilot finally ran her ashore. I pulled down a door, and
- 67. carried it to the main deck aft, while my wife conveyed the child to the same point. The fire was forward, so that we could not leave the boat by the bow, which had been run on shore. I placed my little one upon the door, wrapped in a shawl, with a pillow on each side to keep it from rolling into the water. The captain was to help my wife, while I swam behind the door, holding it with my hands. In this position, partially supported by the raft, I expected to be able to propel it to the shore. My plan was good, and would have been successful, without a doubt, if I had not been intoxicated. When I was about to drop into the water, the stern of the boat suddenly swung around, and I lost my hold upon the raft. I had been lying upon the edge of the deck, with my leg around a stanchion, my head hanging over the water; and I think my position, in addition to the fumes of the liquor I had drank, made me dizzy. I lost the door, and I think I partially lost my senses at the same time. The steamer, as she swung around, slipped from the abrupt shore which held her. This movement created a tremendous excitement, amounting to almost despair, among the passengers and crew. The door was carried away from the steamer, and I lost sight of it. When I was able again to realize my situation, I tried to discover the door, but in vain. I threw a box, which the captain had prepared to support my wife, into the water, and leaped in myself. The current swept the steamer down the river. I paddled my box to the shore, and landed. On which side did you land? I asked. On the north side. I ran on the bank of the river, looking for my child. The glare from the burning steamer lighted up the water, but I could see nothing floating on the surface. I was the only person who had left the boat so far, and I followed her till, two or three miles below the point where I had landed, one of her boilers exploded, and she became a wreck. About one half of the passengers and crew were saved on boxes, barrels, and doors. By the aid of the captain my wife was brought to the shore. I shall never forget her
- 68. agony when I told her that our child was lost. She sank senseless upon the ground; but she came to herself after a time. I wished that I had perished in the flood when I realized the anguish of losing my only child. I could not comfort her; I needed comfort myself. I spent the long night in walking up and down the banks of the river, looking for my lost little boy. Below the place where most of the passengers landed I found many doors and other parts of the boat; but I could not find my child. I reasoned that the current would carry the raft which bore up my child to the same points where other floating articles were found, and I was forced to the conclusion that my darling had rolled from the door and perished in the cold waters. I shuddered to think of it. Before daylight in the morning another steamer appeared, coming down the river. We hailed her, and were taken on board. She proved to be one of my boats, and I caused the most diligent search to be made for my lost little one. About a mile below the point where the Farringford had been run ashore we found a door, with one pillow upon it, aground on the upper end of an island. This discovery was the knell of my last hope. Of course the child had rolled from the door and perished. I wept bitterly, and my wife fainted, though we only realized what seemed inevitable from the first. We discovered this door about daylight, and it was useless to prolong the search. The evidence that my child was lost was too painfully conclusive. My wife wished to return home. We were going on a pleasure excursion, but it had terminated in a burden of woe which can never be lifted from my wife or from me. I drank whiskey to drown my misery. I was seldom sober after this, and I lost all my property in reckless speculations. I became what I am now. My wife never would taste even champagne after that terrible night. She in some measure recovered her spirits, though she can never be what she was before. After I had lost everything, and could no longer provide a home for her, she returned to her father. I have not seen her for five years; but I do not blame her. She was a beautiful woman, and
- 69. worthy of a better husband than I was. You know the whole story now, Philip. These gentlemen knew it before. Not all of it, added Mr. Lamar. And now we can pity and sympathize with you as we could not before. No; I deserve neither pity nor sympathy, groaned my poor father, trembling violently. If I had not been drunk I should have saved my child. Perhaps it is all for the best, since the child was saved, said I. It is impossible! exclaimed Farringford. I cannot believe it. There was no one in that lonely region; and, if my child had reached the shore, it must have perished more miserably of starvation than in the water. You say your wife did not undress the child, because you expected to reach the fort that evening, I continued. Do you know what clothes it had on? I ought to know, for I have tearfully recalled the occasion when I last pressed it to my heart, after supper that awful night. It wore a little white cambric dress, with bracelets of coral on the shoulders. Anything on the neck? Yes; a coral necklace, to which was attached a locket containing a miniature of my wife. In what kind of a shawl was it wrapped when you placed it on the door? I asked, as I unlocked the bureau drawer in which I had placed the precious relics of my childhood. While he was describing it I took the shawl from the drawer. Is this it? Farringford trembled in every fibre of his frame as he glanced at the article.
- 70. It looks like it. I do not know whether it is the same one or not. I trembled almost as much as the poor inebriate in the excitement of the moment. I should hardly consider that sufficient evidence, said Mr. Gray. There are thousands of shawls just like that. I intend to furnish more evidence, I replied, producing the stained and mildewed dress I had brought from the settlement. Do you know that dress, Mr. Farringford? It certainly looks like the one my child wore. It was examined by the gentlemen; but they thought the evidence was not yet conclusive, and I took the bracelets from the drawer. Did you ever see these before? I asked, handing them to the palsied drunkard. You will see the initials P.F. on the clasps. I have seen these, and I know them well. They were given to my child by my brother Philip, replied he, with increasing emotion. There may be some mistake, suggested Mr. Lamar. Hundreds and thousands of just such trinkets have been sold in St. Louis. But these have the initials of my child upon them. P.F. may stand for Peter Fungus, or a dozen other names, replied Mr. Gray. The evidence is certainly good as far as it goes, but not conclusive. What should you regard as conclusive, sir? I asked, rather annoyed at his scepticism, which I regarded as slightly unreasonable. Evidence, to be entirely conclusive, must be susceptible of only one meaning, added Mr. Lamar. The articles you have produced
- 71. may have belonged to some other person, though it is not probable. I don't know that I shall be able to satisfy you, but I will try once more, I replied, taking the locket from the drawer. I handed the locket to Farringford. He grasped it with his shaking hands, and turned it over and over. He examined the necklace with great care, and then tried to open the locket. He trembled so that he could not succeed, and I opened it for him. He glanced at the beautiful face upon which I had so often gazed by the hour together. My wife! exclaimed he, sinking into his chair, and covering his face with his hands, sobbing convulsively like a child. You are my son! Perhaps not, interposed Mr. Lamar, very much to my disgust. But my poor father was satisfied, and sprang forward to embrace me. The excitement was too much for his shattered nerves, and he dropped fainting into my arms. We placed him upon the bed, and I went for Mrs. Greenough.
- 72. T CHAPTER XVI. IN WHICH PHIL STRUGGLES EARNESTLY TO REFORM HIS FATHER. he skilful ministrations of Mrs. Greenough soon restored my father to himself. He had probably eaten nothing since he took his breakfast with me early in the morning, and his frame was not in condition to bear the pressure of the strong emotions which had agitated him. My son! exclaimed he, as the incidents which had just transpired came back to his mind. My father! I replied. He extended his trembling hand to me, and I took it. It would have been a blessed moment to me if I could have forgotten what he was, or if I could have lifted him up from the abyss of disgrace and shame into which he had sunk. I hoped, with the blessing of God, that I should be able to do this in some measure. I determined to labor without ceasing, with zeal and prayer, to accomplish this end. I pity you, my son, said my father, covering his eyes with his hands. It can be no joy to you to find such a father. I should not be sincere, father, if I did not say I wished you were different. Philip,—if that is really your name,—I will reform, or I will die, said he, with new emotion. I have something to hope for now. The good God, who, I believed, had deserted me years ago, has been kinder to me than I deserved.
- 73. He is that to all of us, father. Where did you get this locket, young man? asked Mr. Lamar, who evidently believed there was still a possibility that a mistake had been made. I replied that I had found it in the chest of Matt Rockwood, who had taken me from the door in the river; and I repeated that part of my narrative which I had omitted before. You need not cavil, gentlemen, interposed my father. I am satisfied. I can distinguish the features of my lost son. If you knew my wife, you can see that he resembles her. Look at the portrait, and then look at him. I have seen Mrs. Farringford, but I do not exactly remember her looks, added Mr. Lamar. Matt Rockwood is dead; but there is a living witness who saw the child he found only a day or two after it was picked up, I continued. Who is he? Kit Cruncher; he is at the settlement now, and has known me for eleven years. Mr. Gracewood, whom I expect in St. Louis soon, has known me for six years, and has heard Matt Rockwood tell the story of finding the child. If I am satisfied, no one else need complain, said my father. There are no estates, no property, nor a dollar left, to which any claim is to be established. I am a beggar and a wretch, and an inheritance of shame and misery is all I have for him. But you forget that your wife is still living, Farringford, added Mr. Lamar. Her father is a wealthy man, and his large property, at no very distant day, will be divided among his three children.
- 74. Very true; I did not think of that. I have so long been accustomed to regard her as lost to me that I did not think my boy still had a mother, answered my father, bitterly. But when she sees him, she will not ask that any one should swear to his identity. She will know him, though eleven years have elapsed since she saw him. But where is she? I asked, anxiously. I do not know, Philip. When did you see her last? It is four or five years since we met. But haven't you heard from her? Once, and only once. After she left me, and went back to her father, I tried to see her occasionally, for I have never lost my affection and respect for her. I annoyed Mr. Collingsby, her father, trying to obtain money of him. Three years ago the family moved away from St. Louis, partly, if not wholly, I know, to avoid me, and to take my wife away from the scene of all her misery. Where did they go? To Chicago, where Mr. Collingsby was largely interested in railroad enterprises. Is the family still there? I do not know. They are, added Mr. Gray. But my wife is not there, said my father. Some one told me, a year ago, he had met her in Europe, where she intended to travel for three years with her brother and his wife. Really, Philip, I know nothing more about her. I wish I could lead you to her.
- 75. I was indeed very sad when I thought that years might elapse before I could see her who had given me being. I will make some inquiries, Phil, in regard to the Collingsbys, said Mr. Lamar. Are you satisfied, sir, that I am what I say I am? I asked. I have no doubt you are, though perhaps your case is not absolutely beyond cavil. The old man who died might have found the body of the child, and taken the clothes and trinkets from it; but that is not probable. But I can produce a man who has known me from my childhood, I replied. You can, but you have not, added he, with a smile. I will produce him if necessary. I hope you will see Mr. Gracewood when he arrives. I will, if possible. But, Farringford, was there no mark or scar of any kind on the child which will enable you to identify him? I know of none. Perhaps his mother does, answered my father. But I tell you I am satisfied. I ask for no proof. I know his face now. It all comes back to me like a forgotten dream. Very well; but, Farringford, you have something to live for now, added Mr. Lamar. I have, indeed, replied the trembling sufferer, as he glanced fondly at me. I will try to do better. When you feel able to do anything, we shall be glad to help you to a situation where you can do something to support your boy, said Mr. Gray. I can take care of myself, gentlemen. I am getting three dollars a week now, and I hope soon to obtain more, I interposed.
- 76. Three dollars a week will hardly support you. I shall be able to get along upon that sum for the present. Mrs. Greenough is very kind to me. The two gentleman said all they could to inspire my poor father with hope and strength, and then departed. I was very much obliged to them for the interest and sympathy they had manifested, and promised to call upon them when I needed any assistance. I am amazed, Philip, said my father, when our friends had gone. I knew that you were my father when we met in the evening at the Planters' Hotel, I replied. You remember that you told me you had lost a child on the upper Missouri. I did; I was thinking then what a terrible curse whiskey had been to me. You looked like a bright, active boy, and I desired to warn you, by my own sad experience, never to follow in the path I had trodden. I did not suspect that I was talking to my own son; but all the more would I warn you now. You thrilled my very soul, father, with your words, and I shall never forget them. I shall pray to God to save both you and me from the horrors of intemperance. Philip, I have resolved most solemnly, a hundred times, to drink no more; but I did not keep my promise even twenty-four hours. Is your mind so weak as that? Mind! I have no mind, my son. I haven't a particle of strength, either of body or mind. You must look to God for strength, said Mrs. Greenough, who had listened in silence to our conversation. I have, madam; but he does not hear the prayer of such a wretch as I am.
- 77. You wrong him, Mr. Farringford, replied the widow, solemnly. He hears the prayers of the weakest and the humblest. You have no strength of your own; seek strength of him. My husband was reduced as low as you are. For ten years of his life he was a miserable drunkard; but he was always kind to me. Hundreds of times he promised to drink no more, but as often broke his promise. I became interested in religion, and then I understood why he had always failed. I prayed with my husband, and for him. He was moved, and wept like a child. Then he prayed with me, and the strength of purpose he needed came from God. He was saved, but he never ceased to pray. He redeemed himself, and never drank another drop. Before he died, he had paid for this house, besides supporting us very handsomely for ten years. That is hopeful, madam; but I am afraid I am too far gone. I have no wife to pray with me, said my father, gloomily. I will pray with you. Throwing herself upon her knees before a chair, she poured forth her petition for the salvation of the drunkard with an unction that moved both him and me. I heard my father sob, in his weakness and imbecility. He was as a little child, and was moved and influenced like one. You must pray yourself, Mr. Farringford, said she, when she had finished. You must feel the need of help, and then seek it earnestly and devoutly. I thank you, madam, for all your kindness. I will try to do better. I will try to pray, said he. Could you give me some more of the medicine I took last night and this morning? It helped me very much. Certainly I can. I will do everything in the world for you, if you will only stay here and try to get well.
- 78. She left the room, and went into the kitchen to prepare the soothing drinks which the excited nerves of the patient demanded. I will reform, Philip. I will follow this good lady's advice. Give me your hand, my son, said my father. O, if you only would, father! This world would be full of happiness for us then. We could find my mother, and be reunited forever. God helping me, I will never drink another drop of liquor, said he, solemnly lifting up his eyes, as I held his trembling hand. Mrs. Greenough opportunely returned with the medicines, and with a folded paper in her hand. As my father took his potion, she opened the paper, which was a temperance pledge, on which was subscribed the name of Amos Greenough. This is the pledge my husband signed, with trembling hand, ten years before his death. It was salvation to him here—and hereafter. Will you add your name to it, Mr. Farringford? said Mrs. Greenough. I will. Not unless you are solemnly resolved, with the help of God, to keep your promise, she added. Not unless you are willing to work, and struggle, and pray for your own salvation.
- 79. Phils Father signs the Pledge. Page 193. I am willing; and I feel a hope, even now, madam, that God has heard your prayer for a poor wretch like me. Sign, then; and God bless you, and enable you to keep this solemn covenant with him. She took the writing materials from the bureau, and my father, with trembling hand, wrote his name upon the pledge. May God enable me to keep it! said he, fervently, as he completed the flourish beneath the signature. Amen! ejaculated Mrs. Greenough. May you be as faithful as he was whose name is on the paper with you. Stimulated by his example, and by your kindness, I trust I shall be, said my father.
- 80. Mrs. Greenough then provided a light supper for him, of which he partook, and very soon retired. I told my kind landlady that I had recovered my money, and should now be able to pay my father's board for a time. She had not thought of that matter, and would be glad to take care of him for nothing if she could only save him. As I went to bed I could not but congratulate myself upon finding such a kind and devoted friend as she had proved to be.
- 81. T CHAPTER XVII. IN WHICH PHIL MEETS THE LAST OF THE ROCKWOODS. he next day my father was quite sick; but Mrs. Greenough was an angel at his bedside, and I went to my work as usual. I was filled with hope that the wanderer might yet be reclaimed. Though I longed intensely to see my mother, I think if I had known she was in the city I should not have sought to find her, for I desired to carry to her the joyful news of the salvation of my father. When I could say that he was no longer a drunkard, I should be glad to meet her with this intelligence upon my lips. But she was wandering in distant lands. Plenty and luxury surrounded her, while I was struggling to earn my daily bread, and to take care of my father. The fact that she was in affluence was consoling to me, and I was the more willing to cling to my father in his infirmities. When I went to work that morning I was introduced to a plane and a plank—to test my ability, I supposed, for the men had not yet finished shingling the roof. A plank partition was to be put up in order to make a counting-room in one corner of the storehouse. I had never in my life seen a plane till I came to St. Louis; but I had carefully observed the instrument and its uses. Conant told me how to handle it with ease and effect, and instructed me in setting the iron, so as to make it cut more or less deeply, according to the work to be done. It was hard work, harder than boarding or shingling; but I made it unnecessarily severe for the first hour, and though it was a cool day, the sweat poured off me in big drops. I had not yet got the hang of the thing; but when Conant came from the roof for a bundle of shingles, he looked in to see how I succeeded. A little more instruction from him put me on the right track, and I worked much
- 82. easier; in a word, I learned to use the plane. After removing the rough side from the plank, it was a relief to handle the smoothing- plane, and I polished off the wood to my own satisfaction and that of my employer. In the afternoon I was sent upon the roof again to lay shingles, and we finished that part of the job before night. At six o'clock all the hands were paid off for their week's work. I felt considerable interest in this performance. I had worked three days, and at the price agreed upon I was entitled to a dollar and a half. I shall not want you any longer, Blair, said Mr. Clinch to the young fellow of whom Conant had spoken so disparagingly to me. I owe you six dollars; here is the amount. You don't want me any longer? replied Blair, as he took his wages. No. Why not? You don't suit me. I can't afford to pay you six dollars a week for what you do, answered the employer, bluntly. You don't understand the business, and you don't try to learn it. That boy there does twice as much work in a day as you do. I did not think it right to hear any more of this conversation, and moved away. Though I was pleased with the compliment, I was sorry to have it bestowed upon me at the expense or to the disparagement of another. I walked around the building, but I was soon sent for to receive my wages. Phil, you have done remarkably well, said Mr. Clinch; and I want to use you well. You handle a plane well for one who never saw one before, and I think you were born to be a carpenter. Thank you, sir, I replied. You give me all the credit I deserve.
- 83. And I give you a dollar a day for your work, for you have done twice as much as I expected of you, he added, handing me three dollars. I supposed you would be in the way at first, and I only took you to oblige Captain Davis. I have done the best I knew how, and shall always do so; but I don't ask any more than you agreed to give me. I am entitled to only half of this. Yes, you are. I agreed to give you more if you were worth it. Conant says you have done a man's work most of the time. Of course you can't do that on the average. But you will be worth about a dollar a day to me, now that I have discharged Morgan Blair. Thank you, sir; you are very kind. Kind! Nonsense! I am only doing the fair thing by you. When I think you are worth more than a dollar a day, I shall give it to you. On the other hand, I shall discharge you when I don't want you, or when you are lazy or clumsy. I always speak my mind. I saw that he did, to Blair as well as to me, and I was very thankful for having obtained so good an employer. I was determined to merit his good will by doing my duty faithfully to him. I went home, and found my father more comfortable than in the morning; but he was still very sick, and unable to leave his bed. In the evening I went out to purchase a suit of clothes, which I so much needed. I obtained a complete outfit, which would enable me to attend church the next day, looking like other young men of my age, in the humbler walks of life. Mrs. Greenough had been very particular in urging me to be prepared for church and Sunday school, and had even offered to lend me money to purchase the needed articles. I told her I had never been to church in my life, and I was very glad of the opportunity. When my bundle was ready I turned to leave the store. A young man, whose form and dress looked familiar to me,—though I did not
- 84. see his face, for he was looking at the goods in a glass case,— followed me into the street. Phil, said he; and I recognized the voice of Morgan Blair, the young man who had been discharged that afternoon by Mr. Clinch. I paused to see what he wanted, though I was not very anxious to make his acquaintance after what I knew of him. What is it? I asked. I want to see you about a matter that interests me, he added. What is that? They say you came from way up the Missouri River. Is that so? That's so. Conant said you did. I want to know something about the country up there, and I suppose you can tell me. What do you want to know? I have an uncle up there somewhere, and I want to find him if I can. Do you know in what region he is located? I inquired. I do not; that is what I want to ascertain. Conant told me you came from that country, and I meant to talk with you about it; but you put my pipe out, and I was discharged to-day. I saw you go into that store, and I thought I would wait for you. What do you mean by putting your pipe out? Didn't you put my pipe out? I didn't even know that you smoked. You are rather green, but you have just come from the country. I meant that you caused me to be discharged.
- 85. I did? You heard Clinch say that I did not do half as much work as you did? Yes; I heard that; but it was not my fault. I didn't do any more than I could help, and you put in all you knew how. If you hadn't come, Clinch never would have suspected that I wasn't doing enough for a boy. I don't believe in breaking your back for six dollars a week. But never mind that now. When can I see you and talk over this other matter with you? I can tell you now all I know, I replied. I think I shall go up the Missouri, if I have any chance of finding my uncle. You can't go up this season. No steamers leave so late as this. When did you see your uncle? I never saw him, and I shouldn't know him if I met him to-night. He has been up in the woods for twenty years, I believe. What is his name? Rockwood. Rockwood! I exclaimed, startled by his answer. Yes; my mother was his sister. What was his other name? Matthew. He left Illinois before I was born; but my mother heard from him about ten years ago. Somebody—I don't know who it was—saw him at a wood-yard, and he sent word by this person that he was alive and well, but did not think he should ever come back to Illinois. His name was Matthew Rockwood. Did you ever hear of such a man?
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