Investigation of Surface Waves on Water in Advanced School Laboratory Workshop

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Educational Technology
TECHNOLOGY GROUP
Abstract
Traditionally, in lecture courses on General Physics, the example of waves propagating on the water surface is used for the qualitative description of
wave phenomena. At the same time, in corresponding laboratory courses, there are almost no quantitative tasks on the same topic, the purpose of which
would be to measure the surface wave characteristics. The reason for this difference must be associated with the complexity of the physical mechanism of
generation and propagation of waves on the water surface. In this work, a laboratory problem has been developed to measure the group velocity of circular
waves traveling on the water surface. It uses a standard laboratory water tank equipped with a surface wave generator. The proposed experiment offers
an opportunity for quantitative measurement of surface wave characteristics, a dispute of only qualitative observations of wave phenomena commonly
found in current curricula, due to the introduction of two improvements. First, to produce the water waves’ rich in-contrast representations on a screen,
the optimal generator amplitude is specially set for each experiment; and second, to measure the group wavelength, the device magni ication is preliminary
calculated based on the laws of geometric optics. The tests carried out have shown that the relative accuracy of these measurements is quite acceptable:
within a few percent. The introduction of the problem of surface waves on water into the General Physics laboratory workshop for engineering specialties
of technical universities will contribute to improving the quality of mastering the subject by students.
Problem of Surface Waves
on Water in Higher School
Laboratory Workshop
Chkhartishvili L*, Chikhladze M and Kantaria I
Engineering Physics Department, Georgian Technical University, 77 Merab Kostava
Avenue, Tbilisi, 0160, Georgia
*Correspondence: Levan Chkhartishvili, Professor, Engineering Physics Department,
Georgian Technical University, 77 Merab Kostava Avenue, Tbilisi, 0160, Georgia,
Email: levanchkhartishvili@gtu.ge
Research Article
Introduction
In lecture courses on General Physics the example of
waves, propagating on the water surface is widely used for
the qualitative description of wave phenomena. In contrast,
in corresponding laboratory courses, there are almost no
quantitative tasks aiming to measure various characteristics
of such waves. The reason for this difference we associate
with the complexity of the physical mechanism of formation
and propagation of waves on the water surface.
In general, water waves are a mixture of longitudinal
and transverse surface waves related to forces of surface
tension and gravity, respectively. They are deformations of
the water surface propagating with the wave velocity, while
the water molecules remain at the same positions on average.
Energy, however, moves with the wave velocity as well. There
are known so-called deep- and shallow-water waves. The
distinction between them is determined by the ratio of the
water’s depth d to the half-wavelength λ/2.
When d >> λ/2, one has deep-water waves or short waves.
They reveal the dispersion property, i.e. the velocity of deep-
water waves depends on their wavelength. Every molecule
of a deep-water wave moves in a circular orbit, the diameter
of which decreases with the distance from the surface. The
motion is felt down to a distance of ~λ, where the waves’
energy becomes negligible.
At d << λ/2 shallow-water waves or long waves arise. They
show almost no dispersion. However, the velocity of shallow-
water waves depends on the water’s depth. The molecules of
a deep-water wave move in elliptic orbits.
In this paper, a laboratory problem is developed to
measure by students the group velocity of deep-water surface
waves at low frequencies. It uses a standard laboratory water
tank equipped with a generator of circular surface waves.
The tests carried out have shown that the relative accuracy of
these measurements is quite acceptable: within a few percent.
The introduction of the laboratory problem “Determination of
Group Velocity of Surface Waves on Water” into the General
Physics laboratory workshop for engineering specialties of
technical universities will contribute to improving the quality
of mastering the subject by students.
Literature review
A huge number of studies are devoted to improving the
Article Information
Submitted: September 24, 2024
Approved: October 21, 2024
Published: October 22, 2024
How to cite this article: Chkhartishvili L, Chikhladze M,
Kantaria I. Problem of Surface Waves on Water in Higher
School Laboratory Workshop. IgMin Res. October 22, 2024;
2(10): 826-834. IgMin ID: igmin255; DOI: 10.61927/igmin255;
Available at: igmin.link/p255
Copyright: © 2024 Chkhartishvili L, et al. This is an open
access article distributed under the Creative Commons
Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the
original work is properly cited.
Keywords: Water surface; Group velocity; Laboratory
workshop; General physics course

TECHNOLOGY October 22, 2024 - Volume 2 Issue 10
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quality of teaching physics and closely related issues of other
natural sciences, as well as engineering and mathematics.
In this brief review, we consider only a small part of the
literature recently published on the problems, the solution of
which is especially important for the engineering specialties
of technical universities.
StartingfromtheUSfederallegislationspecifyingequitable
access to education for all students at all levels of education,
including postsecondary education, [1] explored how well the
physics education research community has to serve students
whoinherentlyvaryintheirneeds,abilities,andinterests.Four
research-based curricula (namely, Tutorials in Introductory
Physics, Open Source Tutorials in Physics Sensemaking,
Physics by Inquiry, and Next Generation Physical Science and
Everyday Thinking) were compared with a universal design
for learning framework originated in the education literature
base. It was stated that if the physics education research
community wants instructors to use high-quality, research-
based curricula, curriculum developers should prioritize the
support of all learners.
In the Italian education system, secondary students are
confronted with the foundations of quantum physics during
the inal term (in the preuniversity year) of scienti ic high
school. [2] wishing to illustrate the didactic suggestions
focused on learning motivation that emerged during a long
teaching experience, which could help to tackle the main
problems found. In particular, experimental evidence of the
undulatory nature of matter, postulated by De Broglie, and the
uncertainty principle could conclude the path signi icantly.
The experimental dimension can be further explored with
activities to be carried out not only in the educational
laboratory of the scienti ic high school but also in laboratories
of universities and research institutions, adhering to guidance
projects.
Factors that affect exclusively physical, virtual, and both
Virtual-Physical (VP) and Physical-Virtual (PV) sequences
of manipulatives were explored [3] within a Studio Physics
environment, in which students actively work together to
learn through real-world experimentation and simulation-
based activities as well. Statistics show that comprehensive
learning groups (i.e. any combination of PV or VP sequences)
performed better than any other setting. The study also
pinpoints the importance of the sequence for which virtual or
physical experiments of some speci ic concepts are initially
introduced to students.
Didactic design and outcomes in an educative program,
involving several physics and mathematics university courses
for engineers, based on the use and construction of widgets
generated under Project Oriented Learning and Blended
Learning methodologies was summarized in the work [4]. In
this approach, the widgets generated by teachers are used
by students to appropriate some basic concepts and then
to generate their own widgets to develop complex thinking
skills, applying related concepts but involving alternative
situations. Mathtab and other widget developers were used
to generate activities in terms of their capabilities and course
requirements. Post-test results of students’ value perceptions
indicated positive attitudes toward the use of widgets.
There were different kinds of activities – exhibitions,
educational research, and sessions for students – involving
the Laboratory of Mathematical Machines of the University
of Modena e Reggio Emilia and its collection of instruments,
the mathematical machines [5]. They are copies of historical
instruments, constructed with an educational aim but also
used in the popularization of mathematics.
The “Talk about Chemistry” (TAC) is a series of annual
voluntary meetings to talk about current and stimulating
chemistry topics and discuss how to teach and learn them [6].
TACs are the integrating actions involving undergraduates,
new and senior chemistry teachers, and teacher educators as
well. Their short speeches are clear and accessible, in order
to encourage the audience’s participation. These activities
provide simultaneously initial training and the continuous
education of chemistry teachers. TACs require relatively little
resources and organization but have a quite positive impact
regarding the training of chemistry teachers.
It was identi ied that the majority of chemistry textbooks,
both for high school and higher education, show the key
physical-chemicalconceptofelectronegativityinanahistorical
and decontextualized manner. In this perspective of teaching,
the concept of electronegativity becomes inadequate to the
reality into which it is inserted, because the information
present in the textbooks, generally speaking, does not seek
to explain the relation between electronegativity and other
periodical properties of chemical elements. Besides, textbooks
useonlyone(namely,Pauling’s)de initionofelectronegativity,
not taking into account other possible approaches to the same
concept. Such a strong standardization of characteristics
shown in textbooks, from the point of view of [7], can lead
students to conceptual errors. Thus, it is proposed that the
treatment of the concept of electronegativity be performed
within a historical context establishing connections between
electronegativity and other physical/chemical concepts with
which it is constituted in an inter-relational base.
There was investigated [8] the effect of molecular model
sets on naming simple organic compounds with reference
to the International Union of Pure and Applied Chemistry
(IUPAC) system of naming. A quasi-experimental design using
molecular model sets was used on students in form three
science classes of senior high school. Analysis of the pre-
test and post-test scores showed more improvement in the
performanceoftheexperimentalgroupthanthecontrolgroup.

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A paired sample test, on responses of students’ attitudes,
showed a statistically signi icant difference between the
mean scores for the experimental group before and after the
treatment.Thestudyrevealedthattheintegrationofmolecular
model set approaches to teaching could help reduce, if not
remove completely, the dif iculties students face in naming
organic compounds according to the IUPAC nomenclature.
Furthermore, the study indicated that the integration of
molecular model sets in teaching organic chemistry topics
enhanced the students’ conceptual understanding. Moreover,
the syllabus should not be loaded so that teachers would not
concentrate on the theory aspect alone but would also include
many practical lessons to improve chemistry teaching and
learning in senior high schools.
In the paper [9], responding to 100 years of the Knudsen
gauge, it was described as a corresponding laboratory exercise
for high school physics students. The students were presented
with the outline of the experiment, the necessary equations,
and a collection of parts (including iber suspension, heater,
eddy current damper, mirror, and tension adjustment
assembly) from which it is possible to build the Knudsen gauge
and then take measurements to determine the pressure-force
relationship and the pressure range over which the apparatus
could provide useful results. The vacuum system applied a
diffusion pump and a mercury McLeod gauge was used as the
standard.
After reporting on the impact of students’ preconceptions
on learning, the review [10] highlighted some existing studies
on students’ misconceptions of electricity and magnetism.
This paper also updated physics educators and researchers on
conceptual assessments used to test students’ misconceptions
about electricity and magnetism and suggested strategies for
their remedying. Some educational implications and practical
recommendations for effective teaching and learning in
electricity and magnetism were also outlined.
The purpose of the study [11] was to review the
postgraduate theses on physics education (in Turkey) to
guide the teaching of the sound concept. Both conventional
and contemporary approaches were found to have a positive
effect on the achievement of the sound concept. However,
students and teacher candidates, in general, were not able to
relate their knowledge of sound to daily life and their level of
knowledge was inadequate. The most important output of this
study is the list of possible misconceptions or confusion about
the concept of sound.
A characteristic of teaching and learning approaches used
within the new concept of Learning-by-Being (LBB) was given
[12]. The evolution of educational paradigms from Learning-
by-Doing(LBD)andLearning-by-Understanding(LBU)toward
LBBwasanalyzed.ThebasicideaofLBBisstudents’ownership
of cognitive goals, or the assumption of learning objectives, in
other words – the intrinsic motivation of students. The school
physics lab was examined as an example of a constructivist
learning environment. The basic principles for its organization
in a constructivist manner were enumerated as the provision
of opportunities for students’ own thinking, giving students
the freedom to identify solutions through verbalization of the
problem, a necessity for teachers to know a priori concepts of
students, their effort as a mandatory condition to achieve own
interest.
Theoretical basis
If certain momentum is periodically transmitted to
some point on the lat horizontal surface of a reservoir with
homogeneous water, for example, from a ball that vibrates
in the vertical direction, then the surface waves with a
circular front appear. A reservoir is considered of “in inite
depth” and “in inite area” if these wave disturbances due to
their attenuation with travelling distance practically do not
reach, respectively, the bottom and shores of the reservoir.
In this case, there is no interference of the primary wave
with waves re lected from the bottom and shores. On the
surface of a reservoir of “in inite depth” and “in inite area”
a monochromatic wave, which corresponds to a certain
wavelength λ, propagates with velocity v called wave phase
velocity.
Governing equations for monochromatic waves on the
water surface can be found, e.g., in [13]. They lead the v = v(λ)
dispersion relation for surface tension-gravity progressive
waves on deep-water surface in the following form:
2
2
g
v
 
 
  (1)
where, ρ is the water density, σ is the air-water interface
tension coef icient, and g is the acceleration of gravity.
By the nature of the forces that return the surface of the
water to its initial, i.e. undisturbed or equilibrium state, the
water surface waves are distinguished as gravitational and
capillary waves. They are determined, respectively, by the
force of gravity and the force of the liquid surface tension. It is
clear that in the above formula (1) in the sum under the square
root, the irst term corresponds to gravitational waves and
the second to capillary waves. It is also seen that gravitational
effects dominate at long wavelengths, while capillary ones – at
short wavelengths.
Phase velocity reaches its minimum of
4
min
4g
v


 (2)
at wavelength of
min 2
g

 

 (3)
Note that ρ ≈ 1000 kg/m3
, σ ≈ 0.0727 N/m, and g ≈ 9.81

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m2
/s, so that λmin
≈ 17 mm. Thus, for monochromatic surface
waves, the water can be considered as deep if its depth
satis ies the condition d >> 8.5 mm.
Also note that just at the minimum point (3), the values
of the above two terms are equal each to other. Therefore,
at relatively short wavelengths, λ << λmin
, when waves on the
water surface are mainly of a capillary nature their phase
velocity is determined by the formula:
2
v


 (4)
However, if the surface waves are rather long, λ >> λmin
,
then the gravitational mechanism of the waves’ formation
prevails and the phase velocity is equal to:
2
g
v


 (5)
However, in the general case, a wave propagating on the
water reservoir's surface is not a monochromatic wave, but
a group of monochromatic waves with a frequency of f equal
to the oscillations frequency of the waves’ source (generator).
Therefore, it is too important to determine the group velocity
V of surface waves on water as a function of the group
wavelength. We resolved [14] this problem in analytic form.
The group wavelength Λ can be de ined as the distance that
the intensity maximum of the group of waves travels during
a period of oscillations. Consequently, the following relation
holds:
V
f
  (6)
Let us assume that the width ∆λ of the wavelength range,
which accounts for the most part of the energy carried by a
group of waves, is negligible if compared with the group
wavelength: Δλ << Λ. In this case, the group of waves can
be considered as a so-called wave packet and the velocity of
movement of its intensity maximum, i.e. the group velocity of
waves, can be calculated from the dispersion law v = v(λ) as
follows:
 
 
v
V v


 
 
 

 
 

 
(7)
Substituting into this general relation (7) the expression
(1) written above for the dispersion law of surface waves on
water, we obtain equality for the group velocity:
3
4
2
2
g
V
g

 

 







(8)
The extrema condition, ( )/ 0
dV d
   , implies the
biquadratic equation,
2 4 2
4 2
2 2
24 48
0
g g
   
 
 
 
 (9)
thatdeterminesthepositionsofthegroupvelocityextrema.
This equation has two real and two imaginary roots. From
the real roots, one is positive and the other is negative. It is
obvious that imaginary roots and a real negative root have no
physical meaning. So that the group velocity minimum against
the group wavelength corresponds to the real positive root:
min
(2 3 3)
2 15.97 16
g g g
  

  
 


 (10)
At the group wavelength of Amin
(10), the group velocity (8)
reaches its minimum of
4 4 4
min
3(2 3 3)
1.086
g g g
V
  
  

   (11)
These theoretical results were experimentally veri ied by
the test measurements.
Figure 1 shows the V – Λ dependence calculated for the
above-stated curve parameters σ, and g. The corresponding
coordinates of the weakly expressed minimum, (10) and (11),
are Λmin
= 43.6 mm and Vmin
= 163 mm/s, respectively.
Note that for the relatively short and, therefore, mainly
capillary nature waves, Λ << Λmin
, the group velocity is equal to
3 2
2
V




(12)
whereas for the relatively long, i.e. mainly gravitational
waves, Λ >> Λmin
, we have:
1
2 2
g
V


 (13)
In other words, in these two limiting cases the dispersity
laws of the group velocity, respectively, are the corresponding
phase velocity dispersity laws multiplied by factors 3/2 or
1/2.
Experimental setup
Experimental setups useful for studying the waves on
the water surface usually enable also experiments that are
analogous to those carried out on other types, e.g., light, waves.
Then, all the wave optics laws can be clearly demonstrated by
means of water waves.
Figure 1: Theoretical dependence of water surface waves group velocity on their
group wavelength.

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There were experimentally investigated [15] the large
transient surface wave groups on water produced by the
number of waves focused at one point in space and time.
Results of measurements of the water surface elevation and
the underlying kinematics were compared with theory. A
wave lame was 20 long and 0.3 m wide and had a maximum
working depth of 0.7 m. Waves period was 0.4 – 2.0 s. No
re lected components could be tolerated within the working
section. To achieve this aim, a large-block foam was installed
to provide passive absorption of surface waves.
Dispute the fact that water surface waves are everyday
phenomena they usually are not discussed in textbooks. The
main reason is that surface waves are very different from
most other waves in that their velocity is strongly dependent
on the wavelength. When the wavelength is smaller, the wave
velocity is affected by surface tension: by adding an agent
to change the surface tension, it was noticed that the wave
velocity increases with surface tension. Let us state that as λmin
≈17 mm, at λ < 20 mm the surface tension mechanism has to
dominate, and then the increasing wave velocity with surface
tension seems natural.
Laboratory workshop in General Physics designed by us
[16] for studying in English students of engineering specialties
(but not for “Engineering Physics” specialty itself) commonly
offered by technical universities includes 20 problems, for
which, in particular, are equipped laboratories of the Georgian
Technical University (Tbilisi, Georgia). Currently, the water
surface wave problem is not among them and we have decided
to develop it.
The ripple tank Complete 11260-88 produced by PHYWE
Systeme GmbH & Co. KG, Gottingen, Germany (Figure 2)
serves for the generation and experimental study of 5 – 60 Hz
surface waves in water [17]. The integration in the ripple tank
generator can cause a plug-shaped exciter to be dipped into
the water for the generation of circular surface waves (see the
Figure inserted for the scheme in the laboratory work report
example).
Plastic foam is stuck to the inner sides of the water illed-in
wave tray. This allows an almost complete absorption of the
water waves generated in the tray that hit its sides and thereby
avoids unwanted interference with wave re lections. Water
maximal depth in the tank is about 300 mm, i.e. signi icantly
exceeds group wavelengths of about 10 mm typically observed
in that tank. Thus, the surface waves generated in the ripple
tank should be considered as deep-water ones.
A light emitting diode (LED), which can be considered
as a point source of light, ixed to the bottom of the tank
housing illuminates the wave tray from below and produces a
representationofthewaterwavesthatisrichincontrast.When
the drawing table, i.e. screen is placed on the ripple tank, and
a sheet of white paper is placed on the table, the water waves
can be observed on the sheet of paper. Light from a point
source passes through the wave tray and is partially subject
to de lection according to the structure of the surface waves.
This de lection generates an alternating bright/dark pattern
on the screen. As wave troughs make the light divergent and
crests – convergent, they can be recognized, respectively, as
dark and bright stripes on the observation surface.
In this ripple tank, either continuous or stroboscopic light
can be used. Stroboscopic light allows the propagation velocity
of the water waves to be slowed down to a complete standstill.
The point is that for continuous light, the human eye is only
capable of recognizing the wave image in the lower frequency
range. Stroboscopic light should be used with shorter
wavelengths – higher frequencies, so that wave propagation
is slowed down and made visible to the human eye.
Note that a good contrast is only given at small wave
amplitudes. At large amplitudes, the emitted light rays
combine far before they reach the screen and diverge again
on the long way to it. The result is the disappearance of the
wave image. For this reason, the optimal generator amplitude
should be set for each experiment.
A number of wave phenomena can be observed with
an above-described ripple tank. These are generation,
propagation, re lection, dispersion, refraction, interference,
diffraction, Doppler effect, etc. However, the listed are only
observations, not measurements. Here we propose a way, in
which the same experimental setup can be directly used to
measure the group velocity of surface waves on water.
However, the use of this device for quantitative
measurements requires a preliminary determination of its
magni ication. Let us solve this problem based on the laws
of geometric optics (Figure 3). The light source is considered
Figure 2: Ripple tank with light source Complete 11260-88.

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as a point one and the following values are introduced. Let d
be the distance from the source to the water surface in the
tank, and d' be the distance to the screen. If we assume that
the point light source is located at the bottom of the tank, then
the distance d is nothing more than the depth of the water.
Obviously, their difference
'
d d d
   (14)
coincides with the distance between the water surface
and the screen plane. Let us denote by r and r’ the radii of the
circle of a certain ixed phase of the water surface wave and its
image on the screen. From the similarity of the two triangles
shown in the Figure 3, it follows the equality of the ratios of
their sides:
' '
d r
d r
 (15)
From this, we get the expression for magni ication:
'
1
r d
r d

    (16)
According to our estimates based on the relative positions
of the point light source, water surface, and screen in the ripple
tank, image magni ication  should be approximately 1.36.
Let Λ* is the measured on the screen, i.e., apparent group
wavelength, then the true group wavelength and group
velocity are
*

 

(17)
and
V = Λf (18)
respectively.
Laboratory problem on water surface waves
An example of the proposed laboratory work report is
presented below. It is prepared according to the laboratory
works report template used in the Georgian Technical
University (Tbilisi, Georgia). Template includes the following
sections: Scheme (here Figure 4 shows the circular waves on
the water surface in used ripple tank), Basic Formulas, Table
(here Table 1 represents the results of measurement of group
velocity of surface waves in water in dependence of group
wavelength), Calculations, Curve (here Figure 5 shows the
group velocity of surface waves on water in dependence on
group wavelength both as experimental points and theoretical
curve), and Brief Description of Problem Execution.
Figure 3: Determination of magni ication of ripple tank with light source.
Report
Laboratory Work No: [ number ]
Problem No: [ number ]
Title of Problem: Determination of Group Velocity of Surface
Waves on Water
Faculty: Faculty of Informatics and Control Systems
(Georgian Technical University)
Course: General Physics – 1
Group: [ number ]
Student: [ Name Surname ]
Scheme
Figure 4: Circular waves on the water surface in the ripple tank.
Figure 5: Group velocity of surface waves on water in dependence on group
wavelength: experimental points and theoretical curve.

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Basic formulas
*

 

Vexp
=Λf
theo
3
4
2
2
g
V
g

 

 







theo exp
theo
100 %
V V
V
V


 
Λ group wavelength
Λ* apparent group wavelength
Λ* magni ication
f frequency of wave generator
Vexp
experimental group velocity
Vtheo
theoretical group velocity
g acceleration of gravity
ρ water density
σ water–air interface tension coef icient
δV relative error
The measurements were carried out on the ripple tank
Complete 11260-88. The apparent group wavelength was
measured from the image on the screen, and the true group
wavelength was recalculated taking into account the setup
magni ication. Then based on the detected value of the wave
generator’s oscillation frequency, the experimental value
of surface wave group velocity was determined. For the
same group wavelength, the theoretical group velocity was
calculated as well. The measurement accuracy was estimated
by the relative deviation of the experimental value from the
theoretical one. These experiments were repeated at four
different frequencies. The experimental group velocities
range of 290 – 380 mm/s was found to have corresponded
to the frequency range of 30 – 60 Hz. Their average deviation
from the theoretically calculated values was found to be about
4%.
Lecturer: [ signature ]
Date: [day/month/year]
The oscillation frequency f in the generated surface
waves was detected with an absolute error of about 0.5 Hz,
which for the frequency range 30 – 60 Hz of the performed
measurements corresponds to a relative error of about 1%.
The absolute error in measuring water depth d, distance
∆d between the surface of water and screen, and apparent
wavelength Λ* was 0.05 mm, which at the ripple tank’s given
dimensions (width × length × height = 300 × 370 × 330 mm)
and apparent wavelength of about 10 mm means relative
errors of about 1 and 0.5%, respectively, for magni ication
and apparent wavelength. Summing up these percentages,
we obtain about 2.5% for the error in the experimental
determination of the group velocity of waves on the water
surface. Therefore, the averaged δV deviation of about 4%
from theoretical values being quite small itself nevertheless is
higher than expected.
In this regard, it should be taken into account that the
values of the theoretical curve’s parameters ρ, and σ are
known only approximately. In addition, the errors associated
with the wave packet model description of the surface wave
groups generated in the installation are practically unknown.
This is the reason why the relative error in the above
laboratory work is estimated as averaged relative deviations
of measured values of group velocity from theoretical ones
corresponding same group wavelengths.
Table 1: Results of measurement of group velocity of surface waves in water in
dependence of group wavelength.
#
Frequency
f,
Hz
Apparent
group
wavelength
Λ*,
mm
Image
magni
ication
Γ
Group
wavelength
Λ,
mm
Experimental
group
Velocity
V
exp,
mm/s
Acceleration
of
gravity
g
,
m
2
/s
Water
density
ρ,
kg/m
3
Water–air
interface
tension
coef
icient
σ,
N/m
Theoretical
group
velocity
V
theo
,
mm/s
Relative
error
δV,
%
1
2
3
4
60
50
40
30
8.6
10.0
11.9
13.2
1.36
6.3
7.4
8.8
9.7
379
368
350
291
9.81 1000 0.0727
395
365
332
313
4.1
0.0
5.4
7.0
Table
Calculations
Λ ≈
.
.
≈ 6.32 mm
Λ ≈
.
.
≈ 7.35 mm
Λ ≈
.
.
≈ 8.75 mm
Λ ≈
.
.
≈ 9.71 mm
≈ 6.32 mm ∙ 60 Hz ≈ 379 mm/s
≈ 7.35 mm ∙ 50 Hz ≈ 368 mm/s
≈ 8.75 mm ∙ 40 Hz ≈ 350 mm/s
≈ 9.71 mm ∙ 30 Hz ≈ 291 mm/s
≈
| |
∙ 100 % ≈ 4.1 %
≈
| |
∙ 100 % ≈ 0.0 %
≈
| |
∙ 100 % ≈ 5.4 %
≈
| |
∙ 100 % ≈ 7.0 %
Brief Description of Problem Execution
The purpose of the laboratory problem was to determine
the group velocity of waves propagating on the water surface.
Table 2: Experimental values of group wavelength and corresponding group velocity for
surface waves on water of different depths.
# Group wavelength, mm Group velocity, mm/s Water depth, mm
1
2
3
4
5
6
7
6.3
7.4
8.8
9.7
11.6
15.3
21.4
379
368
350
291
258
243
238
300
300
300
300
40
40
25

DOI: 10.61927/igmin255
2995-8067
ISSN
833
Here, we aimed to introduce the problem of surface waves
on water into the General Physics laboratory workshop.
Achieving this aim has been demonstrated by elaborating on
the Laboratory Problem: “Determination of Group Velocity
of Surface Waves on Water”. Providing explanations of how
students can adjust the experimental parameters, such as
wave frequency and amplitude, to explore other phenomena
related to water surface waves needs further development of
corresponding laboratory problems.
Comparison to published experimental data
Itisexpedienttocompare theobtainedhere andpreviously
published experimental data on surface waves in water to
contextualize the accuracy of the conducted measurements.
In order to observe this interesting phenomenon,
Professors of three secondary schools and the City University
of Hong Kong built [18] a quite large (6 × 4 × 2 ft) water tank to
investigatetherelationbetweenwavelengthandwavevelocity
under different water depths. Measured wave velocities agree
with the theoretical values when the wavelength is large
enough: λ > 20 mm. They have made measurements of the
wave velocity for water depths of 25 and 40 mm. In most
cases, the uncertainty was about 5%. However, occasionally
the uncertainty was up to 15% for waves with highly erratic
waveforms.
Our (for a water depth of 300 mm) and part of their results
corresponding to the lower wavelength are merged in Table
2. One can state that orders of magnitude of the group velocity
measured values, trends of its group wavelength-dependence,
and measurement accuracies are the same.
Conclusion
Based on the analytical expression for the dispersion
relation V = V(Λ) which was obtained theoretically for packets
of surface waves in water, the dependence of the propagation
velocity V of a group of such waves on its group wavelength
Λ has been analyzed. It is concluded that at suf iciently
short group wavelengths, the group velocity should rapidly
decrease, while at suf iciently long waves, on the contrary, it
should increase very slowly. Physically, this behavior of the
dispersion function should be associated, respectively, with
the dominance of capillary and gravitational mechanisms
generating surface waves in water. Between these two
limiting cases there is placed a group velocity minimum,
Vmin
=163 mm/s, which corresponds to the group wavelength
of Λmin
= 43.6 mm. The constructed theoretical curve in
its general shape is in good agreement with the known
experimental curves measured in various water reservoirs
and seas.
In order to make a quantitative comparison between
theory and experiment, the measurements were carried out
in a standard laboratory water tank equipped with a low-
frequency generator of circular surface waves. The frequency
range of measurements, f = 60 – 30 Hz, approximately
correspondstotherangeofgroupwavelengthsofΛ=6–10mm,
which is placed on the decreasing branch (Λ/Λmin
≤ 0.23 << 1)
ofthedispersionfunction.Onaverage,theexperimentalvalues
of group velocity deviate from the theoretical ones by 4%.
The used ripple tank serves for measuring the parameters
of surface waves in water in the frequency range of 5 – 60 Hz,
which corresponds to the group wavelength values of about
10 mm. Figure 1 shows that the experimental study of all the
main features of such waves needs generator modi ication to
widen the group wavelengths range to 1 – 100 mm. However,
to retain the deep-water approximation accuracy at a surface
wavelengthof100mm,oneshouldincreasethewatermaximal
depth in the tank by about 3 times.
Finally, we conclude that the measurements carried out
provide acceptable accuracy and con irm the theoretically
expected dependence of the group velocity of waves on the
water surface on their group wavelength.
The laboratory problem on measuring the group velocity
of waves on the water surface, proposed on the basis of these
studies,beingincludedinthelaboratoryworkshopoftechnical
universities’ course on General Physics will contribute to
improving the students learning quality.
References
1. Scanlon E, Schref ler J, James W, Vasquez E, Chini JJ. Postsecondary
physics curricula and universal design for learning: Planning for diverse
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DOI: 10.61927/igmin255
2995-8067
ISSN
834
10. Mbonyiryivuze A, Yadav LL, Amadalo MM. Students’ conceptual
understanding of electricity and magnetism and its implications: A
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How to cite this article: Chkhartishvili L, Chikhladze M, Kantaria I. Problem of Surface Waves on Water in Higher School Laboratory Workshop. IgMin Res.
October 22, 2024; 2(10): 826-834. IgMin ID: igmin255; DOI: 10.61927/igmin255; Available at: igmin.link/p255
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Tbilisi: Publishing House “Technical University”; 2020. 136 p.
17. PHYWE Systeme. Operating Instructions. Ripple Tank with LED Light
Source, Complete 11260-88. Gottingen (Germany): PHYWE Systeme
GmbH & Co. KG; 2020. 5 p. Available from: www.phywe.com
18. In LW, Lun LY, Kin LM, Cheung H-F. Surface wave on water. Hong Kong:
Physical Society of Hong Kong; 1996. 12 p. Available from: http://www.
phys.ust.hk/genphys/press/cityu.pdf

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