Reflection and Transmission of Mechanical Waves
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The document discusses reflection and transmission of mechanical waves at discontinuities in materials. It explains that waves can be reflected, transmitted, or absorbed depending on whether the wave's frequency matches the object's natural vibration frequencies. Reflection occurs if vibrations are not passed through the material, while transmission occurs if vibrations pass through. The document then applies these concepts to analyze reflection and transmission of pressure waves in arteries at locations where properties change, like narrowing, widening, or bifurcations into branches. Mathematical formulas are presented to calculate reflection and transmission coefficients at such discontinuities.
Reflection and Transmission of Mechanical Waves
- 1. PAPERS Reflection and Transmission of Mechanical Waves authors : Group : 7 1. Millathina Puji Utami (100210102029) 2. Pindah Susanti (100210102056) PHYSICS EDUCATION PROGRAM DEPARTMENT OF MATHEMATICS AND SCIENCE EDUCATION FACULTY OF TEACHER TRAINING AND EDUCATION UNIVERSITY OF JEMBER 2012
- 2. Reflection and Transmission of Mechanical Waves When a light wave with a single frequency strikes an object, a number of things could happen. The light wave could be absorbed by the object, in which case its energy is converted to heat. The light wave could be reflected by the object. And the light wave could be transmitted by the object. Rarely however does just a single frequency of light strike an object. While it does happen, it is more usual that visible light of many frequencies or even all frequencies is incident towards the surface of objects. When this occurs, objects have a tendency to selectively absorb, reflect or transmit light certain frequencies. That is, one object might reflect green light while absorbing all other frequencies of visible light. Another object might selectively transmit blue light while absorbing all other frequencies of visible light. The manner in which visible light interacts with an object is dependent upon the frequency of the light and the nature of the atoms of the object. In this section of Lesson 2 we will discuss how and why light of certain frequencies can be selectively absorbed, reflected or transmitted. Reflection and transmission of light waves occur because the frequencies of the light waves do not match the natural frequencies of vibration of the objects. When light waves of these frequencies strike an object, the electrons in the atoms of the object begin vibrating. But instead of vibrating in resonance at a large amplitude, the electrons vibrate for brief periods of time with small amplitudes of vibration; then the energy is reemitted as a light wave. If the object is transparent, then the vibrations of the electrons are passed on to neighboring atoms through the bulk of the material and reemitted on the opposite side of the object. Such frequencies of light waves are said to be transmitted. If the object is opaque, then the vibrations of the electrons are not passed from atom to atom through the bulk of the material. Rather the electrons of atoms on the material's surface vibrate for short periods of time and then reemit the energy as a reflected light wave. Such frequencies of light are said to be reflected.
- 3. Up to this point we have largely neglected one of the most important features of the arterial system - the complexity of the arterial tree with its myriad bifurcations and frequent anastomoses. These anatomical variations in the arteries mean that the waves propagating along them are continuously altering to the new conditions that they encounter. Any discontinuity in the properties of the artery will cause the wavefronts to produce reflected and transmitted waves according to the type of discontinuity. There are many types of discontinuities in the arterial system; changes in area, local changes in the elastic properties of the arterial wall, bifurcations, etc. We will mainly consider two types of discontinuity: changes in properties in single arteries and bifurcations. Before getting into the mathematical details, here is are sketches of what would happen in the simple wave example if the tube either narrowed or widened at some point. Simple example of a wave in a tube that narrows
- 4. The reflection coefficient is positive so that the leading forward compression wavefront reflects as a backward compression wave and the trailing expansion wave reflects as an expansion wave. The transmitted wave must match the pressure produced by the incident and reflected waves at the discontinuity in area giving a wave of similar form but with an increased amplitude. Simple example of a wave in a tube that widens The reflection coefficient is negative so that the leading forward compression wavefront reflects as a backward expansion wave and the trailing expansion wave reflects as a compression wave. The transmitted wave is similar in form to the incident wave with a reduced amplitude because of the negative reflection. Note the direction of 'circulation' in the different waves. Reflections in a single vessel The mathematical details involved in deriving the exact nature of the reflected and transmitted waves is rather complex but the results are straightforward and will be outlined here. The conservation of mass and energy at
- 5. a discontinuity in an elastic vessel require that an incident wave with a pressure change ΔP must generate a reflected wave with pressure change δP that is given by a reflection coefficient R = δP/ΔP. This definition is familiar from many branches of wave mechanics when it is remembered that pressure has the units of energy per unit volume. The value of the reflection coefficient depends upon the area A and wave speed c upstream 0 and downstream 1 of the discontinuity. For arteries where the velocity is generally much lower than the wave speed the equation for R is valid {mathematical details} so that (A0/c0) - (A1/c1) R = (A0/c0) + (A1/c1) This expression varies with the ratios of areas and wave speeds upstream and downstream. There are two simple limits: 1) closed tube, A1 = 0 for which R = 1 2) open tube, A1 = ∞ for which R = -1 All other cases will lie between these two limits. The transmission coefficient T is simply related to the reflection coefficient T=1+R Physically these limits mean that a wavefront encountering a closed end will be reflected with exactly the same pressure change. Remembering the water water hammer equations dP± = ± ρc dU± this means that the change velocity across the reflected wavefront will be opposite that of the incident wavefront.
- 6. Reflections in a bifurcation If we consider a bifurcation where the parent vessel is 0 and the daughter vessels are 1 and 2, the conservation equations can be solved for the reflection and transmission coefficients. The results for m << 1 are (A0/c0) - (A1/c1) - (A2/c2) R = (A0/c0) + (A1/c1) + (A2/c2) This relationship depends on both the areas and the wave speeds (which depend on the distensibilities of the vessels). If all of these data are know, the reflection coefficient can be easily found. For a general discussion, it is useful to make an assumption about the variation of the wave speed with vessel size so that R can be expressed as a function of the areas of the vessels. A reasonable assumption is that c ~ A-1/4. This follows from the Moens-Korteweg equation for the wave speed in thin-walled, uniform tubes if it is assumed that the product of the elastic modulus and the thickness of the vessel wall are constant. Since arteries are not thin-walled and their wall composition and structure changes from vessel to vessel, this assumption is only an approximation. However, it does fit experimental data for the wave speed in arteries of different diameters reasonably well. Define α as the daughters to parent area ratio α = (A1 + A2)/A0 and γ as the daughter symmetry ratio (we assume without loss of generality that A2 < A1) γ = A2/A1
- 7. The extreme values γ = 0 corresponds to a single vessel with no branches and γ = 1 corresponds to a symmetrical bifurcation. The reflection coefficient can now be expressed in terms of these two area ratios (1 - (α/(1+γ))5/4(1 + γ5/4) R = (1 + (α/(1+γ))5/4(1 + γ5/4) We see for symmetrical bifurcations, γ = 1, that R = 0 for an area ratio α ~ 1.15. For α less than this the bifurcation acts like a partially closed tube and R is positive. For α greater than this value the bifurcation acts more like an open tube and R is negative. Reflection coefficient as a function of area ratio for different symmetry ratios
- 8. For a symmetrical bifurcation that is well-matched in the forward direction, the area ratio for a wave travelling backwards in one of the daughter vessels is approximately &alpha = 2.7. The reflection coefficient for this backward wave is approximately R = -0.5 which means that approximately half of the energy of the backward wave will be reflected back in the forward direction and that this wave will be of the opposite type as the incident wave (i.e. a compression wavefront will be reflected as an expansion wavefront and an expansion wavefront will be reflected as a compression wavefront. This is reasonable physically because the backward wave approaching the bifurcation in one of the daughter vessels (now the parent vessel) will see a bifurcation consisting of its twin vessel and the parent vessel with a net area much larger than its own. The bifurcation will therefore act more like an open-end tube and generate a negative reflection coefficient.