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# Reflection and Transmission of Mechanical Waves

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Reflection and Transmission of Mechanical Waves

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### Reflection and Transmission of Mechanical Waves

1. 1. PAPERS Reflection and Transmission of Mechanical Waves authors : Group : 7 1. Millathina Puji Utami (100210102029) 2. Pindah Susanti (100210102056) PHYSICS EDUCATION PROGRAMDEPARTMENT OF MATHEMATICS AND SCIENCE EDUCATION FACULTY OF TEACHER TRAINING AND EDUCATION UNIVERSITY OF JEMBER 2012
2. 2. Reflection and Transmission of Mechanical Waves When a light wave with a single frequency strikes an object, a number ofthings could happen. The light wave could be absorbed by the object, in whichcase its energy is converted to heat. The light wave could be reflected by theobject. And the light wave could be transmitted by the object. Rarely howeverdoes just a single frequency of light strike an object. While it does happen, it ismore usual that visible light of many frequencies or even all frequencies isincident towards the surface of objects. When this occurs, objects have a tendencyto selectively absorb, reflect or transmit light certain frequencies. That is, oneobject might reflect green light while absorbing all other frequencies of visiblelight. Another object might selectively transmit blue light while absorbing allother frequencies of visible light. The manner in which visible light interacts withan object is dependent upon the frequency of the light and the nature of the atomsof the object. In this section of Lesson 2 we will discuss how and why light ofcertain frequencies can be selectively absorbed, reflected or transmitted. Reflection and transmission of light waves occur because the frequenciesof 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 atomsof the object begin vibrating. But instead of vibrating in resonance at a largeamplitude, the electrons vibrate for brief periods of time with small amplitudes ofvibration; 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 throughthe bulk of the material and reemitted on the opposite side of the object. Suchfrequencies of light waves are said to be transmitted. If the object is opaque, thenthe vibrations of the electrons are not passed from atom to atom through the bulkof the material. Rather the electrons of atoms on the materials surface vibrate forshort periods of time and then reemit the energy as a reflected light wave. Suchfrequencies of light are said to be reflected.
3. 3. Up to this point we have largely neglected one of the most importantfeatures of the arterial system - the complexity of the arterial tree with its myriadbifurcations and frequent anastomoses. These anatomical variations in the arteriesmean that the waves propagating along them are continuously altering to the newconditions that they encounter. Any discontinuity in the properties of the artery will cause the wavefrontsto 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. Wewill mainly consider two types of discontinuity: changes in properties in singlearteries and bifurcations. Before getting into the mathematical details, here is are sketches of whatwould happen in the simple wave example if the tube either narrowed or widenedat some point.Simple example of a wave in a tube that narrows
4. 4. The reflection coefficient is positive so that the leading forward compressionwavefront reflects as a backward compression wave and the trailing expansionwave reflects as an expansion wave. The transmitted wave must match thepressure produced by the incident and reflected waves at the discontinuity in areagiving a wave of similar form but with an increased amplitude.Simple example of a wave in a tube that widensThe reflection coefficient is negative so that the leading forward compressionwavefront reflects as a backward expansion wave and the trailing expansion wavereflects as a compression wave. The transmitted wave is similar in form to theincident wave with a reduced amplitude because of the negative reflection. Notethe direction of circulation in the different waves.Reflections in a single vessel The mathematical details involved in deriving the exact nature of thereflected and transmitted waves is rather complex but the results arestraightforward and will be outlined here. The conservation of mass and energy at
5. 5. a discontinuity in an elastic vessel require that an incident wave with a pressurechange ΔP must generate a reflected wave with pressure change δP that is givenby a reflection coefficient R = δP/ΔP. This definition is familiar from manybranches of wave mechanics when it is remembered that pressure has the units ofenergy per unit volume. The value of the reflection coefficient depends upon thearea A and wave speed c upstream 0 and downstream 1 of the discontinuity. Forarteries where the velocity is generally much lower than the wave speed theequation 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 anddownstream. There are two simple limits: 1) closed tube, A1 = 0 for which R = 1 2) open tube, A1 = ∞ for which R = -1All other cases will lie between these two limits. The transmission coefficient T issimply related to the reflection coefficient T=1+R Physically these limits mean that a wavefront encountering a closed endwill be reflected with exactly the same pressure change. Remembering the waterwater hammer equations dP± = ± ρc dU±this means that the change velocity across the reflected wavefront will be oppositethat of the incident wavefront.
6. 6. Reflections in a bifurcation If we consider a bifurcation where the parent vessel is 0 and the daughtervessels are 1 and 2, the conservation equations can be solved for the reflection andtransmission 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 (whichdepend on the distensibilities of the vessels). If all of these data are know, thereflection coefficient can be easily found. For a general discussion, it is useful tomake an assumption about the variation of the wave speed with vessel size so thatR can be expressed as a function of the areas of the vessels. A reasonableassumption is that c ~ A-1/4. This follows from the Moens-Korteweg equation forthe wave speed in thin-walled, uniform tubes if it is assumed that the product ofthe elastic modulus and the thickness of the vessel wall are constant. Since arteriesare not thin-walled and their wall composition and structure changes from vesselto vessel, this assumption is only an approximation. However, it does fitexperimental data for the wave speed in arteries of different diameters reasonablywell.Define α as the daughters to parent area ratio α = (A1 + A2)/A0and γ as the daughter symmetry ratio (we assume without loss of generality thatA2 < A1) γ = A2/A1
7. 7. The extreme values γ = 0 corresponds to a single vessel with no branchesand γ = 1 corresponds to a symmetrical bifurcation. The reflection coefficient cannow 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 ispositive.For α greater than this value the bifurcation acts more like an open tube and R isnegative.Reflection coefficient as a function of area ratio for different symmetry ratios
8. 8. For a symmetrical bifurcation that is well-matched in the forwarddirection, the area ratio for a wave travelling backwards in one of the daughtervessels is approximately &alpha = 2.7. The reflection coefficient for thisbackward wave is approximately R = -0.5 which means that approximately half ofthe energy of the backward wave will be reflected back in the forward directionand that this wave will be of the opposite type as the incident wave (i.e. acompression wavefront will be reflected as an expansion wavefront and anexpansion wavefront will be reflected as a compression wavefront. This is reasonable physically because the backward wave approaching thebifurcation in one of the daughter vessels (now the parent vessel) will see abifurcation consisting of its twin vessel and the parent vessel with a net area muchlarger than its own. The bifurcation will therefore act more like an open-end tubeand generate a negative reflection coefficient.