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.