H. SAIBI
November 11th 2015
Outline of Lecture:
 Reflection, Transmission, and
Refraction
 Diffraction
 The Doppler Effect
 The Doppler shift and relativity
 Shock Waves
Waves encountering barriers
Reflection, Transmission, and Refraction
Reflection
Coefficient (r)=hr/hin
Transmission
Coefficient ()=ht/hin
Heavier string
Fresnel relations
(valid for light
and sound
waves)
Lighter string
…(1) Reflection and Transmission coefficients
Fig.1. (a) a wave pulse traveling on a string attached to a more massive string in which the
wave speed is half as large. The reflected pulse is inverted, whereas the transmitted pulse is
not. (b) A wave pulse traveling on a string attached to a less massive string in which the wave
speed is twice as large. In this case, the reflected pulse is not inverted.
©2008 by W.H. Freeman and Company
Exercise: Two Soldered Wires
 Two wires, different linear mass
densities. Stretched under a tension FT
(the tension is the same in both wires).
 The wave speed in the first wire is twice
that in the second.
 A harmonic wave traveling in the first
wire is incident on the junction of the
wires.
 (A) if the amplitude of the incident wave
is A, what are the amplitudes of the
reflected and transmitted waves?
 (B) what is the ratio 2/1 of the mass
densities of the wires?
 (C) what fraction of the incident average
power is reflected at the junction and
what fraction is transmitted?
©2008 by W.H. Freeman and Company
Waves encountering barriers
Reflection, Transmission, and Refraction
 Energy conservation gives another relation
between the reflection and transmission
coefficients. This relation is given by:
…(2)
Fraction of the
incident power
that is reflected
Fraction of the
transmitted
 In three dimensions, a boundary between two
regions of differing wave speed is a surface.
 The transmitted ray is bent toward or away from
the normal-depending on whether the wave speed
in the second medium is less or greater than that in
the incident medium.
Fig.2. A wave striking a boundary surface
between two media in which the wave speed
differs. Part of the wave is reflected and part is
transmitted. The change in direction of the
transmitted (refracted) ray is called refraction.
©2008 by W.H. Freeman and Company
Waves encountering barriers
Reflection, Transmission, and Refraction
 Total Internal Reflection:
When the incident angle is
greater than the critical
angle, there is no refracted
ray.
 The amount of energy
reflected from a surface
depends on the surface.
Fig.3. Light from a source in the water is bent
away from the normal when it enters the air.
For angles of incidence above a critical angle
c’ there is no transmitted ray, a condition
known as total internal reflection.
©2008 by W.H. Freeman and Company
Figure 4. In a concert hall, a reflecting shell is placed behind the orchestra, and reflecting
panels are hung from the ceiling to reflect and direct the sound back toward the listeners.
©2008 by W.H. Freeman and Company
Check your understanding:
Balloon Hearing Aid
 If you place the balloon
(filled with carbon
dioxide) between
yourself and a sound
source, the sound gets
louder. Why is that?
 The balloon is to sound
as a magnifying glass is
to light.
©2008 by W.H. Freeman and Company
Diffraction
 If a wavefront is partially blocked by an obstacle,
the unblocked part of the wavefront bends
behind the obstacle. This bending of the
wavefronts is called diffraction.
 Almost all of the diffraction occurs for that part
of the wavefront that passes within a few
wavelengths of the edge of the obstacle. For the
parts of the wavefront that pass farther than a few
wavelengths from the edge, diffraction is
negligible and the wave propagates in straight
lines in the direction of the incident rays.
 When wavefronts encounter a barrier with an
aperture (hole) only a few wavelengths across, the
part of the wavefronts passing through the
aperture all pass within a few wavelengths of an
edge. Thus, flat wavefronts bend and spread out
and become spherical or circular (Fig.5).
Fig.5. Plane waves in a ripple tank
meeting a barrier with an opening that
is approximately one wavelength wide.
Beyond the barrier are circular waves
that are concentric about the opening,
much as if there were a point source at
the opening.
Diffraction
 In contrast, for a beam of particles
falling upon a barrier with an
aperture, the part of the beam
passing through the aperture does
so with no change in the direction of
the particles (Fig.6).
 Diffraction is one of the key
characteristics that distinguish
waves from particles.
©2008 by W.H. Freeman and Company
Fig.6. Comparison of particles and waves
passing through a narrow opening in a barrier.
(a) Transmitted particles are confined to a
narrow-angle beam. (b) Transmitted waves
spread out (radiate widely) from the aperture,
which acts like a point source of circular waves.
Diffraction





Although waves passing through an aperture always
bend, or diffract, to some extent, the amount of
diffraction depends on whether the wavelength is
small or large relative to the width of the aperture.
If the wavelength is greater than or equal to the width
of the aperture (like in Fig.5), the diffraction effects are
large, and the waves speed out as they pass through
the aperture-as if the waves were originating from a
point source.
On the other hand, if the wavelength is small relative
to the aperture, the effect of diffraction is small as
shown in Fig.7.
Near the edges of the aperture the wavefronts are
distorted and the waves appear to bend slightly. For
the most part, however, the wavefronts are not affected
and the waves propagate in straight lines, much like a
beam of particles.
The approximation that waves propagate in straight
lines in the direction of the rays with no diffraction is
known as the ray approximation. Wavefronts are
distorted near the edges of any obstacle blocking part
of the wavefronts. By near we mean within a few
wavelengths of the edges.
Fig.7. Plane waves in a ripple tank meeting a
barrier with an opening width that is large
compared to . The wave continues in the
forward direction, with only a small amount of
spreading into the regions to either side of the
opening.
©2008 by W.H. Freeman and Company
The Doppler effect
 The shift in frequency caused by motion is called the
Doppler effect.
 It occurs when a sound source is moving at speeds less than
the speed of sound.
The Doppler Effect

Consider the source moving with speed us (Fig.8a-b) and a stationary receiver. The source has frequency
fs (and period Ts=1/ fs). The received frequency fr, the number of wave crests passing the receiver per
unit time, is related to wave length  and wave speed v by:
… (3) Stationary receiver

A wave crest leaves the source at time t1 and the next wave crest leaves the source at time t2 (Fig.8c). The
time between these two events is
, and during this time the source and the crest leaving the
source at time t1 travel distances usTs and vTs, respectively. Consequently, at time t2, the distance
between the source and the crest leaving at time t1 equals the wavelength .

Behind the source:

In front of the source:
us<v
… (4)
(-) minus sign is used if =f and
(+) plus sign is used if =b
… (5) Stationary receiver
The Doppler Effect
Fig.8. (a) Waves in a ripple tank produced by a point source moving to the right. The wavefronts are
closer together in front of the source and farther apart behind the source. (b) Successive wavefronts
emitted by a point source moving with speed us to the right. The numbers of the wavferonts
correspond to the positions of the source when the wave was emitted. (c) The source vibrates one cycle
in time Ts. During time ts the source moves a distance usTs and the fifth wavefront travels a distance vTs.
In front of the source the wavelength f=(v-us)Ts, while behind the source b=(v+us)Ts.
©2008 by W.H. Freeman and Company
The Doppler Effect



When the receiver moves relative to the medium, the received
frequency is different simply because the receiver moves past
more or fewer wave crests in a given time.
Let Tr denote the time between arrivals of successive crests for a
receiver moving with a speed ur. Then during the time between
the arrivals of two successive crests, each crest will have traveled a
distance vTr, and during the same time the receiver will have
traveled a distance urTr.
If the receiver moves in the direction opposite to that of the wave
(Fig.9), then during time Tr, the distance a crest moves plus the
distance the receiver moves equals the wavelength. That is,
or
. (if the receiver moves
in the same direction as the wave, then
, so
. Because
, we have:
… (6)

Where, if the receiver moves in the same direction as the wave, the
received frequency is lower, so we choose the negative sign. If the
receiver moves in the direction opposite to that of the wave, the
frequency is higher, so we choose the positive sign. Substituting
for  from Eq.4, we obtain:
Fig.9. The time between arrivals of wave crests
at the receiver is Tr. The wave crests are
represented by orange lines when a wave crest
reaches the receiver, and they are represented
by grey lines when the next crest reaches the
receiver. During time Tr the receiver travels the
distance urTr, while the wave crest travels the
distance vTr.
… (7)
©2008 by W.H. Freeman and Company
The Doppler Effect



The correct choices for the plus or minus signs are most easily determined by remembering that the
frequency tends to increase both when the source moves toward the receiver and when the receiver moves
toward the source.
For example, if the receiver is moving toward the source, the plus sign is selected in the numerator, which
tends to increase the received frequency; if the source is moving away from the receiver, the plus sign is
selected in the denominator so that the equation predicts a decrease in the received frequency.
Equation 7 appears more symmetric, and thus is easier to remember, if expressed in the form:
… (8)

It can be shown that if both us and ur are much smaller than the wave speed v, then the shift in frequency
is given approximately by:
… (9)
where
is the speed of the source relative to the receiver.
 In a reference frame in which the medium is moving (for example, the reference frame of the ground if air is
the medium and if there is a wind blowing), the wave speed v is replaced by
where uw is the
speed of the wind relative to the ground.
 Equations 3 to 9 are valid only in the reference frame of the propagating medium.
The speed of sound
 The speed of sound in air is 343 meters per second
(660 miles per hour) at one atmosphere of pressure
and room temperature (21°C).
 An object is subsonic when it is moving slower than
sound.
The speed of sound
 We use the term supersonic to describe motion at speeds
faster than the speed of sound.
 A shock wave forms where the wave fronts pile up.
 The pressure change across the shock wave is what causes
a very loud sound known as a sonic boom.
Doppler Shift and Relativity




We see from Equations (5-6-7-8) that the magnitude of the Doppler shift in frequency depends on whether it is
the source or the receiver that is moving relative to the medium. For sound, these two situations are physically
different. For example, if you move relative to still air, you feel rushing past you. In your reference frame, there is
no wind. For sound waves in air, therefore, we can tell whether the source or receiver is moving by noting if there
is a wind in the reference frame of the source or the receiver. However, light and other electromagnetic waves
propagate through empty space in which there is no propagating medium. There is no “wind” to tell us whether
the source or receiver is moving.
According to Einstein’s theory of relativity, absolute motion cannot be detected, and all observers measure the
same speed c for light, independent of their motion relative to the source. Thus, Equations 7-8 cannot be correct
for the Doppler shift for light.
Two modifications must be made in calculating the relativistic Doppler effect for light. First, the speed of waves
passing a receiver is c, which is independent of the motion of the receiver. Second, the time interval between the
emission of successive wave crests, which is Ts=1/fs in the reference frame of the source, is different in the
reference frame of the receiver when the two reference frames are in relative motion, because of relativistic time
dilation and length contraction.
The result is that the frequency received depends only on the relative speed of approach (or recession) u, and is
related to the frequency emitted by:
… (10)

Choose the signs that give an up-shift in frequency when the source and receiver are approaching, and vice versa.
Again, when u<<c, f/fs ≈ ± u/c as given by Equation 9.
Shock Waves
 During our derivations of the Doppler-shift expressions, we assumed that the speed u of
the source was less than the wave speed v. if a source moves with speed greater than the
wave speed, then there will be no waves in front of the source.
 Instead, the wave pile up behind the source to form a shock wave. In the case of sound
waves, this shock wave is ahead as a sonic boom when it arrives at the receiver.
 Figure 12 shows a source originally at point P1 moving to the right with speed u. After
some time t, the wave emitted from point P1 has traveled a distance vt. The source has
traveled a distance ut and will be at point P2. The line from this new position of the
source to the wavefront emitted when the source was at P1 makes an angle , called the
Mach angle, with the path of the source, given by:
… (11)
 Thus, the shock wave is confined to a cone that narrows as u increases. The ratio of the
source speed u to the wave speed v is called the Mach number:
… (12)
Shock Waves
Fig.12. (a) Source moving with a speed u that is greater than the wave speed v. The envelope of the wavefronts forms a
cone with the source at the apex.
(b) Waves in a ripple tank produced by a source moving with a speed u>v.
©2008 by W.H. Freeman and Company
Fig.10. Shock waves from a supersonic airplane.
©2008 by W.H. Freeman and Company
Fig.11. Shock waves produced by a bullet traversing a helium balloon.
©2008 by W.H. Freeman and Company
Exercise: A Sonic Boom
A supersonic plane flying due
east at an altitude of 15 km
passes directly over point P.
The sonic boom is heard at
point P when the plane is 22
km east of point P.
What is the speed of the
supersonic plane?
Fig. In the time that the plane moves
a distance ut, the sound moves
distance vt.
©2008 by W.H. Freeman and Company
Doppler Effect
Doppler Effect – 4 cases
 Source moving toward receiver
 Source moving away from receiver
 Receiver (observer) moving towards source
 Receiver (observer) moving away from source.
Source moving case
    vs T
Away:

 1
f  f 
 1  vs
v







c
T
Towards:

 vs 
    v s    1  
c
c 


 1
c
c
f  
f

 v 
 1  vs
 1  s 
c

c 






Receiver (observer) moving case
Towards:
f 
c  vr
 vr 
 f 1  

c 

Away:
f 
c  vr
 v 
 f 1  r 

c 

Source and receiver moving
vr

 1 c
f  f 
v
 1 s
c



 c  vr 
f

 c vS 


 Numerator – Receiver (observer)
 Toward +
 Away –
 Denominator – Source
 Toward –
 Away +