PHY2048, Lecture 23 – Wave Motion
Chapter 14 – Tuesday November 16th
Stephen Hill (covering for Prof. Wiedenhover)
•Review of wave solutions
•The wave equation
•Wave superposition and interference
•Spatial interference and standing waves
•Temporal beating
•Sources of musical sound
•Doppler effect (if time)
•Fifth and final mini exam on Thu. (Chs. 13-15)
•Make up labs Nov. 29 at 12:30pm and 3:30pm in UPL107
•Final exam – Wed. Dec. 8th, 10am to noon, in MOR 104
(Moore Auditorium)
Review - wavelength and frequency
Amplitude
Displacement
Phase
Transverse wave
y ( x, t )  A cos(kx   t   )
angular wavenumber
angular frequency
k
2

2

T
Phase
shift
k is the angular wavenumber.
 is the angular frequency.
frequency
velocity
v

k
1 
f  
T 2


T

f
Traveling waves on a stretched string
-ve curve
a FT
Fnet
FT
a
a
+ve curve
Zero curve
+ve curve
m is the string's linear density, or mass per unit length.
•Tension FT provides the restoring force (kg.m.s-2) in the string.
Without tension, the wave could not propagate.
•The mass per unit length m (kg.m-1) determines the response of the
string to the restoring force (tension), through Newtorn's 2nd law.
•Look for combinations of FT and m that give dimensions of speed
(m.s-1).
v
FT
m
Traveling waves on a stretched string
-ve curve
a FT
Fnet
l
FT
a
a
+ve curve
Zero curve
+ve curve
m is the string's linear density, or mass per unit length.
The Wave Equation
transverse
acceleration
mass
 y

 y
Fnet  FT  2  l    m  l  2  ma y

x

t


Dimensionless
2
2
parameter
 y
 y
proportional to
FT 2  m 2
x
t
curvature
2
2
The wave equation
FT  y  y
 2
2
m x
t
2
2
•General solution:
y( x, t )  ym sin  kx  t 
2 y
2

k
y  x, t 
2
x
-
FT
m
k  -
2
2
or
y( x, t )  ym f  kx  t 
v
v
2 y
2


y  x, t 
2
t
or

2
k
2
v 
2
FT
m
 v
FT
m
The principle of superposition for waves
•It often happens that waves travel simultaneously through the
same region, e.g.
Radio waves from many broadcasters
Sound waves from many musical instruments
Different colored light from many locations from your TV
•Nature is such that all of these waves can exist without altering
each others' motion
•Their effects simply add
•This is a result of the principle of superposition, which applies to all
harmonic waves, i.e., waves that obey the linear wave equation
2
2

y

y
2
v
 2
2
x
t
•And have solutions:
y( x, t )  ym f  kx  t  or
ym sin  kx  t 
The principle of superposition for waves
•If two waves travel simultaneously along the same stretched string,
the resultant displacement y' of the string is simply given by the
summation
y '  x, t   y1  x, t   y2  x, t 
where y1 and y2 would have been the displacements had the waves
traveled alone.
•This is the principle of superposition.
Overlapping waves algebraically add to produce a resultant
wave (or net wave).
Overlapping waves do not in any way alter the travel of
each other
Interference of waves
•Suppose two sinusoidal waves with the same frequency and
amplitude travel in the same direction along a string, such that
y1  ym sin  kx -  t 
y2  ym sin  kx -  t   
•The waves will add.
Interference of waves
Noise canceling
headphones
Interference of waves
•Suppose two sinusoidal waves with the same frequency and
amplitude travel in the same direction along a string, such that
y1  ym sin  kx -  t 
y2  ym sin  kx -  t   
•The waves will add.
•If they are in phase (i.e.  = 0), they combine to double the
displacement of either wave acting alone.
•If they are out of phase (i.e.  = ), they combine to cancel
everywhere, since sin(a) = -sin(a  ).
•This phenomenon is called interference.
Interference of waves
•Mathematical proof:
y1  ym sin  kx -  t 
y2  ym sin  kx -  t   
Then:
y '  x, t   y1  x, t   y2  x, t 
 ym sin  kx -  t   ym sin  kx -  t   
But:
So:
sin a  sin   2sin 12 a    cos 12 a -  
y '  x, t    2 ym cos 12  sin  kx - t  12  
Amplitude
Wave part
Phase
shift
Interference of waves
y '  x, t    2 ym cos 12  sin  kx - t  12  
If two sinusoidal waves of the same amplitude and
frequency travel in the same direction along a stretched
string, they interfere to produce a resultant sinusoidal
wave traveling in the same direction.
•If  = 0, the waves interfere constructively, cos½ = 1 and the wave
amplitude is 2ym.
•If  = , the waves interfere destructively, cos(/2) = 0 and the wave
amplitude is 0, i.e. no wave at all.
•All other cases are intermediate between an amplitude of 0 and 2ym.
•Note that the phase of the resultant wave also depends on the
phase difference.
Adding waves as vectors (phasors) described by amplitude and phase
Interference - Standing Waves
If two sinusoidal waves of the same amplitude and
wavelength travel in opposite directions along a stretched
string, their interference with each other produces a
standing wave.
y '  x, t   y1  x, t   y2  x, t 
 ym sin  kx -  t   ym sin  kx   t   
 2 ym sin  kx  12   cos  t - 12  
x dependence t dependence
•This is clearly not a traveling wave, because it does not have the
form f(kx - t).
•In fact, it is a stationary wave, with a sinusoidal varying amplitude
2ymcos(t).
Link
Reflections at a boundary
•Waves reflect from boundaries.
•This is the reason for echoes - you
hear sound reflecting back to you.
•However, the nature of the reflection
depends on the boundary condition.
•For the two examples on the left, the
nature of the reflection depends on
whether the end of the string is fixed
or loose.
Standing waves
and resonance
•At ordinary frequencies,
waves travel backwards and
forwards along the string.
•Each new reflected wave has
a new phase.
•The interference is basically
a mess, and no significant
oscillations build up.
Standing waves and resonance
•However, at certain special
frequencies, the interference
produces strong standing wave
patterns.
•Such a standing wave is said to be
produced at resonance.
•These certain frequencies are called
resonant frequencies.
Standing waves and resonance
 determined by geometry
•Standing waves occur whenever the
phase of the wave returning to the
oscillating end of the string is
precisely in phase with the forced
oscillations.
•Thus, the trip along the string and
back should be equal to an integral
number of wavelengths, i.e.
2L
2 L  n or  
n
for n  1,2,3...
v
v
f   n , for n  1,2,3...

2L
•Each of the frequencies f1, f1, f1, etc,
are called harmonics, or a harmonic
series; n is the harmonic number.
Standing waves and resonance
•Here is an example of a two-dimensional vibrating diaphragm.
•The dark powder shows the positions of the nodes in the vibration.
Standing waves in air columns
•Simplest case:
- 2 open ends
- Antinode at each end
- 1 node in the middle
1  2 L  2 L /1
•Although the wave is longitudinal,
we can represent it schematically by
the solid and dashed green curves.
Standing waves in air columns
A harmonic series
1  2 L  2 L /1
2
3
4
2L
  , for n  1,2,3,....
n
v
v
f  n ,

2L
for n  1,2,3,...
Standing waves in air columns
A different harmonic series
1
3
5
7
4L
  , for n  1,3,5,....
n
v
v
f  n ,

4L
for n  1,3,5,...
Musical instruments
Flute
Oboe
Saxophone
Link
Musical instruments
v
f 
n
4L
•n = even if B.C. same at
both ends of pipe/string
•n = odd if B.C. different
at the two ends
Wave interference - spatial
Interference - temporal (or beats)
s ( x, t )  sm cos(kx -  t )
•In order to obtain a spatial interference pattern, we placed two
sources at different locations, i.e. we varied the first term in the
phase of the waves.
•We can do the same in the time domain whereby, instead of placing
sources at different locations, we give them different angular
frequencies 1 and 2. For simplicity, we analyze the sound at x = 0.
s  s1  s2  sm  cos1t  cos2t 
cosa  cos   2cos 12 a -   cos 12 a   
s  2 sm cos 12 1 -  2  t cos 12 1   2  t
  2 sm cos  ' t  cos  t
 '  12 1 -  2 
  12 1   2 
Interference - temporal (or beats)
s   2sm cos ' t  cos t
•A maximum amplitude occurs whenever 't has the value 1 or -1.
•This happens twice in each time period of the cosine function.
•Therefore, the beat frequency is twice the frequency ', i.e.
beat  2 '  1 -  2
fbeat  2 f '  f1 - f 2
Link
Doppler effect
•Consider a source of sound at the ‘proper frequency’, f ', moving
relative to a stationary observer.
•The observer will hear the sound with an apparent frequency, f,
which is shifted from the proper frequency according to the
following equation:
 u
f  1   f '
 v
•Here, v is the sound velocity (~330 m/s in air), and u is the
relative speed between the source and detector.
When the motion of the detector or source is towards the
other, use the plus (+) sign so that the formula gives an
upward shift in frequency. When the motion of the
detector or source is away from the other, use the minus
(-) sign so that the formula gives a downward shift.
Mach cone angle:
v
sin  
vS
Energy in traveling waves
y( x, t )  ym sin  kx - t 
Kinetic energy:
vy 
dK 
1
2
dK = 1/2 dm vy2
y
 - ym cos  kx - t 
t
 m dx  - ym  cos2 (kx - t )
2
Divide both sides by dt, where dx/dt = vx
dm = mdx
Similar expression for
elastic potential energy
dK 1
 2 m v x 2 ym2 cos2 (kx - t )
dt
dU 1
 2 m v x 2 ym2 cos2 (kx - t )
dt
Pavg  2  12 m v 2 ym2 cos 2 (kx -  t )  2  12 m v 2 ym2  12  12 m v 2 ym2
Energy is pumped in an oscillatory fashion down the string
Note: I dropped the subscript on v since it represents the wave speed