PHY 113 A General Physics I
9-9:50 AM MWF Olin 101
Plan for Lecture 35:
Chapter 17 & 18 – Physics of wave motion
1. Standing waves
2. Sound waves
3. Doppler effect
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Combinations of waves (“superposition”)
Note that :
sin A  sin B  2 sin  1 2  A  B cos 12  A  B 
 x

yright ( x, t )  y0 sin  2 π  ft  

 λ
 x

yleft ( x, t )  y0 sin  2 π  ft  

 λ
“Standing” wave:
 2 πx 
yright ( x, t )  y left ( x, t )  2 y0 sin 
 cos2 πft 
 λ 
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sin A  sin B  2 sin1 2  A  B cos1 2  A  B 
“Standing” wave:
 2 πx 
yright ( x, t )  y left ( x, t )  2 y0 sin 
 cos2 πft 
 λ 
iclicker exercise:
Why is this superposition result important?
A. Physics instructors like to torture their students.
B. While the trigonometric relation is always true, it
is rarely useful for describing real situations.
C. It can describe the motion of a guitar string.
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2 y 2 2 y
Standing wave solutions to the wave equation :
c
0
2
2
t
x
 2 πx 
ystanding ( x, t )  A sin 
 cos2 πft 
 λ 
 2 ystanding
 2 πx 
2
Check :
 2 πf  A sin 
 cos2 πft 
2
t
 λ 
 2 ystanding
2
 2π 
 2 πx 
Check :
   A sin 
 cos2 πft 
2
x
 λ 
 λ 
2
2
2


 y 2 y
2
π
 2 πx 


2
2
c
  A sin 
 cos2 πft  2 πf   c     0
2
2
t
x
 λ 
 λ  

Equality satisfied iff fλ  c
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 2πnx 
Possible spatial shapes : A sin 

 2L 
n  1,2,3,4.....
 2πx 
Standing wave form : A sin 
cos2πft 
  
2L
nc
 n 
fn 
n  1,2,3,4...........
n
2L
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iclicker question:
Which of the following statements are true
about the string motions described above?
A. The human ear can directly hear the
string vibrations.
B. The human ear could only hear the
string vibration if it occurs in vacuum.
C. The human ear can only hear the string
vibration if it produces a sound wave in
air.
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Comments about waves in materials:
 Solid materials can support both transverse
and longitudinal wave motion
 Fluids, especially gases can support only
longitudinal wave motion
The linearized fluid equations in the ideal gas approximation,
show that the density fluctuations  of air can be written :
2
 2


2
c
0
2
2
t
x
p0
c 
 343m / s
0
2
Alternatively, the sound wave can be expressed in terms of
pressure fluctuations :
 2P 2  2P
c
0
2
2
t
x
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Standing waves in air:
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Coupling standing wave resonances in materials
with sound:
iclicker question:
Suppose an “A” is played on a guitar string.
The standing wave on the string has a frequency
(fg), wavelength (g) and speed (cg) . Which
properties of the resultant sound wave are the
same as wave on the guitar string?
A. Frequency fs
B. Wavelength s
C. Speed cs
D. A-C
E. None of these
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coupling to
sound
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The “Doppler” effect
v=sound velocity
observer moving, source stationary
vt1  d  vO t1
vt 2  T   d  vO t 2
observer
source
1
v
t 2  t1 
T
fO
v  vO
v  vO
fO  f S
v
vS=0
d
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The “Doppler” effect
v=sound velocity
observer stationary, source moving
vt1  d
vt 2  T   vS T  d
v  vS
1
t 2  t1 
T
fO
v
fO  f S
d
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Summary :
v  vO
fO  f S
v  vS
PHY 113 A Fall 2012 -- Lecture 35
v
v  vS
toward
away
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Example:
v  vO
fO  f S
v  vS
Velocity of sound:
v = 343 m/s
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toward
away
In this case :
fO  f S
v  vO
v  vS
f O  f S  vO  vS
fO
vO  v  (v  vS )  40m/s
fS
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iclicker question
Is Doppler radar described by the equations given above for
sound Doppler?
(A) yes
(B) no
Is “ultra sound” subject to the sound form of the Doppler
effect?
(A) yes
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(B) no
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Summary of sound Doppler effect :
v  vO
fO  f S
v  vS
toward
away
Doppler effect for electromagnetic waves :
fO  f S
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v  vR
v  vR
Relative velocity of source
toward observer
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