Ch 16
Wave Motion - I
• Wave Motion Gun http://www.youtube.com/watch?v=ty1zWsXFNs
• Wave Motion Gun music
http://www.youtube.com/watch?v=NNMkyNqA5J4
Hum….
– What mathematical functions are their own
derivatives?
Outline
• The wave equation for waves on a string
– Les Paul starburst, ukelele, fiddle, warshtub bass,
sitar
• Basic shapes of waves
• Adding waves
– AM radio waves
– Standing Waves
• Fourier decomposition
• Review Questions
The wave equation
Dm
Dm ax = T cosq2 – T cosq1
Dm ay = T sinq2 – T sinq1
Dm ax = T 1 – T 1
Dm ay = T q2 – T q1
Dm ax = 0
Dm ay = T ( q2 – q1)
The wave equation
Dm ay
q
dy
=
( q2 – q1)
T
2
Dm
d y
dt

2
T
 tan q  q
dx
2
Dm
d y
dt
Dm
Dx

2
dy 
 dy
|2 
|1 

dx 
 dx
T
 dy
D
 dx
 Dx


2
d y
dt
2

T
 d2y

 dx 2



Dx 







The wave equation
2

d y
d y
 T 
2
2
dt
 dx
Dm
2
Dx
Dm
 d2y
 T 
2
dx

2
d y

2
dt
2
linear density ~
Dm
Dx
d y

dt
1
T

2
2

T d y
 dx
2
d y
dt
2
2
2

d y
dx
2








The wave equation
2
d y
dx
2

T
2
d y
dx
2

dx
2
d y

1
dt
2
2
d y
2
[ ] dt
2
d y
2
1

1
v
2
2
d y
dt
2
The wave equation
2
d y
dx
2

1
v
2
2
d y
dt
2
v 
T
v
T
2


Basic Solutions to the Wave
Equation
y ~ sin[
2
d y
dx
2

1
v
2
2
]
y ~ sin[ x
t]
y ~ sin[ x
vt ]
d y
dt
2
y ~ sin[ k  x  vt ]
y ~ A sin[ k  x  vt ]
Basic Solutions to the Wave
Equation
2
d y
dx
2

1
v
2
2
d y
dt
2
y ~ A sin[ k  x  vt ]
  kv
y  A sin( kx   t )
y  A sin( kx   t )
• What’s the fartherest a point can get away from y=0 ?
• Where is sin() maximum?
• How fast does this maximum location travel?
• Which direction (left/right) does the wave travel?
• Over what difference in phase does a sin() repeat?
• How long does it take a particular point to go down & up?
• Over what distance does a sin() repeat?
Superposition of Waves
y tot  y1  y 2 
A sin   A sin 

2 A cos(
 
2
) sin(
 
2
)
Two waves
same direction, same f, same l,
but init phase difference f
  kx   t
A sin(   f )
A sin(  )
2 A cos(
f
2
2 A cos(
f
2
)
)
sin(  
f
2
sin( kx   t 
)
f
2
)
Two waves
same direction, same f, same l,
but init phase difference f
2 A cos(
f
)
sin( kx   t 
f
2
new amplitude
2
)
traveling part
If f = 0, then max new amplitude = 2A
same f same l
If f = p, then new amplitude = 0  complete cancellation
Two waves
opposite direction, same f, same l,
no phase difference f0
A sin( kx   t )
2 A cos(  t )
A sin( kx   t )
sin( kx )
no traveling part  STANDING WAVE
Two waves
opposite direction, same f, same l,
no phase difference f0
2 A cos(  t )
sin( kx )
no traveling part  STANDING WAVE
nodes
l=L
Standing
Waves
l = 2L/3
l = 2L/4
Standing Waves
fundamental
f n ln  v
;
v fixed by T 
ln  2 L
f n  n f1
n
Fourier Decomposition, or,
What about waves that aren’t purely sine
waves?
Flute
oboe
Fourier Decomposition, or,
What about waves that aren’t purely sine
waves?
traveling waves
y
A
n
sin( k n x   n t )
n
standing waves
y
A
n
n
cos( k n x ) sin(  n t )
Review
16.4.2. The graph shows the vertical displacement as a function of
time at one location in a medium through which a wave is
traveling. What is the amplitude of the wave?
a) 1 m
b) 2 m
c) 4 m
d) 6 m
e) 8 m
16.4.3. The graph shows the vertical displacement as a function of
time at one location in a medium through which a wave is
traveling. What is the period of the wave?
a) 0.5 s
b) 1.0 s
c) 1.5 s
d) 2.0 s
e) 4.0 s
16.5.1. Which one of the following factors is important in determining
the speed of waves on a string?
a) amplitude
b) frequency
c) length of the string
d) mass per unit length
e) speed of the particles that compose the string
16.5.2. Consider the three waves described by the equations below. Which wave(s)
is moving in the negative x direction?
a) A only
b) B only
c) C only
d) A and B
e) B and C
16.5.2. The equation for a certain wave is
y = 4.0 sin [2p(2.5t + 0.14x)]
where y and x are measured in meters and t is measured in seconds.
What is the magnitude and direction of the velocity of this wave?
a) 1.8 m/s in the +x direction
b) 1.8 m/s in the x direction
c) 18 m/s in the x direction
d) 7.2 m/s in the +x direction
e) 0.35 m/s in the x direction
16.5.3. Which one of the following correctly describes a wave described by y = 2.0
sin(3.0x  2.0t) where y and x are measured in meters and t is measured in
seconds?
a) The wave is traveling in the +x direction with a frequency 6p Hz and a
wavelength 3p m.
b) The wave is traveling in the x direction with a frequency 4p Hz and a
wavelength p/3 m.
c) The wave is traveling in the +x direction with a frequency p Hz and a wavelength
3p m.
d) The wave is traveling in the x direction with a frequency 4p Hz and a
wavelength p m.
e) The wave is traveling in the +x direction with a frequency 6p Hz and a
wavelength p/3 m.
16.8.1. A wave is described by the equation y = 0.020 sin (3.0x  6.0t),
where the distances are in meters and time is measured in seconds.
Using the wave equation, determine the speed of this wave?
a) 0.50 m/s
b) 0.75 m/s
c) 1.0 m/s
d) 2.0 m/s
e) 4.0 m/s
16.12.1. What is the frequency of a standing wave with a wave speed
of 12 m/s as it travels on a 4.0-m string fixed at both ends?
a) 2.5 Hz
b) 5.0 Hz
c) 10.0 Hz
d) 15.0 Hz
e) 20.0 Hz
16.13.1. Which one of the following statements explains why a piano
and a guitar playing the same musical note sound different?
a) The fundamental frequency is different for each instrument.
b) The two instruments have the same fundamental frequency, but
different harmonic frequencies.
c) The two instruments have the same harmonic frequencies, but
different fundamental frequencies.
d) The two instruments have the same fundamental frequency and the
same harmonic frequencies, but the amounts of each of the
harmonics is different for the two instruments..
16.13.2. When a wire under tension oscillates in its third harmonic
mode, how many wavelengths are observed?
a) 1/3
b) 1/2
c) 2/3
d) 3/2
e) 2