Chapter 16 – Waves
A traveling wave is a disturbance that propagates in space and time. It can be a single
pulse or a periodic disturbance.
For a transverse wave the disturbance is perpendicular to the direction of motion. For a
longitudinal wave, the disturbance is along the direction of motion.
As an example, a longitudinal wave could be established in a stretched slinky by pushing
and pulling one end in an oscillatory manor. The regions of compression and expansion
would propagate along the length of the slinky.
v
A transverse wave is shown below. Such a wave could be established on a rope by
periodically moving one end up and down.
v
The wave speed is given by
v
F

where F is the tension in the rope in N and  is the mass per unit length of the rope in
kg/m.
Example:
A string clamped at one is held in tension by a 1.5-kg
weight attached to the other, as shown to the right.
The mass of the string is 50 g and the length is 2 m.
What is the speed of a wave on this string?
1
F  Mg  (1.5kg)(9.8m / s 2 )  14.7 N
  m / L  0.05kg / 2m  0.025 kg / m
v
F


14.7 N
 24.2 m / s
0.025kg / m
Wave equation
All traveling waves have the form y ( x, t )  f ( x  vt). If the wave is traveling to the right
(increasing x), then y ( x, t )  f ( x  vt). If it is traveling to the left, then
y ( x, t )  f ( x  vt).
A “sinusoidal” wave traveling to the right can be described by the equation
y  A sin( kx   t   )
where
A = amplitude = maximum vertical displacement (m)
k = 2/ = wave number (m-1)
 = wavelength (m)
 = 2f = 2/T = angular frequency (rad/s)
f = frequency (Hz)
T = period (sec)
 = phase angle (depends on choice of t = 0)
The wave speed can be written as
v
x 
   2  
  f  


t T
 2  k  k
Note: The sinusoidal wave traveling to the right can also be written as
y  A sin[ k ( x 

k
t )   ]  A sin[ k ( x  v t )   ] ,
which has the form y ( x, t )  f ( x  vt).
2
y
A

x
The transverse velocity of the wave can be obtained by differentiating the transverse
displacement.
y  A sin( kx   t   )
dy
vy 
  A cos( kx   t   )
dt
The transverse velocity refers to the up and down motion of the wave as a function of x
and t. This is not the same as the wave speed, which is the speed at which a crest of the
wave moves to the right or left.
Likewise, the transverse acceleration of the wave is given by
ay 
dv y
dt
  2 A sin( kx   t   )
Example:
A wave is described by the equation y  0.05m sin( 2 x  5 t ) . What are the wavelength,
period, and wave speed?
k = 2/ = 2,  =  = 3.14 m
 = 2/T = 5, T = 0.4 = 1.26 s
v = /T = 3.14m/1.26s = 2.5 m/s
What is the transverse velocity at x = 1 m and t = 2 s?
v y   A cos( kx   t   )  (5)(0.05) cos( 2(1)  5(2))
 0.25 cos( 8rad )  0.036 m / s
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Interference and Reflection of Waves
Waves whose amplitudes are not too great obey the principle of superposition. This
means that when two waves meet, the resultant wave is obtained by adding the two waves
point by point. Waves do not collide and bounce off each other as is the case for
particles. They travel through each other and emerge as though they did not interact.
Waves can reflect, however, from the end of a rope. If the end of the rope is tightly
secured, then the wave will invert when reflected.
If the end of the rope is untied and loose, then the wave will reflect back without
inversion. If two ropes with different mass densities are tied together, then a wave will
be partly reflected from the point where they are joined together.
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Standing waves on a string
Waves can interfere either constructively or destructively, depending on their relative
phase when they combine. If waves are generated on a string that is clamped at both
ends, then the waves will be reflected at the clamped ends. The resulting wave will
consist of a sum of waves going left and right. If the waves are sinusoidal and the length
of the string is an integral number of half-wavelengths (L = n /2), then ‘standing waves’
will be formed. These consist of regions where the right and left-going waves interfere
constructively (antinodes) and interfere destructively (nodes).
antinode
node
/2
Allowed standing wave frequencies:

L  n , n  1, 2, 3, ...
2
2L

n
v
v
v
f  
n
 2L / n
2L
F
v
.

fn 
n
2L
F

, n  1, 2, 3, ...
n  1 : f1 
1
2L
n  2 : f 2  2 f1
n  3 : f 3  3 f1
F

1st harmonic (or fundamental )
2nd harmonic (or 1st overtone)
3rd harmonic (or 2nd overtone)
etc.
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Example:
A string 0.8 m long that is clamped at both ends has a tension of 100 N and a linear mass
density of 0.02 kg/m. What are the three lowest allowed standing wave frequencies?
fn 
n
2L
F


n
100
 n 44.2 Hz
2(0.8) 0.02
f1  44.2 Hz , f 2  2 f1  88.4 Hz , f 3  3 f1  132.6 Hz
Power transmitted by a wave
A traveling wave carries energy. If the wave is on a stretched rope, then any piece of the
rope moves up and down like a mass on the end of a spring. It has kinetic energy of
motion and elastic energy associated with its vertical displacement. The total energy is
the sum of its kinetic energy and potential energy. Since the potential energy is zero
when the kinetic energy is a maximum (and vice versa), the total energy of a piece of the
rope it
E  K max  1 mv 2y, max  1 (  x)( A) 2
2
2
where  is the linear mass density of the rope and A is the amplitude of the wave. The
power transmitted is
P
E 1 x 2 2
 
 A
t 2 t
The wave speed is v = x/t. Thus, the power transmitted (watts) is
P  1  v  2 A2
2
Example:
A long rope has a linear density of 0.1 kg/m and subjected to a tension of 40 N. One end
of the rope is moved up and down at a rate of 5 Hz with amplitude of 0.1 m. How much
power is transmitted down the rope?
40 N
 20 m / s
0.1kg / m
v
P  1  v  2 A  1 (0.1kg / m)( 20m / s)( 2 (5Hz )) 2 (0.1m) 2  9.9W
2
2
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