Lecture #1.2
Superposition and Interference of Mechanical Waves
1. Mechanical Waves (Review of the last lecture from the previous semester)
At the end of the last semester we have discussed mechanical waves. Here are the
main results of our discussion.
In contrast to simple harmonic motion, where position velocity and acceleration are
changing as periodic functions of time, but both position and energy of the oscillating
object remain in the same local region of space without traveling anywhere, waves give
us an example of how motion of energy (or information) can be conducted through the
space without actual motion of matter. We have limited out attention to mechanical
waves. This is the most familiar for everyday life type of waves, because we are actually
using them almost every day in the form of sound. We can see them as waves on the
water surface and even sometimes can feel them as seismic waves. All these waves are
governed by Newton's laws and they can only exist in material medium, such as gas,
liquid or solid.
A simple one-dimensional example of the mechanical wave is the wave sent along
the stretched taut string. This becomes possible because the string is under tension. So, if
the string is pulled up and down at one end, it begins to pull up and down the adjacent
section of the string and then the next section and so on. As a result the pulse will travel

along the string with some velocity v . If a person who pulls the string will continue to do
that on periodic basis the continuous wave will travel along this string. If the external
force causing this motion has the form of a simple harmonic motion, then this wave will
have the shape of sinusoidal function and will be called harmonic wave. If we consider
the motion of some element of this string we will see that it moves up and down in the
direction perpendicular to the direction of propagation of the wave. This is why we called
that type of wave transverse wave. Alternatively we can consider a wave produced by a
piston in a long air-filled pipe. In a same way as the person was pulling on the string,
he/she can move the piston rightward and leftward. This motion of the piston can be a
simple harmonic motion. At first it will cause the increase of air pressure right next to the
piston then this pressure increase will move to the next section of the pipe and so on.
When the piston moves back, the pressure will decrease, this decrease will also travel
along the pipe. As a result we have a sound wave in this pipe. However, this wave is
different from the one we considered before, because now different elements of air in the
pipe are moving in the same direction as the direction of wave’s travel. We shall call such
a wave to be longitudinal wave.
Both transverse wave and longitudinal wave are said to be traveling waves, because
they both travel from one point in space to another point in space. It is the wave that
moves but not the material. In both cases elements of the string or elements of the air in
the pipe do not travel anywhere. They just oscillate near the same positions.
If we consider a transverse wave in the string, this is an example of the plane wave,
which propagates in the direction perpendicular to the plane surface. We called this
direction to be x-direction. Since it is transverse wave, the elements of the string are
moving in the direction perpendicular to the x-direction. We called that direction to be ydirection. This means that positions of the elements of the string changes with time and it
will be different for different values of x. To describe this wave we need to know the
b g
function y  y x, t
for the displacement of the elements in the string. We only
considered sinusoidal waves, behaving according to
y  x, t   A sin  kx  t  .
(1.2.1)
In this equation A is called the wave amplitude, k is called the wave number and  is
wave angular frequency. The wave described by equation 1.2.1 is moving in positive x
direction. The phase of the wave is the argument of the sin-function kx  t . If we fix x
in this argument, we can watch how the position of the string changes at certain location
in space. According to equation 1.2.1 it oscillates in a simple harmonic motion. If we fix
time t, we can see the snapshot of this wave in space. It will look like a sin-function,
which changes along x direction. It is periodic function which will reproduce its shape
along axis x. The smallest distance between repetitions of the wave shape along axis x is
called a wavelength  . We have shown that the wave number k and the wavelength are
related according to
k
2

.
(1.2.2)
We have also shown that the time period of the simple harmonic wave is related to
its angular frequency according to

2
.
T
(1.2.3)
If the wave pattern moves in space for distance x during time t , so we can talk
about wave’s speed x t . This speed is also known as a phase speed, because the
wave's phase remains constant, as the wave moves in positive x direction. We have
shown that the wave’s speed is
v

k


T
 f .
(1.2.4)
This means that the wave moves for the distance of one wavelength during the time
interval equal to the wave's period.
We have also shown that the speed of the wave in the string depends on elastic
properties of the string (tension force F) and linear mass density of the string (  is mass
per unit of length) as
v
F

.
(1.2.5)
The other common example of mechanical wave is the sound wave. The sound
wave from the point like source in air is longitudinal spherical wave. When we discussed
kinematics, we introduced the idea of the point-like or particle-like object. The size of
this object is much smaller compared to the distance of its travel. Using the same idea, we
can introduce the point-like source of sound. This is different compared to situation of the
string, where transverse wave was going in one direction of x-axis only. In the case of the
string a wavefront, the surface where oscillations of matter have the same phase is a
planar surface. In the case of the sound wave propagating in all directions along so called
rays in the air, the wavefronts are spherical. Those wavefronts are perpendicular to the
rays, showing the direction of the wave’s propagation. In the case of the longitudinal
wave the oscillations of air are also taking place in the direction of those rays
perpendicular to wavefronts, while for transverse wave they take place in the direction of
the wavefront perpendicular to the rays. If, however, we consider a spherical wave at
very large distance from the source, the wavefront's curvature is almost negligible and we
can treat it as a planar wave.
In the case of the sound wave moving in 3-dimensional medium, the elastic
properties of the medium are related to periodic compressions and expansions of its small
volume elements. The so-called bulk modules
B
p
V V
(1.2.6)
is responsible for these changes. In this equation p is the change of pressure (force
acting per unit of area) and V V is the fractional change of volume. The equation for
speed of sound is then
v
B

,
(1.2.7)
where  is the density of the medium. The speed of sound, as we can see from equation
1.2.7 depends on a substance, since different substances have different densities as well
as on temperature, since density also depends on temperature. In the air under the normal
atmospheric pressure and temperature, the speed of sound is
v  343 m s  770 mi h .
Table 16.1 in the book shows values of speed of sound for some other substances at
different conditions.
If we consider a traveling sound wave in the tube filled by air, this wave appears
due to the motion of the piston at the end of the tube from the left to the right and
backward in the sinusoidal pattern. This motion causes the air elements at the end closer
to the piston change their density according to the same sinusoidal law as a result of
compression and then those changes of density will travel along the tube. Each element
of air with thickness x in the tube will oscillate according to the same sinusoidal law as
the piston does. In the longitudinal wave, those oscillations take place in the same
direction x as this wave propagates. To avoid confusions we used letter s to show this
displacement. If this wave has a form of the cos-function then
s  x, t   A cos  kx  t  ,
(1.2.8)
where A is the amplitude. All other characteristics of the wave, such as k ,  , f ,  , T are
defined in a same way as they were in the case of transverse wave.
Another important characteristic of sound is its intensity, which actually shows how
strong the sound is at a particular point in space. Intensity of sound I at certain surface is
the average power per unit area transferred by this wave trough or onto the surface
I
P
.
A
(1.2.9)
If we consider a point-like source of sound, which emits sound waves isotropically (with
equal intensity in all directions) and if the sound wave's energy is conserved, so it is not
dissipated in space, then we can write that
I
Ps
.
4r 2
(1.2.10)
Here Ps is the power of the source and r is the distance from the source. As you can see,
the sound intensity gets smaller with distance, not even because of the reflections and
scattering from different objects, but because the same energy spreads over larger and
larger areas of the spheres 4r 2 surrounding this point-like source.
We have also introduced the intensity level or sound level, which was defined as
b
g
  10dB log
I
.
I0
(1.2.11)
Here dB is abbreviation for decibel, the unit of sound level, I 0  1012 W m2 stands for
the standard reference intensity, chosen because it is near the low limit of the human
range of hearing. This standard reference level corresponds to zero decibels.
2. Superposition of Waves
Now we will consider a case, when there are several waves coming to the same
point in space from different sources. This could be the case of the sound waves coming
from different objects or the waves on the surface of water coming from different
disturbances.
Since we were discussing transverse waves in the string, let us continue with this
example and consider two waves traveling along the string simultaneously. In this case
the net displacement of the string's element will be an algebraic sum of the two
displacements due to each of the waves
b g b g b g
y  x, t  y1 x, t  y2 x, t .
(1.2.12)
This is known as the principle of superposition of mechanical waves. The overlapping
waves do not in any way alter each other.
The net wave depends on the extent to which the waves are in phase. If these two
waves are exactly in phase, the net wave is just a doubled wave. However, the case, when
there is a finite phase difference  between the waves, is different. We have
y1  x, t   A sin  kx  t 
and
y2  x, t   A sin  kx  t   
These two waves have the same frequency and the same wave number. They both travel
in the positive x direction and have the same amplitude. They are only different by the
phase constant  . In this case the net wave is going to be
y  x, t   y1  x, t   y2  x, t  
A sin  kx  t   A sin  kx  t    

  
2 A cos   sin  kx  t  
2
2 
So, the resultant wave is again a sinusoidal wave traveling in the same positive x
direction. However, this wave has different phase constant  2 and different amplitude
 
2 A cos   . In the case if the phase difference between the two original waves is zero the
2
amplitude of the resultant wave achieves its maximum possible value 2A, which is called
fully constructive interference. If the original phase difference was equal  then the
amplitude of the resultant wave is zero and it is a case of fully destructive interference,
like no wave goes along the string at all. In all other cases we have intermediate
interference.
Now let us consider the case, when the two waves are traveling along the string in
the opposite directions. Let these waves be of the same angular frequency and the same
wave number. In that situation there are places on the string called nodes, where the
string never moves. Half way between the adjacent nodes there are antinodes, where the
amplitude of the resultant wave has its maximum. The pattern like this is called a
standing wave, because this wave does not move in space, but just oscillate with time. Let
us analyze this standing wave combined from
y1  x, t   A sin  kx  t 
and
y2  x, t   A sin  kx  t  ,
which gives
y  x, t   y1  x, t   y2  x, t  
A sin  kx  t   A sin  kx  t  
2 A sin  kx  cos t 
This equation does not describe a traveling wave. Instead it describes a standing wave.
The absolute value of the first term in this equation 2 A sin  kx  gives the amplitude of
oscillations at point x. As you can see this amplitude varies with position. At nodes it is
equal zero
b g
sin kx  0,
kx  n, n  0,1...
n

x
n .
k
2
The last equation shows positions of the nodes along the x-axis. They are separated by
distances equal to the half of wavelength. The position of antinodes, where the amplitude
has its maximum, can be found from the condition
b g
F 1I
kx   G n  J , n  0,1...
H 2K
F
1I F
1I 
x  Gn  J  Gn  J .
H
K
H
k
2
2K 2
sin kx  1,
They are also separated by the half of wavelengths.
The standing wave can be obtained as a result of the reflection of the traveling
wave in the string from the boundary. Indeed, the reflected wave has the same frequency
and wavelength, but travels in the opposite direction. If the end of the string is fixed
(attached to the wall), then the standing wave will have a node at this point. If the end of
the string is fasten to a light ring on a vertical rod, so it can move freely, then this point is
going to be an anti-node for the resultant standing wave.
In the case if both ends of the string are fixed (the guitar string for instance) we can
obtain a series of the standing waves at certain frequencies. Such standing waves are said
to be produced at resonance and the string is resonating at these resonate frequencies.
Since both ends of the string are fixed, they produce nodes for the resultant standing
wave. This is only possible if the distance L between the ends of the string is equal the
integer number of the half wavelengths, which is

2L
, n  1,2...
n
So this wave has frequencies
f 
v

n
v
, n = 1,2...
2L
The oscillation mode with the lowest frequency f=v/2L is called the fundamental mode or
the first harmonic. The collection of all possible modes for n=1,2… is called the
harmonic series and n is called the harmonic number.
Let us now consider more general (not necessarily in one dimension) case of
interference. This may happen with sound waves. It, for instance, takes place, when you
hear sounds coming from different speakers. For simplicity we shall consider the
interference between the two identical sound waves traveling in almost the same
L1
P
direction.
S1
L2
S2
This situation is shown in the picture, where two waves are traveling from the two pointlike sources S1 and S2 to the same point in space P. Both waves are in phase and of
identical wave length. We can also assume that distances from the sources to point P are
much larger than the distance between the sources themselves, so we can say that waves
are traveling in the same direction. Since distance L1 traveled from the first source is
slightly different from the distance L2 traveled from the second source, the answer for
the question, whether or not these waves are in phase at point P, depends strongly on the
path length difference L  L2  L1 . The phase difference between the two waves is
equal

L

2 .
(1.2.13)
The fully constructive interference occurs if the phase difference is equal to the integer
number of 2 , so
L

 0,1,2...
Fully destructive interference occurs when the phase difference between the waves is
equal the odd multiple of  , so
L


1 3 5
, , ,...
2 2 2
One of the most interesting types of sound is the musical sound. In the most part of
cases musical sound is obtained as a result of oscillations of some part of a musical
instrument. For instance, it could be oscillating string of the string instrument such as a
guitar or a violin. Actually, in this case it is even more complicated, because not only
strings but also the bodies of these musical instruments are oscillating. In the case of the
percussion like drums or plates, the oscillating parts are different membranes of the
instruments. In the case of winds, the oscillating parts are not instruments themselves, but
the air columns inside of those instruments.
We have already discussed oscillations of strings. They can occur as a result of the
standing transverse wave in those strings. The wavelength required for such a standing
wave is the one, which has frequency corresponding to the resonant frequency of the
string. In the case of the standing wave, the string will oscillate with a large enough
amplitude to produce a strong hearable sound wave in the air.
In a same way as we can set up the standing wave in a string, we can do it in a pipe
of some wind instrument (for instance organ pipe) filled with air. In this case it is a
longitudinal sound wave, but again it can be obtained as a result of the reflection from the
end of the pipe. This reflection will take place for the pipe with the open end as well as
for the pipe with the closed end. In order to have this standing wave, the wavelength
should be in certain relation to the length of the pipe. These wavelengths correspond to
the resonant frequencies of the pipe. The standing wave in the pipe is very similar to the
standing wave in a string. The nodes of the standing wave correspond to the closed end of
the pipe and antinodes correspond to the open end of the pipe.
In the case of the pipe with two open ends, there are two antinodes at those ends.
The simplest standing wave, which one can set up in such a pipe, is a wave, which only
has one node in between the ends. This is fundamental mode or the first harmonic of that
pipe. It is easy to see that its wavelength   2L , where L is the length of the pipe. More
generally for the pipe with two open ends the resonant frequencies occur, when

2L
, n  1,2,3... ,
n
where n is called harmonic number. The resonant frequencies in this case are
f 
v


nv
, n  1,2,3...
2L
If one has a pipe with only one end opened (which will be the case in your laboratory
experiment next time), it will be slightly different, since it should be a node at one end
and antinode at the other. In the simplest case a wavelength should be 4L which is
fundamental mode for this pipe. In general

4L
, n  1,3,5...
n
In this case harmonic number n is an odd number and frequencies are given by
f 
v


nv
, n  1,3,5...
4L
So, the length of musical instrument reflects the possible values of frequencies it can
generate. The larger the length, the lower frequency it can play.
If one listens to sounds coming from two different sources, but those sources have
very close frequencies, it will be almost impossible to distinguish between the two
sounds. Instead the person will hear a sound which has a frequency equal to the average
of the two combed frequencies. Intensity of this combined sound will change slightly
with time wavering beats with frequency equal to the difference between the two
combined frequencies. Let us consider this beat phenomenon at some point in space,
where waves coming from the two sources have the form of
s1  A cos 1t  ,
s2  A cos 2t  ,
where  1   2 . The superposition principle of two waves gives
s  s1  s2  A  cos 1t   cos 2t   
   2 
   2 
2 A cos  1
t  cos  1
t ,
 2

 2

so we have
s  t   2 A cos t  cos t  ,
where   
1   2
2
and  
1   2
2
. Since frequencies of the two waves are very
close, this means that     and we can treat the first cosine function in the last
equation as the slowly changing wave's amplitude. The angular frequency at which beats
occur is  beat  2    1   2 . This beat-phenomenon is used by musicians to turn their
instruments on the standard frequency until beats disappear.