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In the Chapter on energy, we studied in some detail the behaviour of a single
particle. We found that the concepts of kinetic and potential
energy usefully describe the motion of a particle.
A particle is point-like in the sense that at any instant it occupies
a single point. The natural generalization from a single particle is to
a continuous body.
Continuous bodies are of two kinds, namely rigid and fluid.
A rigid body, even though it is extended over space,
is essentially like a point particle, since all of its points move
together. Fluid bodies are liquid and gaseous. Extended bodies are that are spread over
space are called a medium. Fluid bodies such as the atmosphere, oceans and so on
have new features not present in particles. Fluid bodies have two kinds of motion
that are commonly observed.
Namely, one in which there is a net transport of matter such as in the
flow of river water and the other being periodic motion
such as the motion of a life buoy, which never strays very far from
its position of equilibrium. Elastic bodies lie somewhere between
rigid and fluid media.
In the case of say the flow of a river the entire medium is in motion, and there
is a net physical
transport of the particles of water from the source of the river to the sea. The
study of the motion
of the material forming a medium, water in the case of a river, is studied
in the discipline of fluid mechanics.
to net material movement, a medium has a simpler form of motion.
Anyone who has been to a beach
has seen ocean waves traveling on the surface of the ocean. Such oscillatory motion
of the medium is generically called waves.
Wave motion pervades nature; waves on an ocean or in a bathtub are
familiar to all. What is equally familiar but may not be recognized as
originating in waves, is the whole phenomenon of sound, which is the response of the ear
to pressure waves in air.
There are more complex forms of waves such as earthquakes, radio and
television waves, thermal waves, and non-classical waves that appear
in quantum mechanics called probability waves, and so on.
What distinguishes wave motion
from the behavior of a particle is that a wave is spread out over
space, and it tends to consist of periodic oscillations of some
underlying medium, be it the water of the ocean or the air around
us. For example, a life buoy on the ocean's surface will
bob up and down about
its equilibrium position as the wave passes through it. Similarly,
sound propagates by the air particles oscillating about their equilibrium
When we hear a sound, the energy in the sound wave is deposited on our ear drums, causing
the sensation of sound.
The concept of energy is essential for understanding wave motion, as it is
for so many other central phenomena of physics.
All waves have in common the fact that they are disturbances
of a continuous media - for example air or water - in which energy and
are transferred from one part of space to another without the net physical
transference of matter. The example of waves shows us how subtle and
pervasive are the various forms of energy.
Waves are classified as transverse and longitudinal, depending on the kind of
vibration or oscillation that the underlying media in undergoing.
The simplest possible wave is one that repeats its shape, and allows us to study only
a finite portion of an otherwise infinitely spread out medium. Such a
wave is called a simple wave, and its
fixed pattern repeats itself throughout the medium.
Most waves we see in daily life are far from simple, and look irregular and non-periodic.
There is a branch of mathematics, called Fourier Analysis, that shows
how any arbitrary and complicated wave can be resolved into a sum of simple
waves. Hence, instead of considering complicated
for the entire medium, we need to study only simple wave which consists of only
a finite pattern that repeats itself.
An example of simple wave-motion is what one generates by, say, dropping a
piece of stone into water. What we observe are ripples, which are
waves, created in water. We make the idealization that the pattern of the ripples
propagates forever; this idealization is similar to the one made
in ignoring friction in Newtonian mechanics, and is very useful in
understanding the essential properties of real waves.
Hence, the idealized simple wave has a
fundamental pattern, say the height and length of a single ripple,
that is repeated throughout the medium. How should we describe such a
simple wave as shown in Figure 5.1?
Instead of studying ocean waves, which are ripples on the surface of
the ocean, and
hence two-dimensional, we instead study the simpler case where the
medium is only
one-dimensional, as in the case of a taut string lying along the
-direction, and tied to a distant post.
Waves in general are of many kinds, and for starters we study what
are known as transverse waves.
A transverse wave is one in which the oscillations of the medium
are perpendicular to the motion of the wave.
An accurate example of a transverse wave is
the transverse oscillations
created by shaking the string and making it move along the -direction.
The position of the
string is given by a two dimensional vector , where is
the position of the string along the -direction, and the
coordinate specifies how far the string is stretched from its
equilibrium position, which is when the string is flat. Figure 5.1 shows
such a transverse wave.
The shape of the wave shown in Figure 5.1 is periodic in that it
repeats itself indefinitely. In trigonometry, we have
encountered the sine and cosine function, which are periodic
functions and we briefly discuss them in the box.
Suppose we take a still photograph of
the idealized ocean wave at time ; what we will see is the
shape of the height of the string repeats itself
indefinitely, and is a periodic function of its distance
along the string. The shape of the string is what is
called the (transverse) wave.
Let the distance along the
-direction, and let be the height of the string. The shape
of as one moves along is the what we are
calling a wave.
As shown in Figure 5.1,
a simple wave is a periodic function of , and hence, at
Transverse wave along the -direction
where we have assumed that the amplitude is zero at the origin,
that is for .
Recall both and have dimension of length.
We need quantities and in eq.(5.1) from dimensional
analysis. Since the argument of a sine function is dimensionless,
we need a parameter which has dimension inverse of length, that is
; furthermore, since the sine function is dimensionless,
we need a quantity with dimension to make eq.(5.1)
What is the physical significance of the quantities
and ? The maximum value of the sine function is 1, and hence
from eq.(5.1) we see that , called the amplitude of the
wave, is the maximum height that an string can have, and is shown
in Figure 5.2.
To understand what is , called the wave number, note since
the sine function is periodic,
Amplitude and wavelength of a wave
From the above, we see that if is increased by an amount
the pattern of the wave repeats itself, as shown in Figure 5.2. The
length after which a wave repeats itself is called its wavelength,
and is denoted by . Hence we have
What is the shape of the string at some later time ? We
consider only the simplest case for which the entire wave is
traveling at a constant velocity of ; in other words each point of
the wave propagates at the
same velocity . Hence, a point on the wave
travels to a new position after time . To follow the motion
of means that we initially fix some point in the medium,
say at the value of , and find the value of at this point.We
then increase the value of the
position to reflect the new location of . At , let us fix our attention
on the height of the ocean wave which is at
a distance of m from the shore. We then have
After say sec the distance of the wave is ()m from the shore.
Consequently, since we are following the point on the wave
with the fixed value of ,
the wave at the new position at sec must still have the
same value of . See Figure 5.3.
In other words, paying attention to the
arguments which refer to time and space for the wave, we have
Velocity of propagation of a wave
As the wave propagates, we see from the example above, that we
have to increase the value of m by
an amount of () m, which is the distance covered by the wave in time s.
To understand the general case,let us follow the motion of a point of the wave which is
at at time , and whose value is given by
After time has elapsed, the wave which was at the point has moved
to the new position as shown in Figure 5.3. Hence we have the
following generalization of eq.(5.8)
We see from the above equation that the position and time
coordinates of the wave only appears in the combination .
Hence in general, the shape of the wave for
time and at position is given by
One can easily see that we recover the special case given in eq.(5.8)
We hence have the result that the wave is propagating
in the -direction with a longitudinal velocity .
Figure 5.3 shows the position of the wave at two instances
and . Eq.(5.13) is a fundamental result of wave motion.
We need to analyze eq.(5.13) to deduce the
motion of the underlying medium.From eq.(5.13) we have
We have already mentioned that a simple wave consists of the periodic
motion of the underlying medium. We now deduce the frequency of oscillation
of the medium. Let us fix our attention at a specific point of the medium, say, at
the point , and examine its motion as we vary time. Hence the
does not vary, and we have for the
material element at
We see that the material of the medium at point does not
travel in the -direction, but rather, oscillates back and forth
in the -direction. An important characteristic of a traveling
wave is that the medium oscillates in the transverse -direction
with the same amplitude at all points of the medium. We
will later see that in the case of standing and resonant wave,
this will not be the case.
Recall from our discussion on simple harmonic motion, we had
derived, in eq.(3.87), that a particle subjected to an elastic
potential oscillates with a frequency of about
its equilibrium position .
Comparing eqs.(5.17) and(3.87), we obtain (ignoring
the sign of velocity ) the
important result that the material of the medium undergoes simple harmonic
its equilibrium position, and with a frequency given by
The result we have obtained in eq.(5.19) has a simple interpretation.
Frequency refers to how rapidly the material points of the medium are
oscillating about their equilibrium position. Eq.(5.19) tells us
that in the time that it takes for a point of the wave moves through the distance of
one full wavelength, namely
the material particle, in time interval , undergoes one
Collecting the results for the wave , we obtain the standard
expression for a wave that is given by
Note from the above that the wave at the point
undergoes motion in the -direction - which is perpendicular to the direction of
propagation, as is expected of a transverse wave - with a transverse velocity given
by . The velocity itself varies, since the piece of
string at the point picks up velocity as it moves towards the maximum
displacement for the string given by the amplitude ; once it reaches the maximum
value for its displacement, its velocity slows down to zero and reverses its direction
as the string moves back towards its equilibrium position.
the transverse velocity of the string is varies with time, whereas
the longitudinal velocity of the wave is a constant.
For wave motion there is no net
transfer of the underlying medium along the direction of
propagation, but rather, only energy and momentum propagate. The
energy carried by the wave propagates at velocity .
Since the string is an extended object, it has a certain amount of energy for a unit of length
which is denoted by .
Figure 5.4 is a plot of energy per unit length present at different points of
Figure 5.4 shows that wave propagation from position at time to a
new position at time is in fact
the propagation of energy. We have to constantly supply
energy to the string to keep it propagating; for example, for the case
shown in Figure 5.4, extra energy, indicated
by dashed lines, has to be supplied for the wave to continue its propagation.
The power needed to keep the wave propagating is given from Figure 5.4
Propagation of Energy
We have derived the important result that the amount of
power needed to drive the wave is proportional to the velocity of
propagation . A more detailed discussion on energy propagation
is given in the next section.
The energy for simple harmonic motion has been derived in
eq.(3.90); let the mass per unit length of the string be . We then have
and hence the power generating the wave is given from
The intensity of a wave at some point , denoted by , and is what
is actually measured when a wave passes through. From daily experience, we know
that if sound has a very high intensity, it is painful to the ear. Intensity is defined
to be the amount of energy
that is crossing a unit area around a point in unit time, or equivalently, is equal to
the power flowing per unit area through .
As shown in Figure 5.5, energy flows through an area . Hence,
Intensity of a Wave
where we have used eq(5.28) to obtain the last line.
The most important fact about intensity is that it is
proportional to ; in other words, intensity is proportional
The vibrations of the string that we have analyzed is traveling in a
definite direction, say along the -axis, and with the oscillations of the medium being
in the transverse -direction. This is an example of the general case in which the
vibrations of the medium is perpendicular to the motion of the wave, and is called a
There are two fundamental
types of waves, namely transverse, in which the vibrations of the medium
perpendicular to the direction of motion, and longitudinal, in which the
oscillations of the medium are parallel to the direction of motion. Waves can be
purely longitudinal, purely transverse, or a combination of both.
For example, an ocean wave has a small
longitudinal component as well.
Transverse waves can result from the oscillation of a medium, for example, can be an
ocean wave, or a wave resulting
from applying a transverse vibration to a taut string, or can be an electromagnetic
Longitudinal waves, as shown in Figure 5.6, are the result of the
oscillations of an underlying medium due to its successive
compression and expansion.
Sound waves are a
typical example of longitudinal waves, and consist of a propagating pressure wave
created by the longitudinal compression and expansion of air, and is shown in
We see that for a longitudinal wave the motion of the wave is
along the direction of motion, in this case the pressure of the
medium undergoes oscillations along the -direction.
Finally we have surface waves such as those in earthquakes as well
as ocean waves, which are composed of both longitudinal and
transverse waves, resulting in the circular motion of particles, not on the surface
in the form of outwardly propagating waves, but rather in the
depth of the medium.
Since there is no net transfer of matter in the propagation of
waves, one might ask as to what is it that is being propagated in wave
motion? The answer is that, as mentioned before, there is a flow of energy and momentum
along the direction of motion of the wave.
We examine the energy
flow that takes place for the case of the transverse vibrations of a string.
Energy has to be constantly pumped into the string to keep it
vibrating, and accounts for the energy flowing when the string
undergoes transverse vibrations.
A taut string is characterized by its tension as well as by its
mass per unit length . To simplify matters, and allow us to study the
phenomenon of wave propagation in some detail, we construct the string from
a discrete collection of point particles. We consider the continuous string as
being composed out of
small segments of length , with each segment having a mass of . The position of the -th particle is given by .
The tension of the string is equivalent to having the point
masses with mass being connected by springs, with spring constant being
Each piece of the string has kinetic energy and elastic
potential energy ; the total energy of the wave is the sum of
the energy of each piece of the string. To understand in some
detail the properties of a transverse wave, we replace the
continuous string with a set of discrete particles, each particle
being labeled by integer , having mass and separated by
a distance in the longitudinal -direction. Each particle
is connected by two springs to its two neighboring particles.
What we now have is a set of particles, each subject to an elastic
potential characterized by spring constant , and with
each particle undergoing oscillations in the transverse
*Energy Propagation in Waves
the transverse position of the -th particle is given by .
Each particle, under the force exerted by the springs, is undergoing
simple harmonic motion. Hence, from (3.87), we have
diagram of particles linked by springs
Recall that the phase is fixed by (3.91), and is
determined by the position that the particle occupies at say .
To create the simple wave motion (harmonic) in the string, we have
to start all the different particles in phase. What this
means is that the various particles do not start their
oscillation at with arbitrary positions, but rather, they
are put into a specific configuration, which for the simple wave
has to be chosen to be
If we plot the positions of the various particles at a given
instant as shown in the Figure 5.8, we see that the
particles all start their oscillations from an initial shape of a
Hence, for let , the positions of the particles are then
given, from eqns.(5.32), by
Initial positions of particles
The transverse velocity is given, as in the continuous case of eq.(5.20), by
By comparing with eq.(5.13) we see that the wave is already
looking a lot like the motion of a number of discrete particles.
The energy of this system of -particles undergoing
harmonic oscillations is simply the sum of the energies of the
individual particles, denoted by . That is, energy of wave is
For a small displacement the potential energy of particle is given by
the amount the spring has stretched. The elastic potential energy is given
by the amount that the springs have stretched from the equilibrium length, as shown in
Hence from the Pythagorus
theorem, as shown in Figure5.9, we have
where is the tension of the string.
From eqs.(5.34) and (5.35), we have the important
In other words, each elemental mass at position has a
constant energy, as is expected from our earlier discussion on
simple harmonic motion. Note that the energy being propagated is
proportional to the square of the amplitude and to the
square of the frequency .
For a finite wave, say spread over a distance from to ,
the plot of the energy versus is a constant, the value
of which is given by eq.(5.41). To create the wave, energy
has to be constantly be pumped into the system. If we think of the
particles as being infinitesimal parts of the continuous
taut string, then we need to take the
. The point particles with mass are
interpreted as a small piece of the string of length , and so
we set ; hence
, where is the mass per unit length of the
string. Similarly, the energy is the energy of the
small piece of string of length . From eq.(5.41)
we obtain that the energy expended in the propagation of the wave,
for a time interval , is given by
Hence, the power expended in sustaining wave propagation is given
is the velocity of the wave.
Note we have derived the equation above in eq. (wpwr2) using much
simpler graphical means. In Figure 5.4 we have seen that the power that is
needed to create a propagating wave goes into creating a new piece
of the wave occupying the position from to , and is
the amount of energy per unit time required to create that portion
of the wave.
From dimensional analysis, we can straight away conclude that the
velocity of propagation for a taut string
; a more careful analysis from
eq.(5.48) shows that
So how should we picture the propagation of energy in a wave? For
a wave traveling towards the right, the elemental mass
does work on the elemental mass to its right and
the energy arises from this work.
In summary, we see that the idea of energy is essential in understanding wave motion
since a wave consists of the propagation of
energy and momentum, but without any transport of the material
substance that is undergoing wave motion.
Waves have the rather remarkable property that if there are two or
more waves propagating in the same region of space, there are
simple rules on how to compose them, and the resultant is a single
wave with properties derived from the constituent waves. There is no analogous
property of particles, since if two or more particles are
propagating in space, they continue to maintain their identities.
Consider the case of two transverse waves, both propagating in the
same medium. For example, we can create two different waves
on an ocean; when the propagating waves reach a region where they overlap,
we will have the phenomenon of the superimposition, or, in short, the
superposition of waves.
Suppose we have two waves in the same medium denoted
by and . The resultant wave, denoted by is the
resultant wave. The superposition principle states that
the resultant wave is simply the sum of the two waves. That is
The resultant wave is said to be the result of the interference of the
two waves. The resulting wave has an intensity, from eq.(5.31), given by
Note the above equation is a characteristic feature of wave
behaviour, and will be of great significance in our discussion on quantum
The generelization to the superposition of arbitrary number
waves is straightforward. For simplicity, consider the case of two waves having the
same amplitude and frequency, but which start off with two different initial
shapes. Let the first wave be denoted by
and the second wave be denoted by
The phases and indicate the different starting shapes of the two
Figure 5.10 shows two waves being created, and their
superposition in the space where the two waves overlap. The
intensity of the superposed wave is plotted, and the pattern is
the result of constructive and destructive interference. We give a
derivation of the observed interference pattern.
The net resultant wave is given, as in eq.(5.51), by the
sum of the two waves, namely
Interference: The Superposition of Waves
From the rules of trigonometry, the sum of two sines is given by
Hence, from eq.(5.59), we have for the resultant wave
In general, the resultant
wave is a traveling wave, as can be seen from
its dependence on There are two notable features of .
(a) The intensity of the resultant is shown in Figure 5.10
wave, and is proportional to , and depends on the phase and
is given by phase difference, namely, is a function
(b)The phase of the resultant wave, , is given
Two special cases for the phase difference are given below.
In eq.(5.65) above the component waves are said to in phase for
, and interfere constructively yielding the
amplitude of , which is double of the amplitudes of the
component waves. On the other hand, in eq.(5.67), for
wave is zero, and is said to result from the destructive interference of the
component waves. For arbitrary phases the resultant wave is
in-between the two extreme cases discussed above.
The general case of the superposition of two arbitrary waves with different amplitudes
has a result similar to the special case considered above.
In our discussion on the superposition of waves, we considered the
case when the two waves were both propagating in the same
direction. We can also superpose waves traveling in
opposite directions. Consider a traveling wave in a taut string that hits a boundary,
and is reflected off it. We would then have two waves in the string, namely the
original wave and the reflected wave; the reflected wave, however, would be traveling in
the opposite direction from the initial wave, and would result in a standing wave.
To see how the mathematics works out, consider two waves, similar to eq.(5.59), but
with the direction of propagation of the second wave in the
opposite direction from the first wave. We then have
Note from eq.(5.71) that at the points
, called the nodes
of the standing wave, the
medium is fixed and hence
for all time;
in other words, the nodal points of the wave are
fixed, and do not propagate. The wave is consequently called a standing
wave. Note that, similar to a traveling wave, each point of the medium oscillates about its
equilibrium position; however, unlike a traveling wave, the
amplitude of the wave is not constant, but rather varies from a
maximum value of to the value of at the nodal points.
The incident and reflected wave interfere, and due to
destructive interference all wavelengths do not contribute, as
shown if Figure 5.12. The
reason that only a fixed set of wavelengths are allowed is because
only for these select set of wavelengths is there a constructive interference of the
incident and reflected wave.
Creation of a Standing Wave
We have considered a wave which has only one end fixed at a boundary. What happens
to a wave in a medium, for example a violin string, that
is tied at two ends. Suppose the string has tension given . When the
string is plucked, the string will vibrate, and a standing wave will be created
in the string. The transverse vibrations are given by eq.(5.71), namely
But we are not done; we have to account for the fact that the two ends of
the string are always fixed. Suppose the length of the violin is , with one
fixed end at and the other end at . We then must have
Hence, unlike the case for an arbitrary standing waves,
the violin string can have only wavelengths which
are equal to a multiple of
as given in
eq.(5.75). The reason for this result is easy to understand.
Only waves that have wavelengths that are integer multiples of
can fit into the length of the violin as
standing waves. All other wavelengths, as was the case of a standing
wave, are eliminated by destructive interference.
Furthermore, since wavelength is related to frequency
by eq.(), we have from eq.(5.49)
The frequencies given above are called resonant or harmonic
frequencies, and are the ingredients to all sound that music is based on.
Figure 5.13 shows a few of the resonances for a fixed length
Resonance can also take place for more medium which is more
complex. Consider a circular piece of metal, say a golden bangle.
If one were to strike to bangle, there would be oscillations set
up in the bangle, with wavelength which would exactly be integer
multiples of the circumference of the bangle. If the radius of the
bangle is , the allowed wavelengths for oscillations are then
A particular oscillation of the bangle (ring) is shown in Figure
So far we have considered the superposition of waves that have the
same frequency . What happens when we superpose two waves with
different frequencies? We have experienced such phenomenon
many times in daily life when two sounds with different pitch
(which is a non-technical term for frequency) are heard by us at
the same time. Let the two waves have different frequencies
and . For simplicity, let us assume that we are interested in the oscillations
of the wave at the same point in space (which is the case of the ear receiving two
different sound waves), and hence examine the dependence of the wave only
on time. We denote the waves by and , and have the following
Wave motion in a metallic ring (bangle)
To understand the physical phenomenon that the equation above
refers to, suppose
; we then have
Since the first term
is almost a constant
the sound seems to have a frequency of . However, the maximum amplitude
is now , double of the input waves.
The result is a
sound which has a double the amplitude of , and is consequently much
louder than the component waves, but with
frequency . This waxing and waning of superposed sound wave is
well known in music, and is used for tuning the frequencies of
Consider a train that has a whistle which emanates a sound with frequency .
The sound of the train whistle received by an observer when the train is approaching
has a higher frequency than , and which becomes of a lower frequency when
the train recedes from the observer. In both cases, the effect,
called the Doppler effect, arises from
a moving source of waves, in the case of a moving train.
Let us for concreteness consider the case of sound waves, with stationary air being
the frame of reference with respect to which the source is moving.
Consider first the case of a stationary source which is emitting a signal once every
seconds. The source hence emits signals with a frequency of
. In time the sound propagates a distance of
, and then the second signal is emitted. The wavelength of the sound wave
is the distance between the first and second signal, which for
the stationary case is given by . The speed of
propagation of sound wave is given by eq.(5.19) which relates, for all wave
motion, the frequency and wavelength. Hence we have
Suppose the source is moving at a speed ; then, after emitting
the first signal, it travels a distance of , and then emits
the second signal. Hence, as shown in Figure 5.15 the distance between the
first signal and the second is now a shorter distance of
, and hence the wavelength received by the observer is
Hence, the frequency of sound from a moving source, and received by a stationary observer, is
Clearly, since , we have from eq.(5.91) that ; in other words the
frequency is of
higher pitch, as expected for a source of sound that is
an observer. What happens when the velocity of the source becomes equal to
velocity of sound, that is,
? From eq.(5.91)
it seems that
; this divergence indicates
the onset of a new phenomenon, namely, of shock waves, which is
A similar derivation for a source receding from an
For an observer moving with velocity towards, or away, from a
source receives a frequency given by
Note the switch in the signs of the velocity term in
going from a moving source to a moving observer.
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