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Subsections

Waves

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. In addition 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 position. 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 momentum 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 oscillatory motion 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?

Transverse Waves

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 $x$-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 $y$-direction. The position of the string is given by a two dimensional vector $(x,y)$, where $x$ is the position of the string along the $x$-direction, and the $y$ 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.

Figure 5.1: Transverse wave along the $x$-direction
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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 $t=0$; 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 $x$-direction, and let $y$ be the height of the string. The shape of $y$ as one moves along $x$ is the what we are calling a wave. As shown in Figure 5.1, a simple wave $y$ is a periodic function of $x$, and hence, at $t=0$ we have
\begin{displaymath}
y(0,x)=A \sin(kx)
\end{displaymath} (5.1)

where we have assumed that the amplitude is zero at the origin, that is $y(0,x)=$ for $x=0$. Recall both $x$ and $y$ have dimension of length. We need quantities $A$ and $k$ in eq.(5.1) from dimensional analysis. Since the argument of a sine function is dimensionless, we need a parameter $k$ which has dimension inverse of length, that is $[k]=L^{-1}$; furthermore, since the sine function is dimensionless, we need a quantity $A$ with dimension to make eq.(5.1) consistent. What is the physical significance of the quantities $A$ and $k$? The maximum value of the sine function is 1, and hence from eq.(5.1) we see that $A$, called the amplitude of the wave, is the maximum height that an string can have, and is shown in Figure 5.2.

Figure 5.2: Amplitude and wavelength of a wave
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To understand what is $k$, called the wave number, note since the sine function is periodic, we have
$\displaystyle \sin(kx)$ $\textstyle =$ $\displaystyle \sin(kx+2\pi)$ (5.2)
  $\textstyle =$ $\displaystyle \sin[k(x+\frac{2\pi}{k})]$ (5.3)

From the above, we see that if $x$ is increased by an amount $\displaystyle \frac{2\pi}{k}$ 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 $\lambda$. Hence we have
$\displaystyle y(0,x)$ $\textstyle =$ $\displaystyle y(0,x+\lambda)$ (5.4)
$\displaystyle \Rightarrow A\sin(kx)$ $\textstyle =$ $\displaystyle A\sin[k(x+\lambda)]$ (5.5)
$\displaystyle \Rightarrow k$ $\textstyle =$ $\displaystyle \frac{2\pi}{\lambda}$ (5.6)

What is the shape of the string at some later time $T$? We consider only the simplest case for which the entire wave is traveling at a constant velocity of $v$; in other words each point of the wave propagates at the same velocity $v$. Hence, a point $y$ on the wave travels to a new position after time $T$. To follow the motion of $y$ means that we initially fix some point in the medium, say at the value of $x_0$, and find the value of $y$ at this point.We then increase the value of the position $x$ to reflect the new location of $y$. At $t=0$, let us fix our attention on the height of the ocean wave which is at a distance of $2$m from the shore. We then have
\begin{displaymath}
y(0,2)=A\sin(2k)
\end{displaymath} (5.7)

After say $5$ sec the distance of the wave is ($2+5v$)m from the shore. Consequently, since we are following the point on the wave with the fixed value of $A\sin(2k)$, the wave at the new position at $t=5$ sec must still have the same value of $A\sin(2k)$. See Figure 5.3.

Figure 5.3: Velocity of propagation of a wave
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In other words, paying attention to the arguments which refer to time and space for the wave, we have
\begin{displaymath}
y(5,2+5v)=A\sin(2k)
\end{displaymath} (5.8)

As the wave propagates, we see from the example above, that we have to increase the value of $x=2$ m by an amount of ($vt=5v$) m, which is the distance covered by the wave in time $t=5$ s. To understand the general case,let us follow the motion of a point of the wave which is at $x_0$ at time $t=0$, and whose value is given by
\begin{displaymath}
y(0,x_0)=A\sin(kx_0)
\end{displaymath} (5.9)

After time $T$ has elapsed, the wave which was at the point $x_0$ has moved to the new position $x=x_0+vT$ as shown in Figure 5.3. Hence we have the following generalization of eq.(5.8)
$\displaystyle y(T,x)$ $\textstyle =$ $\displaystyle y(0,x_0)$ (5.10)
$\displaystyle \mathrm{\rm { where }}$      
$\displaystyle x$ $\textstyle =$ $\displaystyle x_0+vT$ (5.11)
$\displaystyle \Rightarrow y(T,x)$ $\textstyle =$ $\displaystyle y(0,x-vT)$ (5.12)

We see from the above equation that the position and time coordinates of the wave only appears in the combination $x-vt$. Hence in general, the shape of the wave for time $t$ and at position $x$ is given by
$\displaystyle y(t,x)$ $\textstyle =$ $\displaystyle y(0,x-vt)$ (5.13)
  $\textstyle =$ $\displaystyle A\sin[k(x-vt)]$ (5.14)

One can easily see that we recover the special case given in eq.(5.8) from eq.(5.13). We hence have the result that the wave is propagating in the $x$-direction with a longitudinal velocity $v$. Figure 5.3 shows the position of the wave at two instances $t=0$ and $t=T$. 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
$\displaystyle y(t,x_m)$ $\textstyle =$ $\displaystyle A\sin[k(x_m-vt)]$ (5.15)
  $\textstyle =$ $\displaystyle A\sin(kx_m-kvt)$ (5.16)

Frequency of a Wave

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 $x_m$, and examine its motion as we vary time. Hence the phase $\phi\equiv kx_m$ does not vary, and we have for the material element at $x_m$
\begin{displaymath}
y(t,x_m)=A\sin(\phi-kvt)
\end{displaymath} (5.17)

We see that the material of the medium at point $x_m$ does not travel in the $x$-direction, but rather, oscillates back and forth in the $y$-direction. An important characteristic of a traveling wave is that the medium oscillates in the transverse $y$-direction with the same amplitude $A$ 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 $f$ about its equilibrium position $x_E$. Comparing eqs.(5.17) and(3.87), we obtain (ignoring the sign of velocity $v$) the important result that the material of the medium undergoes simple harmonic oscillations about its equilibrium position, and with a frequency given by
$\displaystyle 2\pi f$ $\textstyle =$ $\displaystyle kv=\frac{2\pi}{\lambda}v$ (5.18)
$\displaystyle \Rightarrow f\lambda$ $\textstyle =$ $\displaystyle v$ (5.19)

The result we have obtained in eq.(5.19) has a simple interpretation. Frequency $f$ 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 $\displaystyle \frac{\lambda}{v}$, the material particle, in time interval $T=\frac{1}{f}$, undergoes one complete oscillation.


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...)
\\
&=&-2\pi A f \cos(2 \pi \frac{x}{\lambda}-2\pi f t)
\end{eqnarray}
}}}


Collecting the results for the wave $y$, we obtain the standard expression for a wave that is given by

$\displaystyle y(t,x)$ $\textstyle =$ $\displaystyle A\sin[k(x-vt)]$ (5.20)
  $\textstyle =$ $\displaystyle A \sin(2 \pi \frac{x}{\lambda}-2\pi f t)$ (5.21)

Note from the above that the wave at the point $x$ undergoes motion in the $y$-direction - which is perpendicular to the direction of propagation, as is expected of a transverse wave - with a transverse velocity given by $u(t,x)$. The velocity $u(t,x)$ itself varies, since the piece of string at the point $x$ picks up velocity as it moves towards the maximum displacement for the string given by the amplitude $A$; 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. Note that the transverse velocity of the string $u(t,x)$ is varies with time, whereas the longitudinal velocity of the wave is a constant.

Energy, Power and Intensity of a Wave

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 $v$.

Figure 5.4: Propagation of Energy
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Since the string is an extended object, it has a certain amount of energy for a unit of length which is denoted by $\rho(E)$. Figure 5.4 is a plot of energy per unit length $\rho(E)$ present at different points of string $x$. Figure 5.4 shows that wave propagation from position $x$ at time $t$ to a new position $x+\Delta x$ at time $t+\Delta t$ 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 by
$\displaystyle P$ $\textstyle =$ $\displaystyle \frac{\mbox{\rm {Energy expended in time }}\Delta t}{\Delta t}$ (5.22)
  $\textstyle =$ $\displaystyle \frac{\rho(E)\Delta x}{\Delta t}=\rho(E)\frac{\Delta x}{\Delta t}$ (5.23)
  $\textstyle =$ $\displaystyle \rho(E)v$ (5.24)

We have derived the important result that the amount of power needed to drive the wave is proportional to the velocity of propagation $v$. 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 $\mu $. We then have
\begin{displaymath}
\rho(E)=\frac{1}{2}\mu (2\pi f)^2A^2
\end{displaymath} (5.25)

and hence the power generating the wave is given from eq.(5.26) by
\begin{displaymath}
P=\frac{1}{2}\mu (2\pi f)^2A^2v
\end{displaymath} (5.26)

The intensity of a wave at some point $x$, denoted by $I(x)$, 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 $x$ in unit time, or equivalently, is equal to the power $P$ flowing per unit area through $x$. 5.5.

Figure 5.5: Intensity of a Wave
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As shown in Figure 5.5, energy flows through an area $S$. Hence,
$\displaystyle I(x)$ $\textstyle =$ $\displaystyle \frac{\mathrm{Power}}{\mathrm{Unit Area}}$ (5.27)
  $\textstyle =$ $\displaystyle \frac{\mathrm{Power}}{S}$ (5.28)
  $\textstyle \simeq$ $\displaystyle f^2A^2$ (5.29)

where we have used eq(5.28) to obtain the last line. The most important fact about intensity $I$ is that it is proportional to $A^2$; in other words, intensity $I$ is proportional to $y^2$.

Longitudinal Waves

The vibrations of the string that we have analyzed is traveling in a definite direction, say along the $x$-axis, and with the oscillations of the medium being in the transverse $y$-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 transverse wave. There are two fundamental types of waves, namely transverse, in which the vibrations of the medium are 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 wave. 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.

Figure 5.6: Longitudinal wave
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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 Figure [*] 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 $x$-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.


*Energy Propagation in Waves

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 $\tau$ as well as by its mass per unit length $\mu $. 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 $d$, with each segment having a mass of $m=\mu
d$. The position of the $n$-th particle is given by $x_n=nd$. The tension of the string is equivalent to having the point masses with mass $m$ being connected by springs, with spring constant $k$ being given by $\tau=\frac{k}{d}$. Each piece of the string has kinetic energy $T$ and elastic potential energy $U$; 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 $n$, having mass $m$ and separated by a distance $d$ in the longitudinal $x$-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 $k$, and with each particle undergoing oscillations in the transverse $y$-direction.

Figure 5.7: diagram of particles linked by springs
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Suppose the transverse position of the $n$-th particle is given by $y_n$. Each particle, under the force exerted by the springs, is undergoing simple harmonic motion. Hence, from (3.87), we have
\begin{displaymath}
y_n(t)=A\sin(\phi_n - 2\pi f t)
\end{displaymath} (5.30)

Recall that the phase $\phi$ is fixed by (3.91), and is determined by the position that the particle occupies at say $t=0$. 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 $t=0$ with arbitrary positions, but rather, they are put into a specific configuration, which for the simple wave has to be chosen to be
\begin{displaymath}
\phi_n=\frac{2\pi n}{\lambda}d
\end{displaymath} (5.31)

If we plot the positions of the various particles at a given instant $t=0$ as shown in the Figure 5.8, we see that the particles all start their oscillations from an initial shape of a wave.

Figure 5.8: Initial positions of particles
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Hence, for let $t>0$, the positions of the particles are then given, from eqns.(5.32), by
\begin{displaymath}
y_n(t)=A\sin(\frac{2\pi n}{\lambda}d - 2\pi f t)
\end{displaymath} (5.32)

The transverse velocity is given, as in the continuous case of eq.(5.20), by
\begin{displaymath}
u_n(t)=\frac{\partial y_n}{\partial t}(t)=
-2\pi A f \cos( \frac{2 \pi n}{\lambda}-2\pi f t)
\end{displaymath} (5.33)

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 $n$-particles undergoing harmonic oscillations is simply the sum of the energies of the individual particles, denoted by $E_n$. That is, energy of wave is
$\displaystyle E$ $\textstyle =$ $\displaystyle E_1+E_2+....+E_n$ (5.34)
  $\textstyle =$ $\displaystyle \sum_{n=1}^N E_n$ (5.35)

Figure 5.9: Stretched Springs
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For a small displacement $y_n$ the potential energy of particle $n$ 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 Figure5.9. Hence from the Pythagorus theorem, as shown in Figure5.9, we have
$\displaystyle E_n$ $\textstyle =$ $\displaystyle \frac{1}{2}k [\sqrt{d^2+y_n^2}-d]^2+\frac{1}{2}m u_n^2$ (5.36)
  $\textstyle \simeq$ $\displaystyle \frac{1}{2}[\frac{k}{d} y_n^2+m u_n^2] + \mbox{\rm {negligible
terms}}$ (5.37)
  $\textstyle =$ $\displaystyle \frac{1}{2}[\tau y_n^2+m u_n^2]$ (5.38)

where $\tau$ is the tension of the string. From eqs.(5.34) and (5.35), we have the important result
\begin{displaymath}
E_n=\frac{1}{2}m(2\pi f)^2A^2 \mbox{ : constant}
\end{displaymath} (5.39)

In other words, each elemental mass at position $x=nd$ 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 $A$ and to the square of the frequency $f$. For a finite wave, say spread over a distance from $x=0$ to $x=Nd$, the plot of the energy versus $x$ 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 limit of $d \rightarrow 0$. The point particles with mass $m$ are interpreted as a small piece of the string of length $\Delta x$, and so we set $d=\Delta x$; hence the mass $m=\mu \Delta x$, where $\mu $ is the mass per unit length of the string. Similarly, the energy $E_n=\Delta E$ is the energy of the small piece of string of length $\Delta x$. From eq.(5.41) we obtain that the energy expended in the propagation of the wave, for a time interval $\Delta t$, is given by
$\displaystyle \Delta E$ $\textstyle =$ $\displaystyle \frac{1}{2}(2\pi f)^2A^2 \mu \Delta x$ (5.40)
$\displaystyle \Rightarrow \frac{\Delta E}{\Delta t}$ $\textstyle =$ $\displaystyle \frac{1}{2}(2\pi
f)^2A^2\frac{\Delta x}{\Delta t}$ (5.41)

Hence, the power expended in sustaining wave propagation is given by
$\displaystyle \Rightarrow P$ $\textstyle =$ $\displaystyle \frac{\Delta E}{\Delta t}$ (5.42)
  $\textstyle =$ $\displaystyle \frac{1}{2}(2\pi f)^2A^2v$ (5.43)

where $\displaystyle v=\frac{\Delta x}{\Delta t}$ 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 $P$ that is needed to create a propagating wave goes into creating a new piece of the wave occupying the position from $x$ to $x+\delta x$, and $P$ is the amount of energy per unit time required to create that portion of the wave.


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...y}{\partial x})^2+\mu(\frac{\partial y}{\partial
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From dimensional analysis, we can straight away conclude that the velocity of propagation for a taut string $\displaystyle v \propto
\sqrt\frac{\tau}{\mu}$; a more careful analysis from eq.(5.48) shows that

\begin{displaymath}
v=\sqrt\frac{\tau}{\mu}=f\lambda
\end{displaymath} (5.44)

So how should we picture the propagation of energy in a wave? For a wave traveling towards the right, the elemental mass $\mu \Delta x$ does work on the elemental mass $\mu \Delta x$ to its right and the energy $\Delta E$ 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.

Superposition of Waves

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 $y_1(t,x)$ and $y_2(t,x)$. The resultant wave, denoted by $y(t,x)$ is the resultant wave. The superposition principle states that the resultant wave is simply the sum of the two waves. That is
$\displaystyle \mbox{\rm {Resultant wave}}$ $\textstyle =$ $\displaystyle \mathrm{Wave 1}+\mathrm{Wave 2}$ (5.45)
$\displaystyle \Rightarrow y_R(t,x)$ $\textstyle =$ $\displaystyle y_1(t,x)+y_2(t,x)$ (5.46)

The resultant wave $y_R(t,x)$ 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
$\displaystyle I \propto y^2_R(t,x)$ $\textstyle =$ $\displaystyle (y_1(t,x)+y_2(t,x))^2$ (5.47)
  $\textstyle \neq$ $\displaystyle y_1^2(t,x)+y_2^2(t,x)$ (5.48)

Note the above equation is a characteristic feature of wave behaviour, and will be of great significance in our discussion on quantum mechanics. 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
$\displaystyle y_1(t,x)$ $\textstyle =$ $\displaystyle A \sin(P+\phi_1)$ (5.49)
$\displaystyle \mathrm{where}$      
$\displaystyle P$ $\textstyle =$ $\displaystyle 2 \pi \frac{x}{\lambda}-2\pi f t$ (5.50)
  $\textstyle =$ $\displaystyle k(x-vt)$ (5.51)

and the second wave be denoted by
\begin{displaymath}
y_2(t,x)=A \sin(P+\phi_2)
\end{displaymath} (5.52)

The phases $\phi_1$ and $\phi_2$ indicate the different starting shapes of the two waves. 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.

Figure 5.10: Interference: The Superposition of Waves
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The net resultant wave is given, as in eq.(5.51), by the sum of the two waves, namely
$\displaystyle y_R(t,x)$ $\textstyle =$ $\displaystyle y_1(t,x)+y_2(t,x)$ (5.53)
  $\textstyle =$ $\displaystyle A \sin(P+\phi_1)+
A \sin(P+\phi_2)$ (5.54)

From the rules of trigonometry, the sum of two sines is given by
\begin{displaymath}
\sin(A)+\sin(B)=2\sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})
\end{displaymath} (5.55)

Hence, from eq.(5.59), we have for the resultant wave
$\displaystyle y_R(t,x)$ $\textstyle =$ $\displaystyle [2 A\cos(\frac{\phi_1-\phi_2}{2})]
\sin(P+\frac{\phi_1+\phi_2}{2})$ (5.56)
  $\textstyle \equiv$ $\displaystyle A_R\sin(P+\phi_R)$ (5.57)
$\displaystyle \mathrm{where}$      
$\displaystyle A_R$ $\textstyle =$ $\displaystyle 2 A\cos(\frac{\phi_1-\phi_2}{2})$ (5.58)
$\displaystyle \phi_R$ $\textstyle =$ $\displaystyle \frac{\phi_1+\phi_2}{2}$ (5.59)

In general, the resultant wave $y_R(t,x)$ is a traveling wave, as can be seen from its dependence on $P$ There are two notable features of $y_R(t,x)$. (a) The intensity of the resultant is shown in Figure 5.10 wave, and is proportional to $A_R^2$, and depends on the phase and is given by phase difference, namely, is a function of $\displaystyle \phi_1-\phi_2$ (b)The phase of the resultant wave, $\phi_R$, is given by $\displaystyle \frac{\phi_1+\phi_2}{2}$. Two special cases for the phase difference are given below.
$\displaystyle \phi_1=\phi_2; \Rightarrow\phi_R=0$     (5.60)
$\displaystyle \Rightarrow y_R(t,x)$ $\textstyle =$ $\displaystyle 2 A \sin(2 \pi \frac{x}{\lambda}-2\pi ft)$ (5.61)
$\displaystyle \phi_1=\phi_2+\pi; \Rightarrow\phi_R=\pi$     (5.62)
$\displaystyle \Rightarrow y_R(t,x)$ $\textstyle =$ $\displaystyle 0$ (5.63)

In eq.(5.65) above the component waves are said to in phase for $\phi_1=\phi_2$, and interfere constructively yielding the resultant amplitude of $2A$, which is double of the amplitudes of the component waves. On the other hand, in eq.(5.67), for $\phi_1=\phi_2+\pi$ the resultant wave is zero, and is said to result from the destructive interference of the component waves. For arbitrary phases $\phi_1,\phi_2$ 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.

Standing Waves

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.

Figure 5.11: Standing Wave
\begin{figure}
\begin{center}
\epsfig{file=core/standwave1.eps, width=8cm}
\end{center}
\end{figure}

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
$\displaystyle y_{\mathrm{standing}}(t,x)$ $\textstyle =$ $\displaystyle A \sin[k(x-vt)]
+A \sin[k(x+vt)]$ (5.64)
  $\textstyle =$ $\displaystyle [2A\sin(kx)]\cos(2kvt)$ (5.65)
  $\textstyle =$ $\displaystyle [2A\sin(2\pi \frac{x}{\lambda})]\cos(2\pi f t)$ (5.66)

Note from eq.(5.71) that at the points $\displaystyle x=0,\frac{\lambda}{2},\lambda,
\frac{3\lambda}{2}, 2\lambda, ... $, called the nodes of the standing wave, the medium is fixed and hence $y_{\mathrm{standing}}=0$ 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 $2A$ to the value of $0$ 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.

Figure 5.12: Creation of a Standing Wave
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\epsfig{file=core/waves.eps, width=6.845cm}
\end{center}
\end{figure}


Resonance

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 $\tau$. 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
$\displaystyle y_{\mathrm{resonance}}(t,x)=[2A\sin(2 \pi \frac{x}{\lambda})]\cos(2\pi f t)$     (5.67)

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 $L$, with one fixed end at $x=0$ and the other end at $x=L$. We then must have
$\displaystyle y_{\mathrm{resonance}}(t,0)=0$ $\textstyle =$ $\displaystyle y_{\mathrm{resonance}}(t,L)$ (5.68)
$\displaystyle \Rightarrow \sin(2 \pi \frac{L}{\lambda})$ $\textstyle =$ $\displaystyle 0$ (5.69)
$\displaystyle \Rightarrow \lambda$ $\textstyle =$ $\displaystyle \frac{2L}{n}, n=0,1,2,3...$ (5.70)

Hence, unlike the case for an arbitrary standing waves, the violin string can have only wavelengths which are equal to a multiple of $\displaystyle \frac{L}{2}$ 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 $\displaystyle \frac{L}{2}$ 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 $\lambda$ is related to frequency $f$ by eq.([*]), we have from eq.(5.49)
\begin{displaymath}
f=\frac{v}{\lambda}=n\frac{v}{2L}
\end{displaymath} (5.71)

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 $L$.

Figure 5.13: Resonance Wavelengths
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\epsfig{file=core/img7.eps, width=8cm}
\end{center}
\end{figure}

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 $r$, the allowed wavelengths for oscillations are then given by
$\displaystyle \lambda_{\mathrm{Bangle}}$ $\textstyle =$ $\displaystyle 2\pi r n$ (5.72)
$\displaystyle n$ $\textstyle =$ $\displaystyle 0,1,2,....$ (5.73)

A particular oscillation of the bangle (ring) is shown in Figure 5.14.

Figure 5.14: Wave motion in a metallic ring (bangle)
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\begin{center}
\epsfig{file=core/wave5.eps, height=6cm}
\end{center}
\end{figure}

*Beats

So far we have considered the superposition of waves that have the same frequency $f$. 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 $f_1$ and $f_2$. 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 $z_1$ and $z_2$, and have the following
$\displaystyle z_1(t)= A\cos(2\pi f t)$ $\textstyle ;$ $\displaystyle z_1(t)= A\cos(2\pi f t)$ (5.74)
$\displaystyle z(t)$ $\textstyle =$ $\displaystyle A\cos(2\pi f_1 t)+A\cos(2\pi f_2 t)$ (5.75)
  $\textstyle =$ $\displaystyle 2A\cos(2\pi\frac{f_1-f_2}{2}t)\cos(2\pi\frac{f_1+f_2}{2}t)$ (5.76)

To understand the physical phenomenon that the equation above refers to, suppose $\displaystyle f_1 \simeq f_2$; we then have
\begin{displaymath}
z(t)\simeq [2A \cos(2\pi\frac{f_1-f_2}{2}t)]\cos(2\pi\frac{f_1+f_2}{2}t)
\end{displaymath} (5.77)

Since the first term $ \cos(2\pi\frac{f_1-f_2}{2}t)$ is almost a constant the sound seems to have a frequency of $f_1+f_2$. However, the maximum amplitude is now $2A$, double of the input waves. The result is a sound which has a double the amplitude of $2A$, and is consequently much louder than the component waves, but with frequency $2f$. This waxing and waning of superposed sound wave is well known in music, and is used for tuning the frequencies of various instruments.


Doppler Effect

Consider a train that has a whistle which emanates a sound with frequency $f$. The sound of the train whistle received by an observer when the train is approaching has a higher frequency than $f$, 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 $T$ seconds. The source hence emits signals with a frequency of $\displaystyle
f=\frac{1}{T}$. In time $T$ the sound propagates a distance of $d$, 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 $\lambda=d$. The speed of propagation of sound wave $v$ is given by eq.(5.19) which relates, for all wave motion, the frequency and wavelength. Hence we have
$\displaystyle v$ $\textstyle =$ $\displaystyle f\lambda$ (5.78)
  $\textstyle =$ $\displaystyle \frac{\lambda}{T}$ (5.79)

Figure 5.15: Doppler Effect
\begin{figure}
\begin{center}
\input{core/figure50.eepic}
\end{center}
\end{figure}

Suppose the source is moving at a speed $w_s$; then, after emitting the first signal, it travels a distance of $D=w_sT$, 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 $d-D$, and hence the wavelength received by the observer is
$\displaystyle \lambda'$ $\textstyle =$ $\displaystyle d-D$ (5.80)
  $\textstyle =$ $\displaystyle \lambda-w_sT$ (5.81)
  $\textstyle =$ $\displaystyle \lambda-w_s\frac{\lambda}{v}$ (5.82)
  $\textstyle =$ $\displaystyle \lambda(1-\frac{w_s}{v})$ (5.83)

Hence, the frequency of sound from a moving source, and received by a stationary observer, is
$\displaystyle f_s'$ $\textstyle =$ $\displaystyle \frac{v}{\lambda'}$ (5.84)
  $\textstyle =$ $\displaystyle \frac{v}{\lambda(1-\frac{w_s}{v})}$ (5.85)
  $\textstyle =$ $\displaystyle \frac{v}{v-w_s}f$ (5.86)

Clearly, since $w_s>0$, we have from eq.(5.91) that $f'>f$; in other words the frequency is of higher pitch, as expected for a source of sound that is approaching an observer. What happens when the velocity of the source becomes equal to velocity of sound, that is, $w_s\rightarrow v$? From eq.(5.91) it seems that $f'\rightarrow \infty$; this divergence indicates the onset of a new phenomenon, namely, of shock waves, which is discussed below. A similar derivation for a source receding from an observer yields
\begin{displaymath}
f_s'=\frac{v}{v\pm w_s}f \mbox{\rm { :approaching: -; receding: +}}
\end{displaymath} (5.87)

For an observer moving with velocity $w_o$ towards, or away, from a source receives a frequency given by
\begin{displaymath}
f_o'=\frac{v\pm w_o}{v}f \mbox{\rm { :approaching: +; receding: -}}
\end{displaymath} (5.88)

Note the switch in the signs of the velocity term $w_s,w_o$ in going from a moving source to a moving observer.
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Marakani Srikant 2000-09-11