Waves: Class 11 Physics · Chapter 14

14.1 Introduction

Picture a packed cricket stadium during a tense IPL chase. Someone at one end jumps up with arms raised and sits down again, the next person copies a moment later, then the next, and a ripple of standing fans races all the way round the stands. Nobody leaves their seat. Each spectator only rises and sits, yet something clearly travels: a pattern of motion (Fig. 14.1).

Nature is full of the same trick. A leaf floating on a pond rocks up and down as ripples from a dropped stone sweep past, but it is not carried to the bank. A temple bell sets the air around it trembling, and the sound reaches you across the courtyard, yet no gust of air blows from the bell to your ear. A travelling disturbance of this kind, in which a pattern moves through a medium while the medium itself only jiggles about fixed positions, is called a wave (तरंग).

Waves carry energy and information, not matter. Think of a phone call to a cousin in Guwahati. Your voice makes sound waves; the phone's microphone turns them into an electrical signal; the phone sends that signal as radio waves to a tower; it may then travel as light pulses in an optical fibre, before the whole chain runs backwards at the other end. Every step hands the message from one kind of wave to another.

Waves come in three broad families. Mechanical waves (ripples, waves on strings, sound, the tremors of an earthquake) need a material medium; they travel because neighbouring parts of the medium push and pull on one another, so Newton's laws and the medium's elastic properties govern them. Electromagnetic waves (light, radio waves, X-rays) need no medium: starlight crosses near-empty space for centuries before it reaches us. In vacuum they all travel at the same speed, which since 1983 has been fixed exactly by the definition of the metre:

The third family, matter waves, belongs to quantum physics: electrons, protons and even whole atoms show wave behaviour. It sounds abstract, but electron microscopes rely on the waves of electrons to image viruses and the atomic structure of materials. You will meet electromagnetic waves in Class 12; this chapter is about mechanical waves.

How does a disturbance pass along a medium? Picture students standing shoulder to shoulder in a morning-assembly line. Give the first one a quick shove and each student bumps the next before swaying back to their place: the shove runs down the line while every student stays put. A real medium behaves like a chain of small masses joined by springs. In a solid the springs are the bonds between neighbouring atoms of the crystal. Displace one atom and its bonds pull it back while tugging on its neighbours.

Air has no bonds, but it still has springiness. Squeeze a thin slab of air and its density rises a little, so its pressure rises too. The extra pressure pushes on the slab ahead, which then gets squeezed, while the first slab relaxes and even overshoots into a thinner, low-pressure state. Pressure differences thus play the part of the springs, and a pattern of squeezed regions, compressions, and stretched regions, rarefactions, moves through the air. Energy is passed on through these elastic links between neighbouring bits of the medium. Waves in elastic media are therefore close cousins of the oscillators of Chapter 13: each bit of the medium oscillates, and the coupling between neighbours makes the oscillation travel.

14.2 Transverse and longitudinal waves

Mechanical waves are sorted by how the bits of the medium move compared with the direction in which the wave travels. In a transverse wave (अनुप्रस्थ तरंग) each bit moves at right angles to the direction of travel. In a longitudinal wave (अनुदैर्ध्य तरंग) each bit moves back and forth along the direction of travel. A slinky spring shows both (Fig. 14.2).

A rope tied to a gate shows the transverse kind clearly. Jerk your end up and back down once and a single hump, a pulse, runs off towards the gate. Move your hand up and down steadily, like a pendulum bob, and you launch a continuous harmonic (sinusoidal) wave: a train of evenly spaced crests and troughs. Whichever you do, a knot tied in the rope only rises and falls; it never moves towards the gate.

A wave can be pictured in two quite different ways, and you need both. A photograph taken at one instant catches the whole shape spread along the rope. A slow-motion video of a single knot shows how one place moves up and down as time passes. Section 14.3 packs both pictures into one formula.

Sound is the everyday longitudinal wave. Place a loudspeaker at the end of a long tube of air (Fig. 14.3). When the speaker cone jumps forward it crowds the nearby air into a compression; when it pulls back it leaves a rarefaction. A steady tone sends a regular train of compressions and rarefactions down the tube, and each thin layer of air moves to and fro along the tube as they pass.

Stand under a ceiling fan and you feel a breeze: air is being carried across the room. The fan's hum also reaches you, but as a wave, with no air making the trip. Likewise a river current carries water downstream, while a ripple crossing a still tank carries none. Waves like these, whose pattern moves on from one region of the medium to the next, are progressive (travelling) waves (प्रगामी तरंग), transverse or longitudinal.

Why can a stretched string carry transverse waves while the air around it cannot? Picture the medium as thin layers. When a transverse wave goes by, each layer is dragged sideways relative to its neighbours, which is a shear (Chapter 8). A solid pulls back against shear, so it can push the displaced layers back into line. A liquid or a gas simply lets its layers slide, so inside a fluid a transverse wave has no restoring force and dies at once. Squeezing is a different matter: solids, liquids and gases all push back when compressed. That is why longitudinal waves pass through every kind of medium, why a copper rod can carry both kinds, and why sound in air is purely longitudinal. The two kinds rely on different elastic constants, so even in the same solid they generally have different speeds.

Look closely at a cork bobbing on a lake and you will see that it also drifts forward on each crest and back in each trough, tracing a small loop (the third step of Fig. 14.2). Surface water waves therefore mix transverse and longitudinal motion, and the loops shrink with depth. What pulls the surface back level depends on the size of the wave. For tiny ripples less than about two centimetres long, surface tension wins; these are capillary waves. For the long waves of the open sea, from a metre or so to hundreds of metres, the Earth's gravity does the job; these are gravity waves.

14.3 Displacement relation in a progressive wave

One formula has to do two jobs. Read at a fixed time, it must give the shape of the whole string; read at a fixed place, it must give the motion of that bit of string. So it must depend on where you look, $x$, and on when you look, $t$. Let the string lie along the $x$-axis and let $y(x,t)$ be its sideways displacement. The simplest travelling wave, a harmonic one moving towards $+x$, is

Three readings of Eq. (14.2) show that it really is such a wave.

• Follow a crest. A crest is a point where the phase $kx-\omega t+\varphi$ has a particular value. As $t$ increases, $x$ must increase to keep that value, so the crest moves towards $+x$.
• Freeze the clock. At any fixed time $t_0$ the formula is $a\sin (kx+\text{constant})$: the snapshot is a sine curve.
• Watch one place. At any fixed $x_0$ it is $a\sin (\text{constant}-\omega t)$: that bit of string performs simple harmonic motion.
• Flip the sign in front of $\omega t$ and the crest must move the other way, giving a wave towards $-x$:

Table 14.1 collects the symbols, and Fig. 14.4 shows Eq. (14.2) in action: a crest (orange marker) moves steadily to the right, while the bead at $x=0$ only moves down, up and down again on the spot.

14.3.1 Amplitude and phase

The constant $a$ in front of the sine is the amplitude (आयाम): how far, at most, any bit of the medium ever strays from its rest position. Because $\sin$ never exceeds 1 in size, $y$ always stays between $+a$ and $-a$. We choose the sign convention so that $a$ itself is positive, while $y$ is positive above the axis and negative below. The highest points of a snapshot are the crests and the lowest are the troughs.

Think of each bit of string as going round a cycle, like the hand of a clock. The angle $kx-\omega t+\varphi$ says how far through its cycle the bit at $x$ is at time $t$; it is called the phase (कला) and is measured in radians. Once $a$ is known, the phase is all you need to find the displacement. Its value at the origin when the clock reads zero is $\varphi$, the initial phase angle. Shifting where we put $x=0$, or when we start the clock, changes $\varphi$; choosing them well makes $\varphi =0$, and we usually do so from now on.

14.3.2 Wavelength and angular wave number

Two points that are at the same stage of their cycle at the same instant are in phase: two neighbouring crests, say, or two neighbouring troughs. The shortest distance between points in phase is the wavelength $\lambda$ (तरंगदैर्ध्य). To find it from Eq. (14.2), put $\varphi =0$ and look at the snapshot at $t=0$:

Moving along $x$ by $2\pi \mathrm{/}k$ adds $2\pi$ to the angle and brings the sine back to the same value; so do $4\pi \mathrm{/}k$, $6\pi \mathrm{/}k$ and so on. The smallest of these repeat distances is the wavelength:

We call $k$ the angular wave number; some books call it the propagation constant. Its SI unit is the radian per metre; because the radian is a ratio, this is often written simply as ${\mathrm{m}}^{-1}$. Read $k$ as the phase change per metre: $2\pi$ for every wavelength that fits into one metre.

14.3.3 Period, angular frequency and frequency

Instead of a snapshot, now keep watching one place, as in the history graph of step 4 of Fig. 14.4. With $\varphi =0$, the bit at $x=0$ moves as $y(0,t)=a\sin (-\omega t)=-a\sin \omega t$. Its period $T$ is the time for one complete cycle, after which the motion repeats exactly: $\sin \omega t$ and $\sin (\omega t+\omega T)$ must agree for every $t$. The smallest $T$ that does this makes $\omega T$ equal to $2\pi$:

$\omega$ is the angular frequency of the wave, in $\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{-1}$. The frequency $\nu$ (Greek 'nu') counts oscillations per second, measured in hertz (Hz):

Nothing above depended on the wave being transverse. For a longitudinal wave the displacement of each bit is along the direction of travel, and we usually call it $s$ rather than $y$:

All the other symbols keep their meanings. For sound, $s(x,t)$ tells you how far the layer of air that normally sits at $x$ has been shifted forward or back.

14.4 The speed of a travelling wave

How fast does the pattern move? Pick out any feature of the wave, most conveniently a crest, and see how far it goes in a short time $\mathrm{\Delta }t$. In Fig. 14.4 the whole curve slides rigidly to the right, so every point of fixed phase moves with the same speed $v$; if different phases moved at different speeds the shape would not survive.

Read Eq. (14.12) as a statement about timing. While any one bit of the medium completes one cycle, the crest that was above it moves on to where the next crest was: one wavelength. This holds for every travelling wave, of any kind. The equation does not, however, say what the speed is. For mechanical waves the medium decides. Two properties compete: how stiffly the medium resists being deformed (tension in a string, or an elastic modulus: Young's, bulk or shear) and how much inertia it has (mass per metre of string, or density). Once the medium has fixed $v$, choosing a frequency fixes the wavelength. A medium resists shear and compression with different stiffness, which is why transverse and longitudinal waves in it usually differ in speed.

14.4.1 Speed of a transverse wave on a stretched string

What could the speed of a wave on a string depend on? The pull that straightens a kinked string is the tension $T$, so a tighter string should carry waves faster. What has to be set moving is the string's own mass, measured by the linear mass density $\mu =m\mathrm{/}L$, its mass per metre, so a heavier string should be slower. (In this subsection $T$ means tension; elsewhere in the chapter it is the period, and the context always makes clear which.) The full calculation with Newton's second law is beyond this book, but the dimensions of $T$ and $\mu$ almost settle the matter.

Two points about Eq. (14.14) are worth underlining (Fig. 14.6). First, it contains only the string and its tension; the tension is set from outside, by the pegs of a sitar or a hanging weight, not by the material alone. Second, it contains neither the wavelength nor the frequency, so on a given string waves of every frequency travel at the same speed. The frequency is chosen by whatever shakes the string, and the string then forces the wavelength to fit:

Not every medium is so obliging. Where the speed itself changes with frequency, as for light in glass or for waves on deep water, the medium is called dispersive; a pulse, being a blend of many frequencies, then spreads out as it goes.

14.4.2 Speed of a longitudinal wave (speed of sound)

In a longitudinal wave the layers of the medium are alternately squeezed and stretched, so the elastic property that matters is how strongly the medium resists a change of volume. That is the bulk modulus of Chapter 8:

The minus sign makes $B$ positive, because an increase of pressure reduces the volume. $B$ has the dimensions of pressure and is measured in pascals. The inertia is now the density $\rho$, $[\mathrm{M}{\mathrm{L}}^{-3}]$.

For a long thin bar the sides are free to bulge or shrink slightly, so the bar is effectively under a simple longitudinal stress and strain, and the relevant modulus is Young's modulus $Y$. The same dimensional argument and the exact theory give

Table 14.2 lists typical speeds. The rod values come straight from Eq. (14.20) with handbook values of $Y$ and $\rho$; in a large block of the same metal, where the material cannot bulge sideways, the longitudinal speed is somewhat higher.

Liquids and solids are far denser than gases, so why is sound faster in them? Because they are harder still to compress. Compare water and air. Water's bulk modulus, about $2.2\times {10}^9\mathrm{P}\mathrm{a}$, is roughly $15000$ times the effective bulk modulus of air, while water is only about $780$ times denser. The ratio $B\mathrm{/}\rho$ is therefore about $20$ times larger, and the speed about $\sqrt{20}\approx 4.5$ times larger: roughly $1480\mathrm{m}\mathrm{/}\mathrm{s}$ against $331\mathrm{m}\mathrm{/}\mathrm{s}$.

For a gas we can predict the bulk modulus. Newton made the first attempt, and the way it went wrong is instructive.

Try it on dry air at $0\mathrm{°}\mathrm{C}$ and normal pressure, $P=1.013\times {10}^5\mathrm{P}\mathrm{a}$. With the molar mass of air, $M=2.897\times {10}^{-2}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}$, the ideal gas law gives $\rho =PM\mathrm{/}RT=1.29\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^3$, and Newton's formula predicts

Careful measurements give $331\mathrm{m}\mathrm{/}\mathrm{s}$. Newton's value is about $15\mathrm{\%}$ too low, far beyond experimental error. The flaw lay in the assumption, not the algebra (Fig. 14.7). The compressions and rarefactions of a sound wave alternate hundreds of times a second, and the regions involved are centimetres to metres across. Heat cannot flow out of a compressed region, or into a rarefied one, in such a short time. So the compressions warm the gas slightly and the rarefactions cool it: the changes are adiabatic, not isothermal. A gas squeezed adiabatically resists more strongly, because its rising temperature adds to the rise in pressure.

Air is mostly diatomic, with $\gamma =1.40$, so Laplace's formula multiplies Newton's estimate by $\sqrt{1.40}=1.18$: $280\mathrm{m}\mathrm{/}\mathrm{s}\times 1.18\approx 331\mathrm{m}\mathrm{/}\mathrm{s}$, in excellent agreement with experiment.

14.5 The principle of superposition of waves

Two friends at opposite ends of a long slinky each send a pulse at the same moment. In the middle the pulses meet. You might expect a collision, with both pulses bouncing back or wrecking each other, but they simply pass through one another and continue, each with its original shape and speed. While they overlap the slinky takes a new shape, and that shape is easy to predict: at every point, add the displacements the two pulses would cause separately (Fig. 14.8). This rule is the principle of superposition.

Signs matter in the sum. If a crest-shaped pulse meets an identical trough-shaped pulse travelling the other way, there is a moment when they overlap exactly and the string is perfectly straight. The energy has not vanished: at that instant every bit of the string is moving fast, so all the energy is kinetic. A moment later the pulses reappear and move apart. This principle is the key to interference: the combined effect of overlapping waves.

Now let two sources send harmonic waves of the same frequency down the same string, one of them starting a fraction of a cycle later than the other. Both waves then have the same $\omega$, the same $k$ and so the same wavelength and speed, and we take their amplitudes $a$ to be equal. The only difference is a constant phase difference $\varphi$. What does the string do?

Two special cases stand out (Fig. 14.9). When the waves are in phase, $\varphi =0$ (or any whole multiple of $2\pi$), crest meets crest and

the largest possible amplitude, $2a$: constructive interference. When they are exactly out of phase, $\varphi =\pi$, crest meets trough everywhere and

at all points and all times: destructive interference. Between these extremes the amplitude takes every value from $2a$ down to zero; for example $\varphi =2\pi \mathrm{/}3$ gives $2a\cos {60}^{\circ }=a$.

14.6 Reflection of waves

Real media end somewhere. Shout towards a cliff across a valley and the cliff sends your voice back as an echo: sound reflected from a rigid boundary. Knot a light string to a heavy rope and shake the string: part of each wave carries on into the rope and part comes back along the string. In general, a boundary between two media reflects some of a wave and transmits the rest. When the wave strikes the boundary at an angle, the transmitted (refracted) wave bends according to Snell's law and the reflected wave follows the laws of reflection.

Tie a string firmly to a wall and send an upward pulse along it (Fig. 14.10, top). If the wall absorbs none of its energy, the pulse returns with its shape intact but turned upside down. We say it has undergone a phase change of $\pi$ (that is, ${180}^{\circ }$). Think about the forces first. As the pulse arrives it lifts the last bit of string, which tugs upward on the wall; the wall, by Newton's third law, tugs the string downward, and that downward tug is the inverted pulse that runs back. The same answer follows from superposition: the tied end can never move, so whatever comes back must cancel the incoming displacement there at every instant, which only an inverted copy can do.

The opposite case is an end that is completely free. Textbooks model it as a light ring that slides without friction on a smooth vertical rod (Fig. 14.10, bottom). Nothing pulls the ring back down, so the pulse throws it up to about twice its own height, and the reflection comes back the right way up: no phase change. For sound, the open end of a flute or an organ pipe behaves like a free end for the displacement of the air.

To write this down, let a harmonic wave $y_i=a\sin (\omega t-kx)$ (a wave along $+x$) arrive from the left at a boundary at $x=0$. The reflected wave travels along $-x$ and, for the two kinds of end,

At $x=0$ the rigid case gives $y_i+y_r=a\sin \omega t-a\sin \omega t=0$ at all times, exactly the condition of a fixed end. The free case gives $2a\sin \omega t$: the end oscillates with twice the amplitude. (Books write these with different sign conventions; the physics is the same: the sum vanishes at a rigid end and doubles at a free one.)

14.6.1 Standing waves and normal modes

On a guitar string, or in the air inside a flute, there are two boundaries, and a wave is reflected back and forth between them many times. Within a fraction of a second the string holds two wave trains moving in opposite directions, and their sum settles into a steady pattern that seems to go nowhere: a standing wave or stationary wave (अप्रगामी तरंग).

In Eq. (14.37) position and time have separated: one factor depends only on $x$, the other only on $t$, and the moving combination $kx-\omega t$ has gone. Read it as an oscillation $\cos \omega t$, the same for every bit of string, multiplied by an amplitude $\mathrm{\mid }2a\sin kx\mathrm{\mid }$ that is fixed for each bit but changes along the string. Some points, where $\sin kx=0$, never move at all: these are the nodes (निस्पंद). Midway between them the swing is largest: the antinodes (प्रस्पंद). All bits between two neighbouring nodes rise and fall together, while the next section of string moves the opposite way, because $\sin kx$ changes sign at a node. The crests do not march along, hence 'standing'. And since a node never moves, no work is done across it: energy sloshes back and forth within each loop but is not carried along the string.

The nodes are where $\sin kx=0$, that is $kx=n\pi$. With $k=2\pi \mathrm{/}\lambda$:

The antinodes are where $\mathrm{\mid }\sin kx\mathrm{\mid }=1$, that is $kx=(n+\frac{1}{2})\pi$:

Neighbouring nodes are $\lambda \mathrm{/}2$ apart, neighbouring antinodes are $\lambda \mathrm{/}2$ apart, and each antinode lies midway between two nodes, $\lambda \mathrm{/}4$ from each.

Here is the payoff. On an endless string you can send waves of any frequency you like. A string tied down at $x=0$ and at $x=L$ is far choosier, because both ends must be nodes. The end at $x=0$ already is one in Eq. (14.37). For $x=L$ to be a node as well, Eq. (14.38) says the length must hold a whole number of half-wavelengths:

So the allowed wavelengths and frequencies (with $\nu =v\mathrm{/}\lambda$, and $n=1,2,3,\dots$) are

These natural frequencies are the normal modes of the string. The lowest, ${\nu }_1=v\mathrm{/}2L$, is the fundamental mode or first harmonic; $n=2$ gives the second harmonic, $n=3$ the third, and so on, each a whole-number multiple of the fundamental (Fig. 14.12). Here $v=\sqrt{T\mathrm{/}\mu }$ is fixed by the string, which is why tightening a string (larger $T$) or shortening it with a finger (smaller $L$) raises every one of its frequencies.

Pluck a sitar string and you do not get one pure mode. You get many at once, the fundamental and a family of harmonics, each with its own strength. Which harmonics are strong depends on where the string is plucked or bowed: any harmonic that has a node at that point is barely excited. Players use this deliberately, and the resulting blend of harmonics is a large part of what makes a veena, a violin and a sitar sound different on the same note.

Air columns have normal modes too. Blow across the mouth of a test tube or a narrow bottle and you hear the air column inside, closed at the bottom and open at the top. At the closed end the air is pinned against the glass, so the displacement there is always zero (a displacement node); but that layer is squeezed and stretched by the air above it more than anywhere else, so the pressure change is largest there (a pressure antinode). At the open end the reverse holds: the air moves freely and the pressure stays close to that of the room, a displacement antinode and pressure node. With the closed end at $x=0$, a node is already in place, and Eq. (14.39) then asks for an antinode at $x=L$:

so the allowed wavelengths and the normal-mode frequencies, for $n=0,1,2,\dots$, are

The fundamental ($n=0$) is $v\mathrm{/}4L$: a quarter-wavelength fits in the tube. The higher modes are $3v\mathrm{/}4L$, $5v\mathrm{/}4L$ and so on, the odd harmonics only, called the third, fifth, … harmonics. A pipe open at both ends has displacement antinodes at both ends; a whole number of half-wavelengths fits between them, so it has every harmonic, ${\nu }_n=nv\mathrm{/}2L$, just like a string (Fig. 14.13).