Chapter 6: Superposition of Waves
In the late eighteenth century, a German physicist named Ernst Chladni performed an experiment that looked more like art than physics. He took a thin metal plate, clamped it at its centre, and sprinkled fine dry sand evenly over its surface. Then he drew a violin bow along the edge of the plate.
The plate sang out with a clear note — and something remarkable happened to the sand. Instead of scattering randomly, the grains slid across the surface and settled into a sharp, symmetric pattern of curves and straight lines. Change the bowing point, and the note changed; change the note, and an entirely new pattern appeared.
Chladni had made something invisible visible. The plate was not vibrating uniformly. Waves were travelling across it, reflecting from its edges, and returning to cross the waves still coming outward. Where the two sets of waves cancelled each other, the plate stayed perfectly still, and the sand — shaken away from everywhere else — piled up along exactly those motionless lines.
That single observation contains the whole idea of this chapter. Waves do not collide and stop. They pass through one another, add up point by point, and the result can be enormous motion in one place and absolute stillness a centimetre away. Understanding that addition explains musical instruments, tuning, noise cancellation, and much more.
Diagram
Fig. 6.0 – A clamped square metal plate being bowed at its edge, with sand grains gathered along curved motionless lines on its surface.
As you work through this chapter, you will find the physics behind each of these questions, and by the end you will be able to answer all of them yourself.
6.1 Introduction
You already meet waves every day — ripples spreading on the surface of water, sound reaching you from a speaker, and light arriving from a lamp. Water waves, sound waves, light waves, mechanical waves and electromagnetic waves all look very different, yet they share one basic behaviour: a disturbance created at one place travels to another place.
Mechanical Waves and the Role of the Medium
Some waves need a material medium to travel through. A medium is called elastic if its particles, when displaced from their normal (mean) positions, experience restoring forces that pull them back. This elasticity is what allows one particle to disturb the next.
A mechanical wave is a disturbance produced in such an elastic medium by the periodic vibration of its particles about their respective mean positions. Each particle vibrates in place, pushes or pulls its neighbour, and the disturbance moves on.
In this process, energy and momentum are handed from one particle to the next. So a wave carries or transfers energy from one point to another. What it does not do is carry matter: the particles of the medium in which the wave travels are not transported along with the wave.
The cork’s behaviour is easiest to picture when it is drawn alongside the travelling ripple, as shown in Fig. 6.1a.
Diagram
Fig. 6.1a – A ripple travelling across a water surface with a floating cork bobbing vertically in place, arrows showing wave direction horizontal and cork motion vertical.
Waves That Need No Medium
Another type of wave, known as an electromagnetic wave, does not require a material medium for its propagation. Such waves are therefore called non-mechanical waves. Light from the Sun reaching us across empty space is the everyday proof of this.
What This Chapter Will Cover
In earlier classes you have studied sound waves, which are mechanical waves. You have also studied their properties and related phenomena such as echo, reverberation and the Doppler effect.
In this chapter we take that understanding further. We will study mechanical waves and their reflection, the principle of superposition of waves, and the phenomena that follow from that principle — the formation of stationary waves and beats — along with their applications in instruments and technology.
6.2 Progressive Wave
Have you seen the ripples created on the surface of water when a stone is dropped into it? Watching that pattern closely tells you almost everything about how a disturbance travels.
The water is displaced locally, only at the spot where the stone actually falls into it. From there the disturbance slowly spreads outward, and particles that were far away and at rest gradually get disturbed as well, as shown in Fig. 6.2a.
Diagram
Fig. 6.2a – Circular ripples spreading outward from the point where a stone strikes a water surface, with a magnified inset showing one particle oscillating vertically about its rest position.
Notice what happens to any one particle. The wave disturbs it only for a short duration as it passes. The particle oscillates about its position of rest for that short time and then settles again. It is not bodily moved from its position and carried along with the ripple.
A Wave Pulse
The disturbance caused by the stone is a wave pulse. A pulse is a disturbance caused locally, for a short duration — a single travelling hump rather than a continuous train of waves.
From a Pulse to a Progressive Wave
Now suppose the source keeps disturbing the medium instead of striking it once. The disturbance then travels onward continuously in a definite direction. If nothing weakens it and nothing blocks its path, the wave is called a progressive wave or a travelling wave.
Two conditions in that description are worth pausing on. “Without any damping” means the wave does not gradually lose amplitude to resistive forces of the medium. “Without obstruction” means no boundary or obstacle interrupts its onward journey. Both are idealisations, but they let us describe the wave with a single, clean equation.
The sound wave is a familiar example of a progressive wave. It is a pressure wave, consisting of compressions and rarefactions, which travel along the direction of propagation of the wave.
6.2.1 Properties of Progressive Waves
A progressive wave behaves in a very regular way, and the following properties describe that behaviour completely. Read them as a description of what each particle of the medium is doing while the wave passes over it.
Each particle in the medium executes the same type of vibration. The particles vibrate about their mean positions performing simple harmonic motion.
All the vibrating particles of the medium have the same amplitude, the same period and the same frequency.
The phase — that is, the state of vibration of a particle — changes from one particle to another. This is the one quantity that is not shared: each particle starts its motion a little later than the one before it, and that lag is what makes the disturbance appear to move forward.
No particle remains permanently at rest. Each particle comes to rest only momentarily, while at the extreme positions of its vibration.
The particles attain maximum velocity when they pass through their mean positions.
During the propagation of the wave, energy is transferred along the wave. There is no transfer of matter.
The wave propagates through the medium with a certain velocity. This velocity depends upon the properties of the medium, such as its elasticity and density, and not upon how strongly the source was disturbed.
Progressive waves are of two types — transverse waves and longitudinal waves.
In a transverse wave, the vibrations of the particles are perpendicular to the direction of propagation of the wave, and they produce crests and troughs in their medium of travel. In a longitudinal wave, the vibrations of the particles are along the direction of propagation of the wave, and they produce compressions and rarefactions along the direction of propagation of the wave. The two patterns are compared in Fig. 6.2b.
Diagram
Fig. 6.2b – Side-by-side comparison of a transverse wave on a string showing crests and troughs, and a longitudinal wave in a spring showing compressions and rarefactions, with arrows marking particle vibration and wave direction in each.
- Both the transverse as well as the longitudinal mechanical waves can propagate through solids, but only longitudinal waves can propagate through fluids.
Describing a Travelling Disturbance Mathematically
You might recall that when a mechanical wave passes through an elastic medium, the displacement of any particle of the medium at a space point \(x\) at time \(t\) is given by the expression
\[y(x,t) = f(x - vt) \qquad \text{--- (6.1)}\]
where \(v\) is the speed at which the disturbance travels through the medium to the right, that is, towards increasing \(x\). Here \(y\) is the displacement of the particle from its mean position (in metre), \(x\) is the position of that particle along the direction of travel (in metre), and \(t\) is the time (in second).
Why should the combination \((x - vt)\) appear at all? Because the disturbance produced at the point \(x = 0\) at time \(t\) reaches the point \(x = x'\) on the right at time \((t + x'/v)\). Looking at the same fact the other way round, we say that the disturbance of the particle at time \(t\) at position \(x = x'\) actually originated on the left side at the earlier time \((t - x'/v)\).
So \((x - vt)\) is simply a bookkeeping device: it keeps track of the delay with which each point receives what the source did earlier. Thus, Eq. (6.1) represents a progressive wave travelling in the positive \(x\)-direction with a constant speed \(v\).
The function \(f\) depends on the motion of the source of disturbance. If the source of disturbance is performing simple harmonic motion, the wave is represented as a sine or cosine function of \((x - vt)\), multiplied by a term which will make \((x - vt)\) dimensionless. This last requirement matters because the argument of a sine must be a pure number, not a length.
Generally we represent such a wave by the following equation
\[y(x,t) = A\sin(kx - \omega t) \qquad \text{--- (6.2)}\]
where \(A\) is the amplitude of the wave, \(k = 2\pi/\lambda\) is the wave number, \(\lambda\) and \(\omega\) are the wavelength and the angular frequency of the wave, and \(v = \omega/k\) is the speed. The SI units of \(k\), \(\lambda\) and \(\omega\) are rad m\(^{-1}\), m and rad s\(^{-1}\) respectively.
If \(T\) is the time period of oscillation, then
\[n = \frac{1}{T} = \frac{\omega}{2\pi}\]
is the frequency of oscillation, measured in Hz (s\(^{-1}\)).
Read physically, Eq. (6.2) says that \(k\) tells you how much phase is gained per metre travelled along the wave, while \(\omega\) tells you how much phase is gained per second at a fixed point. Their ratio \(\omega/k\) is therefore metres per second — the wave speed.
If the wave is travelling to the left, i.e., along the negative \(x\)-direction, then the equation for the disturbance is
\[y(x,t) = A\sin(kx + \omega t) \qquad \text{--- (6.3)}\]
6.3 Reflection of Waves
A progressive wave does not always continue forward for ever. When a wave travelling through a medium reaches an interface — the boundary surface separating two different media — a certain part of the wave energy comes back into the same medium. The wave changes its direction of travel. This is called reflection of a wave from the interface, as shown in Fig. 6.3a.
Diagram
Fig. 6.3a – A wave approaching the boundary between two media, with one arrow showing the reflected part returning into the first medium and a smaller arrow showing the part transmitted into the second medium.
For sound in particular, reflection is the phenomenon in which the sound wave travelling from one medium to another comes back in the original medium, with slightly different intensity and energy.
Before looking at the examples, one piece of vocabulary is worth fixing. When we compare two media for a wave, the one that is heavier or offers more inertia to the wave — and in which the wave travels more slowly — is called the denser medium, while the lighter one is called the rarer medium. A rigid wall, a heavy string, or the closed end of a pipe behaves as a denser medium for a wave arriving at it.
What happens at the moment of reflection is not always the same. Sometimes the returning wave keeps its original form, and sometimes it comes back turned upside down. To understand the reflection of waves and to see when each of these occurs, we will consider three examples below.
6.3.1 Reflection of a Transverse Wave
The clearest way to study the reflection of a transverse wave is to send a single pulse along a string and watch what returns. Three simple arrangements are enough, and each one changes only the nature of the boundary at the far end.
Reflection from a Rigid Support
The reflected pulse in this arrangement is shown in Fig. 6.1.
Diagram
Fig. 6.1 – Reflection of a wave pulse sent as a crest from a rarer medium to a denser medium: a string AB fixed at rigid support B, upper part showing a crest moving right towards B, lower part showing a trough moving left away from B.
Reflection from a Free End
Diagram
Fig. 6.2 – Reflection of a wave pulse sent as a crest from a denser medium to a rarer medium: a string whose end B carries a frictionless ring on a vertical rod, showing an incident crest and a reflected crest.
Reflection at the Junction of Two Strings
Explaining What You Observed at a Fixed End
In the first arrangement, when the crest moves along the string towards B, it pulls the particles of the string in the upward direction. Each particle in turn tugs its neighbour upward, and that is how the crest advances.
Similarly, when the crest reaches B at the rigid support, it tries to pull the point B upwards. But being a rigid support, B remains at rest. It cannot move, so an equal and opposite reaction is produced on the string according to Newton’s third law of motion, and the string is pulled downwards.
That downward pull is itself a disturbance, and it travels back along the string. Thus a crest gets reflected as a trough (Fig. 6.1), or a trough gets reflected as a crest.
You have learnt in Std. X and Std. XI that there is a phase difference of \(\pi\) radian between the particles at a crest and at a trough. Turning a crest into a trough is therefore the same statement as shifting the phase by \(\pi\).
Explaining What You Observed at a Free End
In the second arrangement, we observe that when the crest reaches the point B, it pulls the ring upwards and causes the ring to move upward. The boundary is free to respond, so no opposing reaction of the earlier kind is set up.
The wave is seen to get reflected back as a crest, and no phase change occurs on reflection from a rarer medium (Fig. 6.2).
Explaining What You Observed at the Junction
In the third arrangement, we find that a crest travelling from the heavy string gets reflected as a crest from the lighter string. In other words, reflection at the surface when a wave is travelling from a denser medium to a rarer medium causes a crest to be reflected as a crest, as seen in Fig. 6.3 (a).
Diagram
Fig. 6.3 (a) – A crest travelling along a heavy string towards the junction O with a light string, and the reflected pulse returning along the heavy string as a crest.
But in the same arrangement, when a crest travels from the lighter string to the heavy string, the crest is reflected as a trough, and vice versa, as seen in Fig. 6.3 (b).
Diagram
Fig. 6.3 (b) – A crest travelling along a light string towards the junction O with a heavy string, and the reflected pulse returning along the light string as a trough.
The pattern behind all three arrangements is now clear. What decides the result is not the string alone, but which of the two media is denser. A wave arriving at a denser medium comes back inverted; a wave arriving at a rarer medium comes back erect. The rigid support and the frictionless ring are only the two extreme cases of this same rule.
6.3.2 Reflection of a Longitudinal Wave
A longitudinal wave carries no crests or troughs, so we must follow pressure instead of sideways displacement. In a longitudinal wave, a compression is a high pressure region while a rarefaction is a low pressure region. Reflection is then described by asking whether the returning pulse is still a high pressure region or has become a low pressure one.
Reflection from a Denser Medium
Consider a longitudinal wave travelling from a rarer medium to a denser medium.
When the compression reaches the denser medium, it tries to push the particles of that medium. But the energy of the particles in the rarer medium is not sufficient to compress the particles of the denser medium, which are heavier and more tightly bound.
According to Newton’s third law of motion, an equal and opposite reaction comes into play. As a result, the particles of the rarer medium get compressed instead, and this newly formed high pressure region travels backwards, as shown in Fig. 6.4.
Diagram
Fig. 6.4 – Reflection of a longitudinal wave from a denser medium, showing an incident compression at the boundary and the reflected compression returning into the rarer medium.
Thus, when the longitudinal wave travels from a rarer medium to a denser medium, a compression is reflected as a compression and a rarefaction is reflected as a rarefaction. There is no change of pressure phase during this reflection.
Reflection from a Rarer Medium
When a longitudinal wave travels from a denser medium to a rarer medium, a compression is reflected as a rarefaction, as shown in Fig. 6.5. Here reversal of pressure phase takes place, that is, the pressure phase changes by \(\pi\) radians.
Diagram
Fig. 6.5 – Reflection of a longitudinal wave from a rarer medium, showing an incident compression at the boundary and the reflected rarefaction returning into the denser medium.
The reason becomes clear if we follow the particles. When the compression reaches a rarer medium from a denser medium, it pushes the particles of the rarer medium. Due to this, the particles of the rarer medium get compressed and move forward, and a rarefaction is left behind at the boundary. Thus a compression gets reflected as a rarefaction. Similarly, a rarefaction gets reflected as a compression (Fig. 6.5).
6.4 Superposition of Waves
Suppose you wish to listen to your favourite music. Is it always possible, particularly when there are many other sounds from the surroundings disturbing you? How can the background sounds be blocked?
Of course, the mobile lover generation uses headphones and enjoys listening to its favourite music. But even then you cannot avoid the background sound completely. Why should that be so, when the earpiece is already pressed against the ear?
The answer lies in how two waves behave when they arrive at the same place. They do not block each other, and they do not bounce off each other. They simply add.
Two Waves at the Same Point
We know that sound waves are longitudinal waves propagating through an elastic medium. When two waves travelling through a medium cross each other, each wave travels in such a way as if there is no other wave present at all.
Each wave sets the particles of the medium into simple harmonic motion. Thus each particle of the medium is set into two simple harmonic motions at the same time, due to the two waves.
A particle, however, can only be at one place at one instant. So the total displacement of the particles, at any instant of time during the travelling of these waves, is the vector sum of the two displacements, as illustrated in Fig. 6.4a.
Diagram
Fig. 6.4a – Two waves arriving at a common point P from opposite sides, with the individual displacements drawn as arrows and the resultant displacement shown as their vector sum.
This happens according to the principle of superposition of waves.
As displacement is a vector, we must add the individual displacements by considering their directions. Two upward displacements of 3 mm and 2 mm give 5 mm, but an upward displacement of 3 mm together with a downward displacement of 2 mm gives only 1 mm. Ignoring the sign is the commonest error in numerical problems on this topic.
This principle applies to all types of waves, like sound waves, light waves, waves on a string, and so on, and we say that interference of waves has taken place.
Superposition is therefore not one more property of waves; it is the single idea from which the rest of this chapter grows. Adding two identical pulses gives us constructive and destructive interference, adding two identical waves travelling in opposite directions gives us stationary waves, and adding two waves of slightly different frequency gives us beats.
6.4.1 Superposition of Two Wave Pulses of Equal Amplitude and Same Phase Moving towards Each Other
Let us now put the principle of superposition to work. We consider the superposition of two wave pulses in two different ways, and the first of these is the case of two pulses that are on the same side of the string — both crests, say — travelling towards each other.
The propagation of the approaching wave pulses, their successive positions after every second, their superposition, and their propagation after superposition are shown in Figs. 6.6 (a) to 6.6 (g).
Diagram
Fig. 6.6 – Superposition of two wave pulses of equal amplitude and same phase moving towards each other, drawn as seven successive frames (a) to (g) at one-second intervals, individual pulses dashed and the resultant in a full line.
Suppose two waves cross each other between \(t = 2\) s and \(t = 4\) s, as shown in Figs. 6.6 (c), (d) and (e). Here the two wave pulses superpose, and the resultant displacement is equal to the sum of the displacements (full line) due to the individual wave pulses (dashed lines).
Because both pulses displace the particles in the same direction, the sum is larger than either pulse alone. This is constructive interference.
The displacements due to the wave pulses after crossing, at \(t = 5\) s and \(t = 6\) s, are shown in Figs. 6.6 (f) and (g). After crossing each other, both the wave pulses continue to maintain their individual shapes.
That last observation is worth noticing carefully. The tall combined hump seen during the crossing is temporary; it belongs to the string, not to the pulses. Neither pulse is enlarged, delayed or distorted by the encounter.
6.4.2 Superposition of Two Wave Pulses of Equal Amplitude and Opposite Phases Moving towards Each Other
The second case is the more surprising one. Here the two pulses are of equal amplitude but lie on opposite sides of the string — one is a crest and the other a trough — and they travel towards each other.
The propagation of the approaching wave pulses, their successive positions after every second, their superposition and propagation after superposition are shown in Fig. 6.7 (a) to Fig. 6.7 (e).
Diagram
Fig. 6.7 – Superposition of two wave pulses of equal amplitude and opposite phases moving towards each other, drawn as five successive frames (a) to (e) at one-second intervals, individual pulses dashed and the resultant in a full line.
These wave pulses superimpose at \(t = 2\) s, and the resultant displacement (full line) is zero, due to the individual displacements (dashed lines) differing in phase exactly by \(180^\circ\). This is destructive interference.
The displacement due to one wave pulse is cancelled by the displacement due to the other wave pulse when they cross each other (Fig. 6.7 (c)). For that instant the string looks perfectly straight, as though no wave were present on it anywhere.
After crossing each other, both the wave pulses continue and maintain their individual shapes. Cancellation, like reinforcement, lasts only as long as the overlap does.
6.4.3 Amplitude of the Resultant Wave Produced due to Superposition of Two Waves
So far we have added pulses picture by picture. We now do the same addition with algebra, for two continuous waves, and obtain a formula for the amplitude of the resultant wave.
Consider two waves having the same frequency but different amplitudes \(A_1\) and \(A_2\). Let these waves differ in phase by \(\phi\). The phase difference tells us how far behind one wave is in its cycle compared with the other; it arises, for instance, when the two waves have travelled different distances before meeting.
To keep the algebra clean we watch one fixed particle of the medium, the one at \(x = 0\). The displacement of each wave at \(x = 0\) is given as
\[y_1 = A_1 \sin \omega t\]
\[y_2 = A_2 \sin (\omega t + \phi)\]
where \(\omega\) is the angular frequency in rad s\(^{-1}\), \(t\) is the time in second, and the amplitudes \(A_1\) and \(A_2\) are in metre.
According to the principle of superposition of waves, the resultant displacement at \(x = 0\) is
\[y = y_1 + y_2\]
or,
\[y = A_1 \sin \omega t + A_2 \sin (\omega t + \phi)\]
Separating the Time Part from the Phase Part
Expanding the second term using \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\), we get
\[y = A_1 \sin \omega t + A_2 \sin \omega t \cos \phi + A_2 \cos \omega t \sin \phi\]
Collecting the \(\sin \omega t\) terms together,
\[y = \left(A_1 + A_2 \cos \phi\right)\sin \omega t + A_2 \sin \phi \, \cos \omega t\]
Notice the structure of this expression. It is a \(\sin \omega t\) term and a \(\cos \omega t\) term, each multiplied by a constant that does not involve time. Any such combination can itself be written as a single sine wave, and that is exactly what we now do.
If we write
\[A \cos \theta = A_1 + A_2 \cos \phi \qquad \text{--- (6.4)}\]
and
\[A \sin \theta = A_2 \sin \phi \qquad \text{--- (6.5)}\]
we get
\[y = A \cos\theta \sin \omega t + A \sin\theta \cos \omega t\]
\[\therefore \; y = A \sin(\omega t + \theta) \qquad \text{--- (6.6)}\]
This is the equation of the resultant wave. It has the same frequency as that of the interfering waves — superposition changes the amplitude and the phase, never the frequency. The constant \(\theta\) is the phase by which the resultant leads the first wave.
Finding the Resultant Amplitude
The resultant amplitude \(A\) is given by squaring and adding Eqs. (6.4) and (6.5).
\[A^2\cos^2\theta + A^2\sin^2\theta = \left(A_1 + A_2\cos\phi\right)^2 + \left(A_2 \sin\phi\right)^2\]
\[A^2 = A_1^{\,2} + 2A_1A_2\cos\phi + A_2^{\,2}\cos^2\phi + A_2^{\,2}\sin^2\phi\]
Since \(\cos^2\theta + \sin^2\theta = 1\) on the left and \(\cos^2\phi + \sin^2\phi = 1\) on the right,
\[\therefore \; A = \sqrt{A_1^{\,2} + 2A_1A_2\cos\phi + A_2^{\,2}} \qquad \text{--- (6.7)}\]
Equation (6.7) is worth reading physically. The resultant amplitude is not simply \(A_1 + A_2\); it depends on the phase difference through the middle term \(2A_1A_2\cos\phi\), which can be positive, zero or negative. Geometrically, the amplitudes add like two vectors inclined at an angle \(\phi\), as shown in Fig. 6.7a.
Diagram
Fig. 6.7a – Two arrows of lengths A1 and A2 inclined at angle phi added by the triangle law, with the resultant arrow labelled A and its inclination to A1 labelled theta.
Special Cases
1. When \(\phi = 0\), i.e., the waves are in phase, \(\cos\phi = 1\) and the resultant amplitude is
\[A = \sqrt{A_1^{\,2} + 2A_1A_2\cos 0 + A_2^{\,2}} = \sqrt{(A_1 + A_2)^2} = A_1 + A_2\]
The resultant amplitude is maximum when \(\phi = 0\).
If the amplitudes of the waves are equal, i.e., \(A_1 = A_2 = A_0\) (say), then the resultant amplitude is \(2A_0\).
2. When \(\phi = \pi\), i.e., the waves are out of phase, \(\cos\phi = -1\) and the resultant amplitude is
\[A = \sqrt{A_1^{\,2} + 2A_1A_2\cos \pi + A_2^{\,2}} = \sqrt{(A_1 - A_2)^2} = |A_1 - A_2|\]
The resultant amplitude is minimum when \(\phi = \pi\).
If the amplitudes of the waves are equal, i.e., \(A_1 = A_2 = A_0\) (say), then the resultant amplitude is zero.
Thus, the maximum amplitude is the sum of the two amplitudes when the phase difference between the two waves is zero, and the minimum amplitude is the difference of the two amplitudes when the phase difference between the two waves is \(\pi\).
These are exactly the two situations you met as pictures in the previous two subsections, now obtained from a single formula.
From Amplitude to Intensity
What the ear or the detector actually registers is intensity, not amplitude. The intensities of the waves are proportional to the squares of their amplitudes.
Hence, when \(\phi = 0\),
\[I_{\max} \propto (A_{\max})^2 \propto (A_1 + A_2)^2 \qquad \text{--- (6.8)}\]
and when \(\phi = \pi\),
\[I_{\min} \propto (A_{\min})^2 \propto (A_1 - A_2)^2 \qquad \text{--- (6.9)}\]
Therefore intensity is maximum when the two waves interfere in phase, while intensity is minimum when the two waves interfere out of phase.
You will learn more about superposition of waves in Chapter 7 on Wave Optics.
6.5 Stationary Waves
We have seen the superposition of two wave pulses, having the same amplitudes and either the same phase or opposite phases, and the changes in the resultant amplitude, pictorially in Section 6.4. We have also derived the mathematical expression for the resultant displacement when two waves of the same frequency superimpose, as given by Eqs. (6.4) to (6.6).
In both of those studies the two disturbances met once, crossed, and moved apart. Suppose instead that the overlap never ends — that two waves of the same amplitude and the same frequency occupy the same stretch of medium continuously, but travel in opposite directions. This is exactly what happens whenever a wave is sent along a bounded medium, because the reflected wave returns through the incoming wave, as you saw in Section 6.3.
Now we are going to study this example of superposition of waves having the same amplitude and the same frequency travelling in opposite directions. The result is a pattern that does not travel at all, and it is called a stationary wave.
Before taking up the mathematics, work through the following example, which uses only the ideas of period and wavelength from Section 6.2.1 and prepares the ground for what follows.
6.5.1 Formation of Stationary Waves
Imagine a string stretched between two fixed points. If the string is pulled at the middle and released, we get what is known as a stationary wave.
Why should a single pull produce two waves? Because the disturbance created at the middle has nowhere to go except outwards along the string, in both directions at once. Releasing of the string therefore produces two progressive waves travelling in opposite directions.
These waves then travel to the two ends, where they cannot continue further. They are reflected at the fixed ends, and the reflected waves travel back into the string through the waves still coming towards them.
The waves produced in the string initially and their reflected waves combine to produce stationary waves, as shown in Fig. 6.8 (a).
Diagram
Fig. 6.8 (a) – Formation of stationary waves on a string fixed at both ends, with two-sided arrows indicating the motion of the particles of the string.
Notice the two-sided arrows drawn in Fig. 6.8 (a). They are there to remind you that the particles of the string are genuinely moving up and down; it is only the pattern of that motion — the location of the still points and the location of the violently moving points — that stays fixed.
6.5.2 Equation of Stationary Wave on a Stretched String
Consider two simple harmonic progressive waves of equal amplitudes (\(a\)) and wavelength (\(\lambda\)) propagating on a long uniform string in opposite directions. Remember from Section 6.2.1 that \(2\pi/\lambda = k\) and \(2\pi n = \omega\), so the two ways of writing a wave equation used here and there are the same thing in different dress.
The equation of the wave travelling along the \(x\)-axis in the positive direction is
\[y_1 = a \sin 2\pi\left(nt - \frac{x}{\lambda}\right) \qquad \text{--- (6.10)}\]
The equation of the wave travelling along the \(x\)-axis in the negative direction is
\[y_2 = a \sin 2\pi\left(nt + \frac{x}{\lambda}\right) \qquad \text{--- (6.11)}\]
Here \(a\) is the amplitude of each individual wave in metre, \(n\) is the frequency in hertz, \(\lambda\) is the wavelength in metre, \(x\) is the position of a particle along the string in metre and \(t\) is the time in second.
Adding the Two Waves
When these waves interfere, the resultant displacement of the particles of the string is given by the principle of superposition of waves as
\[y = y_1 + y_2\]
\[y = a \sin 2\pi\left(nt - \frac{x}{\lambda}\right) + a \sin 2\pi\left(nt + \frac{x}{\lambda}\right)\]
By using
\[\sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)\]
with \(C = 2\pi\left(nt - \dfrac{x}{\lambda}\right)\) and \(D = 2\pi\left(nt + \dfrac{x}{\lambda}\right)\), the half-sum leaves only the time term and the half-difference leaves only the position term. We get
\[y = 2a \sin 2\pi n t \, \cos \frac{2\pi x}{\lambda}\]
or,
\[y = 2a \cos \frac{2\pi x}{\lambda} \, \sin 2\pi n t \qquad \text{--- (6.12)}\]
Using \(2a \cos \dfrac{2\pi x}{\lambda} = A\) in Eq. (6.12), we get
\[y = A \sin (2\pi n t)\]
As \(\omega = 2\pi n\), we get
\[y = A \sin \omega t\]
This is the equation of a stationary wave, which gives the resultant displacement due to two simple harmonic progressive waves.
Why This Wave Does Not Travel
It may be noted that the terms in position \(x\) and time \(t\) appear separately, and not as a combination \(2\pi(nt \pm x/\lambda)\).
Hence, the wave is not a progressive wave. In a progressive wave the two variables were locked together inside one bracket, and that locking was exactly what made the pattern march forward. Here they have come apart.
The variable \(x\) is present only in the expression for the amplitude. The amplitude of the resultant wave is given as
\[A = 2a \cos \frac{2\pi x}{\lambda}\]
It is a periodic function of \(x\), i.e., the amplitude is varying periodically in space.
The amplitudes are different for different particles, but each point on the string oscillates with the same frequency \(\omega\) — the same as that of the individual progressive wave. All the particles of the string pass through their mean positions simultaneously twice during each vibration.
The string as a whole is vibrating with frequency \(\omega\) with different amplitudes at different points. The wave is not moving either to the left or to the right. We therefore call such a wave a stationary wave or a standing wave.
The particles move so fast that the visual effect is the formation of loops. It is therefore customary to represent stationary waves as loops. In the case of a string tied at both the ends, loops are seen when a stationary wave is formed, because each progressive wave on a string is a transverse wave.
Condition for Node
Nodes are the points of minimum displacement. This is possible if the amplitude is minimum (zero), i.e.,
\[2a \cos \frac{2\pi x}{\lambda} = 0\]
Since \(2a\) is not zero,
\[\cos \frac{2\pi x}{\lambda} = 0\]
or,
\[\frac{2\pi x}{\lambda} = \frac{\pi}{2}, \; \frac{3\pi}{2}, \; \frac{5\pi}{2}, \; \ldots\]
\[\therefore \; x = \frac{\lambda}{4}, \; \frac{3\lambda}{4}, \; \frac{5\lambda}{4}, \; \ldots\]
i.e.,
\[x = (2p - 1)\frac{\lambda}{4} \quad \text{where } p = 1, 2, 3, \ldots\]
Subtracting any one of these positions from the next shows that the distance between two successive nodes is \(\dfrac{\lambda}{2}\).
Condition for Antinode
Antinodes are the points of maximum displacement, i.e.,
\[A = \pm 2a\]
\[\therefore \; 2a \cos \frac{2\pi x}{\lambda} = \pm 2a\]
or,
\[\cos \frac{2\pi x}{\lambda} = \pm 1\]
\[\therefore \; \frac{2\pi x}{\lambda} = 0, \; \pi, \; 2\pi, \; 3\pi, \ldots\]
or,
\[x = 0, \; \frac{\lambda}{2}, \; \lambda, \; \frac{3\lambda}{2}, \; \ldots\]
i.e.,
\[x = p\frac{\lambda}{2} \quad \text{where } p = 0, 1, 2, 3, \ldots\]
The distance between two successive antinodes is \(\dfrac{\lambda}{2}\).
Nodes and antinodes are formed alternately. Therefore, the distance between a node and an adjacent antinode is \(\dfrac{\lambda}{4}\).
The Motion of the String at Different Instants
The equation also tells us what the whole string looks like at a chosen instant. When \(\sin \omega t = 1\), at that instant of time, all the particles for which \(\cos kx\) is positive have their maximum displacement in the positive direction. At the same instant, all the particles for which \(\cos kx\) is negative have their maximum displacement in the negative direction.
When \(\sin \omega t = 0\), all the particles cross their mean positions, some of them moving in the positive direction and some in the negative direction. At that instant the string is momentarily straight, though it is far from being at rest.
Stationary Waves in a Pipe
Longitudinal waves, for example sound waves travelling in a tube or pipe of finite length, are reflected at the ends in the same way as transverse waves along a string are reflected at the fixed ends.
Interference between these waves travelling in opposite directions gives rise to standing waves, as shown in Fig. 6.8 (b).
Diagram
Fig. 6.8 (b) – A longitudinal stationary wave in a pipe closed at one end, drawn on the left in the conventional loop representation and on the right as the actual back-and-forth oscillation of the material particles, with antinodes marked A and nodes marked N.
We represent a longitudinal stationary wave by a loop, but the actual motion of the particles is along the length of the loop and not perpendicular to it. The loop is only a convenient way of drawing how large the vibration is at each place; it is not a picture of the path of any particle.
6.5.3 Properties of Stationary Waves
The equation derived in the previous subsection contains all the behaviour of a stationary wave. The following properties simply state that behaviour in words, so that it can be recalled and applied quickly.
Stationary waves are produced due to the superposition of two identical waves — either transverse or longitudinal waves — travelling through a medium along the same path in opposite directions.
If two identical transverse progressive waves superimpose or interfere, the resultant wave is a transverse stationary wave, as shown in Fig. 6.8 (a).
- When a transverse stationary wave is produced on a string, some points on the string are motionless. The points which do not move are called nodes.
- There are some points on the string which oscillate with the greatest amplitude (say \(A\)). They are called antinodes.
- Points between the nodes and antinodes vibrate with values of amplitudes between 0 and \(A\).
- If two identical longitudinal progressive waves superimpose or interfere, the resultant wave is a longitudinal stationary wave. Figure 6.8 (b) shows a stationary sound wave produced in a pipe closed at one end.
- The points at which the amplitude of the particles of the medium is minimum (zero) are called nodes.
- The points at which the amplitude of the particles of the medium is maximum (say \(A\)) are called antinodes.
- Points between the nodes and antinodes vibrate with values of amplitudes between 0 and \(A\).
The distance between two consecutive nodes is \(\dfrac{\lambda}{2}\), and the distance between two consecutive antinodes is \(\dfrac{\lambda}{2}\).
Nodes and antinodes are produced alternately. The distance between a node and an adjacent antinode is \(\dfrac{\lambda}{4}\).
The amplitude of vibration varies periodically in space. All points vibrate with the same frequency.
Though all the particles (except those at the nodes) possess energy, there is no propagation of energy. The wave is localised and its velocity is zero. Therefore, we call it a stationary wave.
All the particles between adjacent nodes, i.e., in one loop, vibrate in phase. There is no progressive change of phase from one particle to another. All the particles in the same loop are in the same phase of oscillation, which reverses for the adjacent loop.
Reading the Properties Together
Properties 7 and 8 are the two that carry the most physics, and they are worth restating.
Property 7 explains the name. Energy is present everywhere in the medium, but it is trapped between the boundaries — it sloshes back and forth within each loop instead of being carried onward, so the wave transports nothing from one end to the other.
Property 8 explains the appearance. Because a whole loop moves together, the eye sees one smooth swinging segment rather than a rippling pattern, and because adjacent loops are in opposite phase, one loop is at the top of its swing exactly when its neighbour is at the bottom.
Musical instruments such as the violin and the tanpura are based on the principle of formation of stationary waves or standing waves.
6.5.4 Comparison of Progressive Waves and Stationary Waves
Both kinds of wave involve the same particles performing the same simple harmonic motion, so the differences between them lie entirely in how that motion is distributed in space and in what the wave does with energy. The following comparison collects those differences.
In a progressive wave, the disturbance travels from one region to the other with a definite velocity. In stationary waves, the disturbance remains in the region where it is produced, and the velocity of the wave is zero.
In progressive waves, the amplitudes of all the particles are the same, but in stationary waves, the amplitudes of the particles are different.
In a stationary wave, all the particles cross their mean positions simultaneously, but in a progressive wave this does not happen.
In progressive waves, all the particles are moving, while in stationary waves the particles at the position of the nodes are always at rest.
Energy is transmitted from one region to another in progressive waves, but in stationary waves there is no transfer of energy.
All the particles between two consecutive nodes are moving in the same direction and are in phase, while those in adjacent loops are moving in opposite directions and differ in phase by \(180^\circ\), in stationary waves. But in a progressive wave, the phases of adjacent particles are different.
The Single Idea Behind the Whole Table
If you can remember only one line of the comparison, remember this one: a progressive wave shares its phase around and keeps its amplitude uniform, while a stationary wave shares its phase within a loop and hands out different amplitudes to different places.
Every other entry follows from that. Different amplitudes at different points immediately produce permanently still points, which is entry 4. Permanently still points block the passage of energy, which is entry 5, and a wave that carries no energy forward cannot travel, which is entry 1.
This comparison is asked very frequently as a short descriptive question in board examinations, and marks are usually awarded point by point. It is therefore worth learning the six entries in matched pairs, always stating what happens in the progressive wave and what happens in the stationary wave in the same sentence.
6.6 Free and Forced Vibrations
The frequency at which an object tends to vibrate when hit, plucked or somehow disturbed is known as its natural frequency. In these vibrations, the object is not under the influence of any outside force.
The natural frequency is decided by the body itself — by its dimensions, its tension or stiffness, and its mass. You cannot change it by disturbing the body more strongly; a harder pluck gives a louder note, not a different one.
Free Vibrations
When a simple pendulum is pulled aside and released, it performs free vibrations with its natural frequency. Similarly, when a string of a guitar is plucked at some point, it performs free vibrations with its natural frequency.
In free vibration, the body at first is given an initial displacement and the force is then withdrawn. The body starts vibrating and continues the motion on its own. No external force acts on the body further to keep it in motion.
In the case of free vibrations, a body continuously loses energy due to the frictional resistance of the surrounding medium. Therefore, the amplitude of vibrations goes on decreasing, the vibrations of the body eventually stop, and the body comes to rest.
Forced Vibrations
The vibrations in a drill machine and in a washing machine are forced vibrations. Here the motor keeps applying a periodic push, and the body has no choice about the rate at which it moves.
Also, the vibrations produced in the table top due to tuning forks of two different frequencies are different, as they are forced vibrations due to two tuning forks of different frequencies. The table top does not insist on its own natural frequency; it takes up whichever frequency is imposed on it.
In forced vibrations, an external periodic force is applied on a body whose natural period is different from the period of the force. The body is made to vibrate with a frequency equal to that of the externally impressed force.
The amplitude of forced vibrations depends upon the difference between the frequency of the external periodic force and the natural frequency of the body. If this difference is small, the amplitude of forced vibrations is large, and vice versa. This dependence is shown in Fig. 6.6a.
Diagram
Fig. 6.6a – Graph of amplitude of forced vibration against driving frequency, showing a sharp peak where the driving frequency equals the natural frequency of the body.
Resonance
If the frequencies exactly match, it is termed as resonance, and the amplitude of vibration is maximum.
An object vibrating with its natural frequency can cause another nearby object to vibrate. The second object absorbs the energy transmitted by the first object and starts vibrating, if the natural frequencies of the two objects match. You have seen the example of two simple pendula supported from a string in the previous chapter. The second object is said to undergo forced vibrations.
Strings or air columns can also undergo forced oscillations if the frequency of the external source of sound is close to the natural frequency of the system. Resonance is said to occur, and we hear a louder sound.
6.7 Harmonics and Overtones
When a string or an air column is set into vibrations by some means, the waves are reflected from the ends and stationary waves can be formed.
An important condition to form stationary waves depends on the boundary conditions that constrain the possible wavelengths or frequencies of vibration of the system. A boundary condition is simply a requirement that the medium must satisfy at its ends — a fixed end of a string must stay at rest, an open end of a pipe must be free to move — and only those waves that fit the ends exactly can survive.
Because of this fitting requirement, a bounded system cannot vibrate at just any frequency. It possesses a set of allowed frequencies, and these are called the natural frequencies of the normal modes of oscillations.
Naming the Allowed Frequencies
The minimum of these frequencies is termed the fundamental frequency or the first harmonic. The corresponding mode of oscillations is called the fundamental mode or fundamental tone.
The term overtone is used to represent higher frequencies. The first frequency higher than the fundamental frequency is called the first overtone, the next higher frequency is the second overtone, and so on. This ordering is shown in Fig. 6.7a.
Diagram
Fig. 6.7a – A vertical frequency ladder for a system whose allowed frequencies are n, 2n, 3n and 4n, labelled on the left as first, second, third and fourth harmonic and on the right as fundamental, first overtone, second overtone and third overtone.
The term ‘harmonic’ is used when the frequency of a particular overtone is an integral multiple of the fundamental frequency.
How the Two Names Differ
In strings and air columns, the frequencies of overtones are integral multiples of the fundamental frequencies, hence they are termed as harmonics.
But all harmonics may not be present in a given sound. The overtones are only those multiples of the fundamental frequency which are actually present in a given sound. The harmonics may or may not be present in the sound so produced.
So the two words answer two different questions. “Harmonic” answers the arithmetical question — how many times the fundamental is this frequency? “Overtone” answers the practical question — where does this frequency stand in the list of tones the system is actually producing? Harmonics are counted whether or not they occur; overtones are counted only among those that occur.
To understand the concept of harmonics and overtones, let us study the vibrations of an air column.
6.7.1 End Correction
When an air column vibrates, either in a pipe closed at one end or in a pipe open at both ends, the boundary conditions demand that there is always an antinode at the open end or ends, since the particles of the medium are comparatively free there, and a node at the closed end, since there is hardly any freedom for the particles to move at a rigid boundary.
That is the ideal statement of the boundary condition. In a real pipe it needs a small correction.
Why the Antinode Lies Outside the Pipe
The antinode is not formed exactly at the open end, but it is slightly beyond the open end, as air is more free to vibrate there in comparison to the air inside the pipe.
Also, as the air particles in the plane of the open end of the pipe are not free to move in all directions, reflection takes place at a plane a small distance outside the pipe. The wave, in effect, does not know that the pipe has ended until it has travelled a little past the rim, as shown in Fig. 6.7b.
Diagram
Fig. 6.7b – The open end of a pipe drawn in section, with the antinode marked slightly outside the rim and the small gap between the rim and the antinode labelled as the end correction e.
The distance between the open end of the pipe and the position of the antinode is called the end correction.
According to Reynold, to the first approximation, the end correction at an end is given by
\[e = 0.3\,d\]
where \(d\) is the inner diameter of the pipe, and \(e\) and \(d\) are both measured in metre in SI units.
Notice that the correction depends on the width of the pipe and not on its length or on the frequency being sounded. A wide pipe therefore needs a larger correction than a narrow one of the same length.
Corrected Length of the Air Column
Thus the length \(L\) of the air column is different from the length \(l\) of the pipe. It is \(L\), not \(l\), that must be used in every formula for the modes of vibration, because the stationary wave is set up between the node and the true position of the antinode.
For a pipe closed at one end
The corrected length of the air column \(L\) = length of the air column in the pipe \(l\) + end correction at the open end.
\[\therefore \; L = l + e \qquad \text{--- (6.13)}\]
For a pipe open at both ends
The corrected length of the air column \(L\) = length of the air column in the pipe \(l\) + end corrections at both the ends.
\[\therefore \; L = l + 2e \qquad \text{--- (6.14)}\]
6.7.2 Vibrations of Air Column in a Pipe Closed at One End
Consider a long cylindrical tube closed at one end. It consists of an air column with a rigid boundary at one end. When a vibrating tuning fork is held near the open end of the closed pipe, sound waves are sent by the fork inside the tube.
Longitudinal waves travelling along a pipe of finite length are reflected at the ends, just as transverse waves are reflected at the fixed ends of a string. The phase of the reflected wave depends on whether the end of the pipe is open or closed, and on how wide or narrow the pipe is in comparison to the wavelength of a longitudinal wave such as a sound wave.
The incident and reflected waves then travel through each other in opposite directions, and by the principle of superposition a stationary wave is set up inside the pipe. Which stationary waves are allowed is decided entirely by the two ends.
The Boundary Conditions
At the closed end there is the least freedom for the motion of air particles, because the rigid surface will not let them move along the pipe. Thus, there must be a node at the closed end.
The particles a little beyond the open end are the most free to vibrate. As a result, an antinode must be formed a little beyond the open end, at the distance \(e\) discussed in the previous subsection.
The length \(l\) of the pipe and the length \(L\) of the air column are shown separately in all the figures — refer to Figs. 6.9 and 6.10 — because it is \(L\) that enters the formulae.
The Fundamental Mode
The first mode of vibrations of the air column closed at one end is as shown in Fig. 6.9 (a). This is the simplest mode of vibration of an air column closed at one end, known as the fundamental mode. It has just one node, at the closed end, and one antinode, at the open end, and therefore fits only a quarter of a wave into the pipe.
Diagram
Fig. 6.9 (a) – Set-up for generating vibrations of the air column in a pipe closed at one end, with a tuning fork held above the open end, a node at the closed end and an antinode slightly beyond the open end; the distance of the antinode from the open end of the pipe has been exaggerated.
\(\therefore\) Length of air column
\[L = \frac{\lambda}{4} \quad \text{and} \quad \lambda = 4L\]
where \(\lambda\) is the wavelength of the fundamental mode of vibrations in the air column, measured in metre. If \(n\) is the fundamental frequency, we have
\[v = n\lambda \qquad \text{--- (6.15)}\]
\[\therefore \; n = \frac{v}{\lambda}\]
\[\therefore \; n = \frac{v}{4L} = \frac{v}{4(l+e)} \qquad \text{--- (6.16)}\]
Here \(v\) is the speed of sound in air in m s\(^{-1}\), \(l\) is the length of the pipe in metre and \(e\) is the end correction in metre.
The fundamental frequency is also known as the first harmonic. It is the lowest frequency of vibration in an air column in a pipe closed at one end.
The First Overtone
The next mode of vibrations of the air column closed at one end is as shown in Fig. 6.9 (b). Here the air column is made to vibrate in such a way that it contains a node at the closed end, an antinode at the open end, with one more node and antinode in between.
Diagram
Fig. 6.9 (b) – First overtone for vibrations of the air column in a pipe closed at one end, showing two nodes and two antinodes; the distance of the antinode from the open end of the pipe has been exaggerated.
If \(n_1\) is the frequency and \(\lambda_1\) is the wavelength of the wave in this mode of vibrations in the air column, we have, the length of the air column
\[L = \frac{3\lambda_1}{4}\]
\[\therefore \; \lambda_1 = \frac{4L}{3} = \frac{4(l+e)}{3} \qquad \text{--- (6.17)}\]
The velocity in the second mode is given as \(v = n_1\lambda_1\)
\[\therefore \; n_1 = \frac{v}{\lambda_1} = \frac{3v}{4L} = \frac{3v}{4(l+e)}\]
\[\therefore \; n_1 = 3n \qquad \text{--- (6.18)}\]
This frequency is the third harmonic. It is the first overtone. Remember that the overtones are always numbered sequentially, in the order in which they actually occur, while the harmonic number counts multiples of the fundamental.
The Second Overtone
The next higher mode of vibrations of the air column closed at one end is as shown in Fig. 6.9 (c). Here the same air column is made to vibrate in such a way that it contains a node at the closed end, an antinode at the open end, with two more nodes and antinodes in between.
Diagram
Fig. 6.9 (c) – Second overtone for vibrations of the air column in a pipe closed at one end, showing three nodes and three antinodes; the distance of the antinode from the open end of the pipe has been exaggerated.
If \(n_2\) is the frequency and \(\lambda_2\) is the wavelength of the wave in this mode of vibrations in the air column, we have
Length of air column
\[L = \frac{5\lambda_2}{4}\]
\[\therefore \; \lambda_2 = \frac{4L}{5} = \frac{4(l+e)}{5} \qquad \text{--- (6.19)}\]
The velocity in this mode is given as \(v = n_2\lambda_2\)
\[\therefore \; n_2 = \frac{v}{\lambda_2} = \frac{5v}{4L} = \frac{5v}{4(l+e)}\]
\[\therefore \; n_2 = 5n \qquad \text{--- (6.20)}\]
This frequency is the fifth harmonic. It is the second overtone.
The General Result
Continuing in a similar way, for the \(p\)th overtone we get the frequency \(n_p\) as
\[n_p = (2p+1)\,n \qquad \text{--- (6.21)}\]
where \(p = 1, 2, 3, \ldots\) and \(n\) is the fundamental frequency.
Thus for a pipe closed at one end only odd harmonics are present and even harmonics are absent.
6.7.3 Vibrations of Air Column in a Pipe Open at Both Ends
In this case the boundary conditions are such that an antinode is present at each open end. When a source of sound like a tuning fork is held near one end of the pipe, it sends the waves inside the pipe.
Even though both the ends of the pipe are open, the air inside the pipe is still bound by the wall of the tube. As a result, the air inside the pipe is a little denser than the air outside.
That small difference is enough to cause a reflection. When the waves travel to the other open end, there is partial reflection at the open end. The partially reflected waves superimpose with the incident waves. Under suitable conditions, stationary waves will be formed.
There is maximum freedom for the motion of the air column at both the ends, as the pipe is open at both ends, and this is why an antinode must sit at each end.
Following One Compression Through the Pipe
Suppose a compression produced by a tuning fork travels through the air column. It gets reflected as a rarefaction at the open end, as you saw in Section 6.3.2 for reflection from a rarer medium.
The rarefaction moves back and gets reflected as a compression at the other end. It suffers a second reflection at the open end near the source, and then interferes with the wave coming in, by a path difference of \(2L\).
The different modes of vibrations of the air column in a pipe open at both ends are shown in Fig. 6.10 (a), (b) and (c).
The Fundamental Mode
The fundamental tone or mode of vibrations of an air column open at both ends is as shown in Fig. 6.10 (a). There are two antinodes at the two open ends and one node between them.
Diagram
Fig. 6.10 (a) – Fundamental mode of vibration of the air column in a pipe open at both ends, showing an antinode just beyond each open end and a single node at the middle; the distance of the antinodes from the open ends has been exaggerated.
\(\therefore\) Length of air column
\[L = \frac{\lambda}{2} \quad \text{or,} \quad \lambda = 2L\]
and
\[v = n\lambda = 2nL \qquad \text{--- (6.22)}\]
\[\therefore \; n = \frac{v}{\lambda} = \frac{v}{2L} = \frac{v}{2(l + 2e)} \qquad \text{--- (6.23)}\]
where \(v\) is the speed of sound in air in m s\(^{-1}\), \(l\) is the length of the pipe in metre and \(e\) is the end correction in metre.
This is the fundamental frequency or the first harmonic. It is the lowest frequency of vibration.
The First Overtone
The next possible mode of vibrations of an air column open at both ends is as shown in Fig. 6.10 (b). Three antinodes and two nodes are formed.
Diagram
Fig. 6.10 (b) – First overtone of the air column in a pipe open at both ends, showing three antinodes and two nodes.
\(\therefore\) Length of air column \(= L = \lambda_1\)
i.e., \(\lambda_1 = L = (l + 2e)\) — (6.24)
If \(n_1\) and \(\lambda_1\) are the frequency and wavelength of this mode of vibration of the air column respectively, then
\[v = n_1\lambda_1\]
\[n_1 = \frac{v}{\lambda_1} = \frac{v}{L} = \frac{v}{(l + 2e)}\]
\[\therefore \; n_1 = 2n \qquad \text{--- (6.25)}\]
This is the frequency of the second harmonic or the first overtone. Notice at once what has happened: the second harmonic, which was forbidden in the closed pipe, is perfectly allowed here.
The Second Overtone
In the next possible mode of vibrations of an air column open at both ends, as shown in Fig. 6.10 (c), four antinodes and three nodes are formed.
Diagram
Fig. 6.10 (c) – Second overtone of the air column in a pipe open at both ends, showing four antinodes and three nodes.
\(\therefore\) Length of air column
\[= L = \frac{3\lambda_2}{2}\]
\[\therefore \; \lambda_2 = \frac{2L}{3} = \frac{2(l + 2e)}{3} \qquad \text{--- (6.26)}\]
If \(n_2\) and \(\lambda_2\) are the frequency and wavelength of this mode of vibration of the air column respectively, then \(v = n_2\lambda_2\)
\[\therefore \; n_2 = \frac{v}{\lambda_2} = \frac{3v}{2L} = \frac{3v}{2(l + 2e)}\]
\[\therefore \; n_2 = 3n \qquad \text{--- (6.27)}\]
This is the frequency of the third harmonic or the second overtone.
The General Result
Thus all harmonics are present as overtones in the modes of vibration of an air column open at both ends.
Continuing in this manner, the frequency \(n_p\) for the \(p\)th overtone is
\[n_p = (p+1)\,n \qquad \text{--- (6.28)}\]
where \(n\) is the fundamental frequency and \(p = 0, 1, 2, 3\ldots\)
It may be noted that
Sound produced by an open pipe contains all harmonics. Its quality is richer than that produced by a closed pipe.
The fundamental frequency of vibration of an air column in an open pipe is double that of the fundamental frequency of vibration in a closed pipe of the same length.
Using These Formulae in the Laboratory
Using the formula and knowing the values of \(n\), \(l\) and the end correction, the velocity of sound in air at room temperature can be calculated.
As discussed earlier, the antinodes are formed a little beyond the open ends of the pipe. It is however not possible to locate the positions of the antinodes precisely. Therefore, in experiments, the length of the pipe is measured and end corrections are incorporated.
6.7.4 Practical Determination of End Correction
The formula \(e = 0.3d\) is only a first approximation, and in careful work the end correction is measured rather than assumed. The trick is to use the same pipe diameter twice, so that \(e\) is the same in both measurements and can be eliminated.
An exact method to determine the end correction, using two pipes of the same diameter but different lengths \(l_1\) and \(l_2\), with fundamental frequencies \(n_1\) and \(n_2\) respectively, is as follows.
For a Pipe Open at Both Ends
The speed of sound in air is the same for both pipes, so we may equate the two expressions for it.
\[v = 2n_1L_1 = 2n_2L_2 \quad \text{using Eq. (6.22)}\]
\[\therefore \; n_1L_1 = n_2L_2\]
Substituting the corrected lengths from Eq. (6.14),
\[\therefore \; n_1(l_1 + 2e) = n_2(l_2 + 2e)\]
\[\therefore \; e = \frac{n_1l_1 - n_2l_2}{2(n_2 - n_1)} \quad \text{or} \quad \frac{n_2l_2 - n_1l_1}{2(n_1 - n_2)} \qquad \text{--- (6.29)}\]
For a Pipe Closed at One End
\[v = 4n_1L_1 = 4n_2L_2\]
\[\therefore \; n_1L_1 = n_2L_2\]
Substituting the corrected lengths from Eq. (6.13),
\[\therefore \; n_1(l_1 + e) = n_2(l_2 + e)\]
\[\therefore \; e = \frac{n_1l_1 - n_2l_2}{(n_2 - n_1)} \quad \text{or} \quad \frac{n_2l_2 - n_1l_1}{(n_1 - n_2)} \qquad \text{--- (6.30)}\]
Both forms of each result are the same expression; only the order of subtraction has been reversed in the numerator and the denominator together, so you may use whichever keeps the arithmetic positive.
Diagram
Fig. 6.7c – Resonance tube set-up: a glass tube clamped vertically with its lower end dipping into a cylinder of water and a vibrating tuning fork held just above its open upper end, with the air column length marked as l = λ/4.
6.7.5 Vibrations Produced in a String
Consider a string of length \(l\) stretched between two rigid supports. The linear density, that is, the mass per unit length of the string, is \(m\), measured in kg m\(^{-1}\), and the tension \(T\), measured in newton, acts on the string due to stretching.
If it is made to vibrate by plucking or by using a vibrator like a tuning fork, a transverse wave can be produced along the string.
When the wave reaches the fixed ends of the string, it gets reflected with a change of phase by \(\pi\) radians, exactly as described in Section 6.3.1. The reflected waves interfere with the incident wave and stationary waves are formed along the string. The string vibrates with different modes of vibrations.
Since both ends are rigidly held, both ends must be nodes. This single requirement decides every mode the string can have.
The Fundamental Mode
If a string is stretched between two rigid supports and is plucked at its centre, the string vibrates as shown in Fig. 6.11 (a). It consists of an antinode formed at the centre and nodes at the two ends, with one loop formed along its length.
Diagram
Fig. 6.11 (a) – Fundamental mode of a stretched string fixed at both ends, showing one loop with nodes at the two ends and an antinode at the centre, the length marked as l = λ/2.
If \(\lambda\) is the wavelength and \(l\) is the length of the string, we get
Length of loop \(= \dfrac{\lambda}{2} = l\)
\[\therefore \lambda = 2l\]
The frequency of vibrations of the string,
\[n = \frac{v}{\lambda} = \frac{1}{2l}\sqrt{\frac{T}{m}} \qquad \left(\because \; v = \sqrt{\frac{T}{m}}\right)\]
The relation \(v = \sqrt{T/m}\) says that the speed of a transverse wave on a string is decided by how tightly it is stretched and how heavy it is per metre — pulling harder speeds the wave up, and loading the string down slows it.
This is the lowest frequency with which the string can vibrate. It is the fundamental frequency of vibrations or the first harmonic.
The First Overtone
If the centre of the string is prevented from vibrating by touching it with a light object, and the string is plucked at a point midway between one of the segments, the string vibrates as shown in Fig. 6.11 (b).
Diagram
Fig. 6.11 (b) – First overtone of a stretched string, showing two loops with a node at the centre in addition to the nodes at the two ends, the length marked as l = λ.
Two loops are formed in this mode of vibrations. There is a node at the centre of the string and at its both ends. If \(\lambda_1\) is the wavelength of vibrations, the length of one loop
\[= \frac{\lambda_1}{2} = \frac{l}{2}\]
\[\therefore \; \lambda_1 = l\]
Thus, the frequency of vibrations is given as
\[n_1 = \frac{1}{\lambda_1}\sqrt{\frac{T}{m}}\]
\[n_1 = \frac{1}{l}\sqrt{\frac{T}{m}}\]
Comparing with the fundamental frequency, we get that \(n_1 = 2n\).
Thus the frequency of the first overtone or second harmonic is equal to twice the fundamental frequency.
The Second Overtone
The string is made to vibrate in such a way that three loops are formed along the string, as shown in Fig. 6.11 (c).
Diagram
Fig. 6.11 (c) – Second overtone of a stretched string, showing three loops with two nodes between the fixed ends, the length marked as l = 3λ/2.
If \(\lambda_2\) is the wavelength here, the length of one loop is
\[\frac{\lambda_2}{2} = \frac{l}{3}\]
\[\therefore \; \lambda_2 = \frac{2l}{3}\]
Therefore the frequency of vibrations is
\[n_2 = \frac{1}{\lambda_2}\sqrt{\frac{T}{m}}\]
\[n_2 = \frac{3}{2l}\sqrt{\frac{T}{m}}\]
Comparing with the fundamental frequency, we get that \(n_2 = 3n\).
Thus the frequency of the second overtone or third harmonic is equal to thrice the fundamental frequency.
The General Result
Similarly, for higher modes of vibrations of the string, the frequencies of vibrations are \(4n\), \(5n\), \(6n\) and so on.
Thus all harmonics are present in the case of a stretched string, and the frequencies are given by
\[n_p = p\,n \qquad \text{--- (6.31)}\]
In this respect a stretched string behaves like a pipe open at both ends and unlike a pipe closed at one end, even though the string carries transverse waves and has nodes — not antinodes — at its two ends. What matters is only that the two ends are alike.
6.7.6 Laws of a Vibrating String
The fundamental frequency of a vibrating string under tension is given as
\[n = \frac{1}{2l}\sqrt{\frac{T}{m}} \qquad \text{--- (6.32)}\]
where \(l\) is the vibrating length in metre, \(T\) is the tension in newton and \(m\) is the linear density in kg m\(^{-1}\).
This single formula contains three separate statements, because it involves three quantities that can be varied independently. Holding two of them fixed and changing the third gives the three laws of a vibrating string.
From this formula, three laws of a vibrating string can be given as follows.
1) Law of length: The fundamental frequency of vibrations of a string is inversely proportional to the length of the vibrating string, if the tension and mass per unit length are constant.
\[n \propto \frac{1}{l}, \quad \text{if } T \text{ and } m \text{ are constant.} \qquad \text{--- (6.33)}\]
2) Law of tension: The fundamental frequency of vibrations of a string is directly proportional to the square root of the tension, if the vibrating length and mass per unit length are constant.
\[n \propto \sqrt{T}, \quad \text{if } l \text{ and } m \text{ are constant.} \qquad \text{--- (6.34)}\]
3) Law of linear density: The fundamental frequency of vibrations of a string is inversely proportional to the square root of the mass per unit length (linear density), if the tension and vibrating length of the string are constant.
\[n \propto \frac{1}{\sqrt{m}}, \quad \text{if } T \text{ and } l \text{ are constant.} \qquad \text{--- (6.35)}\]
Frequency in Terms of Radius and Density
The linear density can be re-expressed in terms of quantities that are easier to measure for a wire, namely its radius and the density of its material.
If \(r\) is the radius and \(\rho\) is the density of the material of the string, linear density is given as
Linear density = mass per unit length
= volume per unit length \(\times\) density
\[= \left(\frac{\pi r^2 l}{l}\right)\rho = \pi r^2 \rho\]
As \(n \propto \dfrac{1}{\sqrt{m}}\), if \(T\) and \(l\) are constant, we get
\[n \propto \frac{1}{r\sqrt{\pi\rho}}\]
\[\therefore \; n \propto \frac{1}{\sqrt{\rho}} \quad \text{and} \quad n \propto \frac{1}{r} \qquad \text{--- (6.36)}\]
Thus the fundamental frequency of vibrations of a stretched string is inversely proportional to (i) the radius of the string and (ii) the square root of the density of the material of the vibrating string.
Note the difference in the two dependences. The frequency falls in direct proportion to the radius, but only as the square root of the density. Doubling the radius therefore halves the frequency, while doubling the density lowers it by a factor of only about 1.41.
6.8 Sonometer
The three laws of the previous section were obtained by reasoning from a formula. A sonometer is the instrument that allows them to be tested directly in the laboratory, by letting us change one quantity at a time and measure the effect.
A sonometer consists of a hollow rectangular wooden box called the sound box. The sound box is used to make a larger mass of air vibrate, so that the sound produced by the vibrating string — a metal wire in this case — gets amplified. A thin wire by itself moves very little air and is barely audible; the box, driven by the wire, moves a great deal of it.
The same principle is applied in stringed instruments such as the violin, guitar, tanpura and so on.
Construction
There are two bridges P and Q along the width of the box, which can be moved parallel to the length of the box. A metal wire of uniform cross-section runs along the length of the box over the bridges.
The wire is fixed at one end, and its other end passes over a pulley. A hanger with suitable slotted weights can be attached to the free end of the wire. The complete arrangement is shown in Fig. 6.12.
Diagram
Fig. 6.12 – Experimental set-up of a sonometer: a hollow wooden sound box carrying a wire fixed at one end, passing over two movable bridges P and Q and over a pulley at the far end to a hanger with slotted weights, with a paper rider R placed on the wire midway between the bridges.
By changing the weights, the tension in the wire can be varied. The movable bridges allow us to change the vibrating length AB of the wire. Between them, these two adjustments give us independent control of \(T\) and \(l\), which is exactly what the laws require.
How Stationary Waves Are Set Up
If the wire is plucked at a point midway between the bridges, transverse waves are produced in the wire. Stationary waves are produced between the two bridges due to the reflection of the transverse wave at the bridges and their superposition.
Thus the portion AB of the wire between the two bridges P and Q is the vibrating length. The bridges act as the rigid supports of Section 6.7.5, so nodes are formed there.
The wire can also be made to vibrate by holding a vibrating tuning fork near it. The frequency of vibration is then the same as that of the tuning fork, since these are forced vibrations. If this frequency happens to be one of the natural frequencies of the wire, standing waves with large amplitude are set up in the wire, since the two vibrate in resonance.
The Paper Rider
To identify the resonance, a small piece of paper, known as the rider R, is placed over the wire at a point in the middle of the length AB, as determined by the position of the bridges P and Q.
If the frequency of the tuning fork and that of the fundamental mode of vibration of the wire match — this is achieved by adjusting the length AB of the wire using the bridges P and Q — the paper rider happens to be at the antinode and flies off the wire.
A sonometer can be used to verify the laws of a vibrating string.
Verification of the First Law of a Vibrating String
By measuring the length of the wire and its mass, the mass per unit length (\(m\)) of the wire is determined. Then the wire is stretched on the sonometer and the hanger is suspended from its free end. A suitable tension (\(T\)) is applied to the wire by placing slotted weights on the hanger.
The length of wire (\(l_1\)) vibrating with the same frequency (\(n_1\)) as that of the tuning fork is determined as follows.
A light paper rider is placed on the wire midway between the bridges. The tuning fork is set into vibrations by striking it on a rubber pad. The stem of the tuning fork is held in contact with the sonometer box, so that the vibrations are passed to the wire through the box.
By changing the distance between the bridges without disturbing the paper rider, the frequency of vibrations of the wire is changed. When the frequency of vibrations of the wire becomes exactly equal to the frequency of the tuning fork, the wire vibrates with maximum amplitude and the paper rider is thrown off.
In this way a set of tuning forks having different frequencies \(n_1\), \(n_2\), \(n_3\), … are used, and the corresponding vibrating lengths of wire are noted as \(l_1\), \(l_2\), \(l_3\), … by keeping the tension \(T\) constant. We will observe that
\[n_1 l_1 = n_2 l_2 = n_3 l_3 = \ldots = \text{constant}\]
for a constant value of tension (\(T\)) and mass per unit length (\(m\)).
\[\therefore \; nl = \text{constant}\]
i.e., \(n \propto \dfrac{1}{l}\), if \(T\) and \(m\) are constant.
Thus, the first law of a vibrating string is verified by using a sonometer.
Verification of the Second Law of a Vibrating String
The vibrating length (\(l\)) of the given wire of mass per unit length (\(m\)) is kept constant for the verification of the second law. By changing the tension, the same length is made to vibrate in unison with different tuning forks of various frequencies.
If tensions \(T_1\), \(T_2\), \(T_3\), … correspond to frequencies \(n_1\), \(n_2\), \(n_3\), … etc., we will observe that
\[\frac{n_1}{\sqrt{T_1}} = \frac{n_2}{\sqrt{T_2}} = \frac{n_3}{\sqrt{T_3}} = \ldots = \text{constant}\]
or
\[\frac{n}{\sqrt{T}} = \text{constant}\]
\(\therefore \; n \propto \sqrt{T}\), if \(l\) and \(m\) are constant. This is the second law of a vibrating string.
Verification of the Third Law of a Vibrating String
For the verification of the third law of a vibrating string, two wires having different masses per unit length \(m_1\) and \(m_2\) (linear densities) are used.
The first wire is subjected to a suitable tension and made to vibrate in unison with a given tuning fork. The vibrating length is noted as (\(l_1\)). Using the same fork, the second wire is made to vibrate under the same tension and the vibrating length (\(l_2\)) is determined.
Thus the frequency of vibration of the two wires is kept the same under the same applied tension \(T\). It is found that
\[l_1\sqrt{m_1} = l_2\sqrt{m_2}\]
\[l\sqrt{m} = \text{constant}\]
But by the first law of a vibrating string, \(n \propto \dfrac{1}{l}\).
Therefore we get that \(n \propto \dfrac{1}{\sqrt{m}}\), if \(T\) and \(l\) are constant. This is the third law of a vibrating string.
In this way, the laws of a vibrating string are verified by using a sonometer.
6.9 Beats
This is an interesting phenomenon based on the principle of superposition of waves. So far we have superposed waves of the same frequency; now we allow the two frequencies to differ slightly.
When there is superposition of two sound waves, having the same amplitude but slightly different frequencies, travelling in the same direction, the intensity of sound varies periodically with time. This phenomenon is known as the production of beats.
The reason for the variation is that the two waves cannot stay in step. At one moment they arrive in phase and reinforce each other; a little later one has gained a full cycle over the other, so they arrive out of phase and cancel; then they fall into step again, and the cycle repeats.
The occurrences of maximum intensity are called waxing, and those of minimum intensity are called waning. One waxing and a successive waning together constitute one beat.
The number of beats heard per second is called the beat frequency.
6.9.1 Analytical Method to Determine Beat Frequency
Consider two sound waves, having the same amplitude and slightly different frequencies \(n_1\) and \(n_2\). Let us assume that they arrive in phase at some point \(x\) of the medium.
The displacement due to each wave at any instant of time at that point is given as
\[y_1 = a\sin 2\pi\left(n_1 t - \frac{x}{\lambda_1}\right)\]
\[y_2 = a\sin 2\pi\left(n_2 t - \frac{x}{\lambda_2}\right)\]
Let us assume for simplicity that the listener is at \(x = 0\). This costs us nothing, because we only wish to know how the sound at one fixed place changes with time.
\[\therefore \; y_1 = a\sin(2\pi n_1 t)\]
and
\[y_2 = a\sin(2\pi n_2 t)\]
Superposing the Two Waves
According to the principle of superposition of waves,
\[y = y_1 + y_2\]
\[\therefore \; y = a\sin 2\pi n_1 t + a\sin 2\pi n_2 t\]
or,
\[y = 2a\, \sin\left[2\pi\left(\frac{n_1 + n_2}{2}\right)t\right]\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] \qquad \text{--- (6.37)}\]
by using the formula
\[\sin C + \sin D = 2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)\]
Rearranging the above equation, we get
\[y = \left[2a\cos\left(2\pi\left(\frac{n_1 - n_2}{2}\right)t\right)\right]\sin\left[2\pi\left(\frac{n_1 + n_2}{2}\right)t\right]\]
Substituting
\[2a\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] = A\]
and
\[\frac{n_1 + n_2}{2} = n,\]
we get
\[y = A\sin 2\pi n t \qquad \text{--- (6.38)}\]
This is the equation of a progressive wave having frequency \(n\) and amplitude \(A\). The frequency \(n\) is the mean of the frequencies \(n_1\) and \(n_2\) of the arriving waves, while the amplitude \(A\) varies periodically with time.
The listener hears a note of the average frequency \(n\), and it is only the loudness of that note which swells and fades. This is exactly what your ear reports when two nearly-tuned strings are sounded together, as illustrated in Fig. 6.13.
Diagram
Fig. 6.13 – Superposition of two harmonic waves of nearly equal frequencies, drawn one below the other, with the resultant beneath them showing a slowly varying envelope that produces the formation of beats.
Condition for Waxing
The intensity of sound is proportional to the square of the amplitude. Hence the resultant intensity will be maximum when the amplitude is maximum.
For maximum amplitude (waxing),
\[A = \pm 2a\]
\[\therefore \; 2a\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] = \pm 2a\]
or,
\[\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] = \pm 1\]
i.e.,
\[2\pi\left(\frac{n_1 - n_2}{2}\right)t = 0, \; \pi, \; 2\pi, \; 3\pi, \ldots\]
\[\therefore \; t = 0, \; \frac{1}{n_1 - n_2}, \; \frac{2}{n_1 - n_2}, \; \frac{3}{n_1 - n_2}, \ldots\]
Thus, the time interval between two successive maxima of sound is always \(\dfrac{1}{n_1 - n_2}\).
Hence the period of beats is
\[T = \frac{1}{n_1 - n_2}\]
The number of waxings heard per second is the reciprocal of the period of waxing.
\[\therefore \; \text{frequency of beats, } N = n_1 - n_2 \qquad \text{--- (6.39)}\]
This is a strikingly simple result, and it is what makes beats so useful: by merely counting how many swells of loudness occur per second, you measure the difference of two frequencies without measuring either of them.
Condition for Waning
The intensity of sound will be minimum when the amplitude is zero (waning).
For minimum amplitude, \(A = 0\),
\[\therefore \; 2a\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] = 0\]
or,
\[\cos\left[2\pi\left(\frac{n_1 - n_2}{2}\right)t\right] = 0\]
\[\therefore \; 2\pi\left(\frac{n_1 - n_2}{2}\right)t = \frac{\pi}{2}, \; \frac{3\pi}{2}, \; \frac{5\pi}{2}, \ldots\]
\[\therefore \; t = \frac{1}{2(n_1 - n_2)}, \; \frac{3}{2(n_1 - n_2)}, \; \frac{5}{2(n_1 - n_2)}, \ldots\]
Therefore the time interval between two successive minima is also \(\dfrac{1}{(n_1 - n_2)}\), which is expected.
By comparing the instances of successive waxing and waning, we come to know that waxing and waning occur alternately with equal frequency. Each waning falls exactly halfway between two waxings.
What Beats Really Are
The variation in the loudness of sound that goes up and down is the phenomenon of formation of beats. It can be considered as the superposition of waves and the formation of standing waves in time, at one point in space where waves of slightly different frequencies are passing.
The two waves are in and out of phase, giving constructive and destructive interference in turn. The interval between two maximum sound intensities is the time period of beats.
6.9.2 Applications of Beats
Because the beat frequency is exactly the difference of two frequencies, beats turn the ear — and the electronic detector — into a very sensitive comparison instrument. The following applications all use that idea.
1] Matching the frequencies of musical instruments. The phenomenon of beats is used for matching the frequencies of different musical instruments by artists. They go on tuning until no beats are heard by their sensitive ears.
When the beat frequency becomes equal to zero, the musical instruments are in unison with each other, i.e., their frequencies are identical, and the effect of playing such instruments together gives pleasant music.
2] Determining the speed of an aeroplane. The speed of an aeroplane can be determined by using Doppler RADAR.
A microwave signal (pulse) of known frequency is sent towards the moving aeroplane. The principle of the Doppler effect, giving the apparent frequency when the source and observer are in relative motion, applies twice — once for the signal sent by the microwave source and received by the aeroplane, and a second time when the signal is reflected by the aeroplane and is received back at the microwave source.
The phenomenon of beats, arising due to the difference in the frequencies produced by the source and received at the source after reflection from the aeroplane, allows us to calculate the velocity of the aeroplane.
Doppler ultrasonography and the echo cardiogram work on a similar principle. Doctors use an analogous set-up to assess the direction and speed of blood flow in the human body and to identify circulation problems. Measurement of the dimensions of the blood vessels can be used to estimate the volume flow rate. Ultrasound beams also determine phase shifts to diagnose vascular problems in arteries and veins.
3] Determining an unknown frequency. The unknown frequency of a sound note can be determined by using the phenomenon of beats.
Initially the sound notes of known and unknown frequency are heard simultaneously. The known frequency, from a source of adjustable frequency, is adjusted in such a way that the beat frequency reduces to zero.
At this stage the frequencies of both the sound notes become equal. Hence the unknown frequency can be determined.
Notice why the adjustment is pushed all the way to zero beats. A non-zero beat frequency tells you only the size of the difference and not its sign, so the unknown frequency could lie on either side of the known one. Driving the beats to zero removes that ambiguity completely.
6.10 Characteristics of Sound
Two sounds may differ from each other in three independent ways, and every description of a sound — however casual — refers to one of them. Sound has three characteristics: loudness, pitch and quality.
Loudness
Loudness is the human perception of the intensity of sound.
We know that when a sound wave travels through a medium, there are regions of compressions and rarefactions. Thus there are changes in pressure. When a sound is heard, say by a human, the wave exerts pressure on the human ear.
The pressure variation is related to the amplitude and hence to the intensity. Depending on the sound produced, the variation in this pressure is from 28 Pa for the loudest tolerable sound, to \(2.0 \times 10^{-5}\) Pa for the feeblest sound, like a whisper, that can be heard by a human.
That is a range of more than a million to one in pressure, which is the first hint that the ear cannot be responding in simple proportion to what reaches it.
Intensity is a measurable quantity, while the sensation of hearing or loudness is very subjective. It is therefore important to find out how a sound of intensity \(I\) affects a detectable change \(\Delta I\) in the intensity for the human ear to note. It is known that the value of such \(\Delta I\) depends linearly on the intensity \(I\), and this fact allows humans to deal with a large variation in intensity.
In other words, the ear notices changes in proportion, not changes in amount. Adding a given amount of intensity to a quiet sound is very noticeable, while adding the same amount to a loud sound may not be noticed at all.
The response of the human ear to sound is exponential and not linear. It depends upon the amount of energy crossing unit area around a point per unit time.
Intensity is proportional to the square of the amplitude. It also depends upon various other factors, like the distance of the source from the listener, the motion of air, the density of the medium, the surface area of the sounding body, and so on. The presence of other resonant objects around the sounding body also affects the loudness of sound.
The Decibel Scale
Scientifically, sound is specified not by its intensity but by the sound level \(L\), expressed in decibels (dB), defined as
\[L = 10 \log_{10}\left(\frac{I}{I_0}\right) \qquad \text{--- (6.40)}\]
where \(I_0\) is a minimum reference intensity, \(10^{-12}\) W/m\(^2\), that a normal human ear can hear, and \(I\) is the intensity of the given sound in W m\(^{-2}\).
Sound levels are then expressed in decibel (dB). When \(I = I_0\), \(L = 0\); thus the standard reference intensity has the measure of sound level 0 dB.
The unit of difference in loudness is the bel. You have studied about this unit in Std. XI.
\[1 \text{ decibel} = \frac{1}{10} \text{ bel}\]
As mentioned above, the minimum audible sound is denoted by 0 dB, while whispering and normal speech have levels 10 dB and 60 dB respectively, at a distance of approximately 1 m from the source. The intensity level of the maximum tolerable sound for a human ear is around 120 dB.
Loudness and Frequency
Loudness is different at different frequencies, even for the same intensity. Two sounds carrying the same energy per second may not seem equally loud if one of them lies at a frequency to which the ear is less sensitive.
For measuring loudness, the unit phon is used. Phon is a measure of loudness. It is equal to the loudness in decibel of any equally loud pure tone of frequency 1000 Hz.
Pitch
Pitch is a sensation of sound which helps the listener to distinguish between a high frequency and a low frequency note.
Pitch is the human perception of frequency — a higher frequency denotes a higher pitch. The pitch of a female voice is higher than that of a male voice.
Keep the pairing straight: loudness goes with intensity, and pitch goes with frequency. Playing a note more forcefully changes its loudness but not its pitch, as you saw with the freely vibrating string in Section 6.6.
Quality or Timbre
Normally, sound generated by a source has a number of frequency components with different amplitudes.
Quality of sound is that characteristic which enables us to distinguish between two sounds of the same pitch and loudness. We can recognise the voice of a person or an instrument due to its quality of sound.
Quality depends on the number of overtones present in the sound along with a given frequency.
Musical Sound and Noise
A sound which produces a pleasing sensation to the ear is a musical sound. It is produced by regular and periodic vibrations without any sudden change in loudness.
Musical sound has certain well-defined frequencies with sizable amplitude; these are normally harmonics of a fundamental frequency.
A mixture of sounds of different frequencies which do not have any relation with each other produces what we call a noise. Noise therefore is not pleasant to hear. If, in addition, it is loud, it may cause headaches.
The distinction is thus not one of loudness but of order. A soft sound with unrelated frequencies is still noise, and a loud sound built on harmonics of one fundamental is still music.
Musical Scale
A sequence of frequencies which have a specific relationship with each other is called a musical scale.
Normally, both in Indian classical music and in western classical music, eight frequencies, in specific ratios, form an octave, each frequency denoting a specific note.
In a given octave the frequency increases along sa re ga ma pa dha ni sä, as well as along Do Re Mi Fa So La Ti Do. An example of values of frequencies is 240, 270, 300, 320, 360, 400, 450, 480 Hz respectively.
Notice that the last frequency in this list is exactly twice the first. That doubling is what defines an octave, and it is the same factor of two that appeared when we compared an open pipe with a closed pipe of the same length in Section 6.7.3.
6.11 Musical Instruments
Audible waves originate in vibrating strings, vibrating air columns, and vibrating plates and membranes. Accordingly, musical instruments are classified into three main types: (a) stringed instruments, (b) wind instruments, and (c) percussion instruments.
Notice that this classification is really a classification of vibrating systems, and each of the three has already been studied in this chapter.
Stringed Instruments
Stringed instruments consist of stretched strings. Sound is produced by plucking of the strings. The strings are tuned to certain frequencies by adjusting the tension in them, exactly as the law of tension in Section 6.7.6 predicts.
They are further of three different types.
Plucked string type: In these instruments the string is plucked by the fingers, e.g., tanpura, sitar, guitar, veena, etc.
Bowed string type: In these instruments, a string is played by bowing, e.g., violin, sarangi.
Struck string type: The string is struck by a stick, e.g., santoor, piano.
Wind Instruments
These instruments consist of an air column. Sound is produced by setting up vibrations of the air column, which then behaves as the open or closed pipe of Sections 6.7.2 and 6.7.3.
They are further of three different types.
Freewind type: In these instruments free brass reeds are vibrated by air. The air is either blown or compressed, e.g., mouth organ, harmonium, etc.
Edge type: In these instruments air is blown against an edge, e.g., flute.
Reedpipes: They may consist of single or double reeds, and also instruments without reeds, e.g., saxophone and clarinet (single reed), bassoon (double reed), bugle (without reed).
Percussion Instruments
In these instruments sound is produced by setting up vibrations in a stretched membrane, e.g., tabla, drum, dhol, mridangam, sambal, daphali, etc.
These instruments sometimes also consist of metal plates which produce sound when they are struck against each other or with a beater, e.g., cymbals (i.e., jhanja), xylophone, etc.
Why Instruments Sound Different
A blow on the membrane or plate, or the plucking of a string, produces vibrations with one fundamental and many overtones.
The superposition of several natural modes of oscillations, with different amplitudes and hence intensities, characterises different musical instruments. We can thus distinguish the instruments by their sounds.
This is the quality or timbre of Section 6.10, now traced back to its physical origin: not to the fundamental, which the instruments may share, but to the particular recipe of overtones each one adds to it.
Stationary Waves in Practice
The production of different notes by a musical instrument depends on the creation of stationary waves.
For a stringed instrument such as a guitar or a sitar, the two ends of the string are fixed. Depending on where the string is plucked, stationary waves of various modes can be produced. Plucking at the midpoint produces the minimum frequency, or the fundamental mode of vibration, because the midpoint is the antinode of that mode.
In wind instruments, the air column is made to vibrate by blowing. By changing the length of the air column, the note can be changed. In a wind instrument like the flute, holes can be uncovered to change the vibrations of the air column; this changes the pattern of nodes and antinodes.
In practice, the sound produced is made up of several stationary waves having different patterns of nodes and antinodes. A musician’s skill lies in stimulating the string or the air column so as to produce the desired mixture of frequencies.
6.12 Summary
Waves and Progressive Waves
A mechanical wave is a disturbance produced in an elastic medium due to the periodic vibrations of its particles about their mean positions. It transfers energy and momentum from one point to another, but there is no transfer of matter. Electromagnetic waves need no material medium and are non-mechanical waves.
In a progressive wave the disturbance travels continuously in a given direction, without damping and without obstruction, from one particle to another. A wave pulse is a disturbance produced locally and for a short duration.
In progressive waves all particles perform simple harmonic motion with the same amplitude, period and frequency, but the phase changes from particle to particle. Energy is transferred along the wave; matter is not.
Transverse waves have particle vibrations perpendicular to the direction of propagation and produce crests and troughs. Longitudinal waves have particle vibrations along the direction of propagation and produce compressions and rarefactions. Both can travel through solids, but only longitudinal waves can travel through fluids.
The displacement of a particle in a progressive wave travelling in the positive \(x\)-direction is \[y(x,t) = f(x - vt) \qquad (6.1)\]
For a simple harmonic source, \[y(x,t) = A\sin(kx - \omega t) \qquad (6.2)\] and for a wave travelling in the negative \(x\)-direction, \[y(x,t) = A\sin(kx + \omega t) \qquad (6.3)\] where \(A\) is the amplitude, \(k = 2\pi/\lambda\) is the wave number in rad m\(^{-1}\), \(\lambda\) is the wavelength in m, \(\omega\) is the angular frequency in rad s\(^{-1}\), and \(v = \omega/k\) is the wave speed. Also \(n = 1/T = \omega/2\pi\), measured in Hz.
Reflection of Waves
At an interface between two media, part of the wave energy returns into the original medium with slightly different intensity and energy. This is reflection.
Transverse wave, reflection from a denser medium (rigid support): a crest is reflected as a trough and a trough as a crest; there is a phase change of \(\pi\) radian.
Transverse wave, reflection from a rarer medium (free end): a crest is reflected as a crest; there is no phase change.
Longitudinal wave, rarer to denser medium: a compression is reflected as a compression and a rarefaction as a rarefaction; there is no change of pressure phase.
Longitudinal wave, denser to rarer medium: a compression is reflected as a rarefaction and a rarefaction as a compression; the pressure phase changes by \(\pi\) radians.
Superposition of Waves
Principle of superposition: when two or more waves passing through a medium arrive at a common point, each produces its own displacement independent of the other, and the resultant displacement is the vector sum of the individual displacements. The individual waves are unchanged in shape after crossing.
Two pulses of equal amplitude and the same phase give constructive interference; two pulses of equal amplitude and opposite phase give destructive interference.
For two waves of the same frequency, amplitudes \(A_1\) and \(A_2\) and phase difference \(\phi\), superposing at \(x = 0\): \[A\cos\theta = A_1 + A_2\cos\phi \qquad (6.4)\] \[A\sin\theta = A_2\sin\phi \qquad (6.5)\] \[y = A\sin(\omega t + \theta) \qquad (6.6)\] \[A = \sqrt{A_1^{\,2} + 2A_1A_2\cos\phi + A_2^{\,2}} \qquad (6.7)\]
When \(\phi = 0\), \(A = A_1 + A_2\) (maximum); when \(\phi = \pi\), \(A = |A_1 - A_2|\) (minimum). For equal amplitudes \(A_0\), these become \(2A_0\) and zero.
Intensity is proportional to the square of the amplitude: \[I_{max} \propto (A_1 + A_2)^2 \qquad (6.8)\] \[I_{min} \propto (A_1 - A_2)^2 \qquad (6.9)\]
Stationary Waves
Two identical waves of equal amplitude and frequency travelling along the same path in opposite directions superpose to give a stationary wave: \[y_1 = a\sin 2\pi\left(nt - \frac{x}{\lambda}\right) \qquad (6.10)\] \[y_2 = a\sin 2\pi\left(nt + \frac{x}{\lambda}\right) \qquad (6.11)\] \[y = 2a\cos\frac{2\pi x}{\lambda}\,\sin 2\pi nt \qquad (6.12)\] which may be written as \(y = A\sin\omega t\) with \(A = 2a\cos(2\pi x/\lambda)\).
Position and time appear separately, so the wave does not travel. The amplitude varies periodically in space, while every particle vibrates with the same frequency \(\omega\).
Nodes are points of zero amplitude, at \(x = (2p-1)\dfrac{\lambda}{4}\), with \(p = 1, 2, 3, \ldots\)
Antinodes are points of maximum amplitude, at \(x = p\dfrac{\lambda}{2}\), with \(p = 0, 1, 2, 3, \ldots\)
The distance between successive nodes is \(\lambda/2\), between successive antinodes is \(\lambda/2\), and between a node and an adjacent antinode is \(\lambda/4\).
In a stationary wave all particles in one loop vibrate in phase, adjacent loops are \(180^\circ\) out of phase, particles at nodes are always at rest, and there is no transfer of energy — the wave velocity is zero. In a progressive wave, all amplitudes are equal, all particles move, and energy is transmitted.
Free and Forced Vibrations
The natural frequency is the frequency at which a body vibrates when disturbed and left to itself, with no outside force.
In free vibration the body is displaced once and then released; it vibrates at its natural frequency, loses energy to friction, and finally comes to rest.
In forced vibration an external periodic force drives the body, which vibrates at the frequency of the applied force. The amplitude is larger when the driving frequency is closer to the natural frequency.
Resonance occurs when the driving frequency exactly equals the natural frequency; the amplitude is then maximum.
Harmonics and Overtones
The lowest natural frequency is the fundamental frequency or first harmonic. Frequencies higher than the fundamental are overtones, numbered in the order in which they occur. A harmonic is any integral multiple of the fundamental. All harmonics need not be present in a given sound.
End correction: \(e = 0.3\,d\), where \(d\) is the inner diameter of the pipe. \[L = l + e \quad \text{(pipe closed at one end)} \qquad (6.13)\] \[L = l + 2e \quad \text{(pipe open at both ends)} \qquad (6.14)\]
Pipe closed at one end — node at the closed end, antinode at the open end: \[v = n\lambda \qquad (6.15)\] \[n = \frac{v}{4L} = \frac{v}{4(l+e)} \qquad (6.16)\] \[\lambda_1 = \frac{4L}{3} = \frac{4(l+e)}{3} \qquad (6.17)\] \[n_1 = 3n \qquad (6.18)\] \[\lambda_2 = \frac{4L}{5} = \frac{4(l+e)}{5} \qquad (6.19)\] \[n_2 = 5n \qquad (6.20)\] \[n_p = (2p+1)n \qquad (6.21)\] Only odd harmonics are present; even harmonics are absent.
Pipe open at both ends — antinode at each open end: \[v = n\lambda = 2nL \qquad (6.22)\] \[n = \frac{v}{2L} = \frac{v}{2(l+2e)} \qquad (6.23)\] \[\lambda_1 = L = (l+2e) \qquad (6.24)\] \[n_1 = 2n \qquad (6.25)\] \[\lambda_2 = \frac{2L}{3} = \frac{2(l+2e)}{3} \qquad (6.26)\] \[n_2 = 3n \qquad (6.27)\] \[n_p = (p+1)n \qquad (6.28)\] All harmonics are present, the sound is richer, and the fundamental frequency is double that of a closed pipe of the same length.
End correction determined from two pipes of the same diameter and lengths \(l_1\), \(l_2\) with fundamental frequencies \(n_1\), \(n_2\): \[e = \frac{n_1l_1 - n_2l_2}{2(n_2 - n_1)} \quad \text{or} \quad \frac{n_2l_2 - n_1l_1}{2(n_1 - n_2)} \quad \text{(open pipe)} \qquad (6.29)\] \[e = \frac{n_1l_1 - n_2l_2}{(n_2 - n_1)} \quad \text{or} \quad \frac{n_2l_2 - n_1l_1}{(n_1 - n_2)} \quad \text{(closed pipe)} \qquad (6.30)\]
Vibrating Strings
For a string of length \(l\), tension \(T\) and linear density \(m\), with nodes at both fixed ends, the speed of the transverse wave is \(v = \sqrt{T/m}\) and all harmonics are present: \[n_p = p\,n \qquad (6.31)\] \[n = \frac{1}{2l}\sqrt{\frac{T}{m}} \qquad (6.32)\]
Laws of a vibrating string: \[n \propto \frac{1}{l} \quad \text{if } T, m \text{ constant} \qquad (6.33)\] \[n \propto \sqrt{T} \quad \text{if } l, m \text{ constant} \qquad (6.34)\] \[n \propto \frac{1}{\sqrt{m}} \quad \text{if } T, l \text{ constant} \qquad (6.35)\]
With linear density \(m = \pi r^2\rho\), \[n \propto \frac{1}{\sqrt{\rho}} \quad \text{and} \quad n \propto \frac{1}{r} \qquad (6.36)\]
A sonometer, with its sound box, movable bridges P and Q, hanger and paper rider, is used to verify all three laws. Resonance is detected when the paper rider placed at the antinode is thrown off.
Beats
Beats are the periodic variations in intensity produced by superposing two sound waves of the same amplitude and slightly different frequencies travelling in the same direction. One waxing plus the successive waning is one beat.
For a listener at \(x = 0\): \[y = 2a\sin\left[2\pi\left(\frac{n_1+n_2}{2}\right)t\right]\cos\left[2\pi\left(\frac{n_1-n_2}{2}\right)t\right] \qquad (6.37)\] \[y = A\sin 2\pi nt \qquad (6.38)\] with \(n = (n_1+n_2)/2\) and \(A = 2a\cos[2\pi((n_1-n_2)/2)t]\).
The period of beats is \(T = 1/(n_1 - n_2)\), and the beat frequency is \[N = n_1 - n_2 \qquad (6.39)\]
Beats are audible only if the frequency difference is small, practically less than 6 to 7 Hz for a normal human ear.
Applications: tuning musical instruments to unison, Doppler RADAR for the speed of aeroplanes and vehicles, sonar for submarines, Doppler ultrasonography and echo cardiography for blood flow, and determination of an unknown frequency by reducing the beat frequency to zero.
Characteristics of Sound
Loudness is the perception of intensity. The pressure variation ranges from 28 Pa for the loudest tolerable sound to \(2.0 \times 10^{-5}\) Pa for the feeblest audible sound. The response of the ear is exponential, not linear.
Sound level: \[L = 10\log_{10}\left(\frac{I}{I_0}\right) \qquad (6.40)\] with \(I_0 = 10^{-12}\) W/m\(^2\); 1 decibel = 1/10 bel. Whisper is about 10 dB, normal speech about 60 dB at about 1 m, and the maximum tolerable level about 120 dB. Loudness at different frequencies is measured in phon, referred to a pure tone of 1000 Hz.
Pitch is the perception of frequency; higher frequency means higher pitch.
Quality or timbre distinguishes two sounds of the same pitch and loudness, and depends on the overtones present.
Musical sound arises from regular, periodic vibrations with harmonically related frequencies; noise is a mixture of unrelated frequencies. An octave contains eight frequencies in specific ratios, for example 240, 270, 300, 320, 360, 400, 450, 480 Hz.
Musical Instruments
- Stringed instruments (plucked, bowed and struck string types), wind instruments (freewind, edge and reedpipe types) and percussion instruments (stretched membranes and metal plates) all produce notes by setting up stationary waves, and are distinguished from one another by the mixture of overtones they generate.
6.13 Objective Questions
- Choose the correct option.
- When an air column in a pipe closed at one end vibrates such that three nodes are formed in it, the frequency of its vibrations is ……. times the fundamental frequency. (A) 2 (B) 3 (C) 4 (D) 5
- If two open organ pipes of length 50 cm and 51 cm sounded together produce 7 beats per second, the speed of sound is. (A) 307 m/s (B) 327 m/s (C) 350 m/s (D) 357 m/s
- The tension in a piano wire is increased by 25%. Its frequency becomes ….. times the original frequency. (A) 0.8 (B) 1.12 (C) 1.25 (D) 1.56
- Which of the following equations represents a wave travelling along the y-axis? (A) \(x = A\sin(ky - \omega t)\) (B) \(y = A\sin(kx - \omega t)\) (C) \(y = A\sin(ky)\cos(\omega t)\) (D) \(y = A\cos(ky)\sin(\omega t)\)
- A standing wave is produced on a string fixed at one end with the other end free. The length of the string (A) must be an odd integral multiple of \(\lambda/4\). (B) must be an odd integral multiple of \(\lambda/2\). (C) must be an odd integral multiple of \(\lambda\). (D) must be an even integral multiple of \(\lambda\).
6.14 Check Your Concepts
Both transverse and longitudinal mechanical waves travel through solids, but only longitudinal waves travel through liquids and gases. Explain the reason in terms of the elastic properties of the medium.
A wave pulse sent as a crest along a string returns as a trough from a rigid support, but returns as a crest from a ring that slides freely on a rod. Explain both observations, and state the phase change in each case.
For a longitudinal wave, a compression is reflected as a compression from a denser medium but as a rarefaction from a rarer medium. Explain both results by following the motion of the particles at the boundary.
Two identical pulses of opposite phase meet on a string, and for an instant the string is perfectly straight. Has the energy of the pulses disappeared? Justify your answer.
In the equation of a stationary wave, the position \(x\) and the time \(t\) appear in separate factors, whereas in a progressive wave they appear together in one bracket. What physical difference does this mathematical difference express?
Why is there no transfer of energy along a stationary wave, even though every particle except those at the nodes possesses energy?
Distinguish between free vibrations and forced vibrations, and state the condition under which the amplitude of forced vibrations becomes maximum.
Why are the even harmonics completely absent in the vibrations of an air column in a pipe closed at one end? Answer using the boundary conditions at the two ends.
An open pipe and a closed pipe have exactly the same length. Which of them produces the lower fundamental note, and in what ratio do their fundamental frequencies stand? Which of the two produces the richer sound, and why?
What is meant by end correction? Why is the correction applied twice for a pipe open at both ends and only once for a pipe closed at one end? On what property of the pipe does its value depend?
A sonometer wire is vibrating in its fundamental mode. State, with reasons based on the laws of a vibrating string, what happens to its frequency if (a) the vibrating length is halved, (b) the tension is made four times, and (c) the wire is replaced by one of the same material and same length but twice the radius.
Why is the paper rider in a sonometer experiment always placed midway between the two bridges, and why is it thrown off exactly at resonance?
A musician tunes two instruments by playing them together until the beats disappear, instead of judging the two notes separately by ear. Explain why this method is far more precise.
Two sound sources differ in frequency by 40 Hz. Will a listener hear beats? Give the reason for your answer.
An unknown fork gives 4 beats per second with a standard fork. Explain why this observation alone cannot fix the unknown frequency, and describe how loading the unknown fork with wax settles the question.
A flute and a sitar play the same note with the same loudness, and yet a listener can tell them apart at once. Which characteristic of sound is responsible, and on what does it depend?
Why is the sound level defined using a logarithm of the intensity ratio rather than the intensity itself? What does a rise of 20 dB mean in terms of intensity?
A guitarist touches a vibrating string lightly at its exact midpoint. Explain why the string does not fall silent, and state which harmonic is heard after the touch.
Answer in brief.
- A wave is represented by an equation \(y = A\sin(Bx + Ct)\). Given that the constants A, B and C are positive, can you tell in which direction the wave is moving?
- A string is fixed at the two ends and is vibrating in its fundamental mode. It is known that the two ends will be at rest. Apart from these, is there any position on the string which can be touched so as not to disturb the motion of the string? What will be the answer to this question if the string is vibrating in its first and second overtones?
- What are harmonics and overtones?
- For a stationary wave set up in a string having both ends fixed, what is the ratio of the fundamental frequency to the second harmonic?
- The amplitude of a wave is represented by \(y = 0.2\sin 4\pi\left[\dfrac{t}{0.08} - \dfrac{x}{0.8}\right]\) in SI units. Find (a) wavelength, (b) frequency and (c) amplitude of the wave.
State the characteristics of progressive waves.
State the characteristics of stationary waves.
Derive an expression for the equation of a stationary wave on a stretched string.
Find the amplitude of the resultant wave produced due to interference of two waves given as \(y_1 = A_1\sin\omega t\) and \(y_2 = A_2\sin(\omega t + \varphi)\).
State the laws of vibrating strings and explain how they can be verified using a sonometer.
Show that only odd harmonics are present in the vibrations of an air column in a pipe closed at one end.
Prove that all harmonics are present in the vibrations of the air column in a pipe open at both ends.
6.15 Practice with Numericals
A progressive wave is represented by \(y = 0.02\sin(8\pi x - 400\pi t)\), where \(x\) and \(y\) are in metre and \(t\) is in second. Find its amplitude, wave number, wavelength, angular frequency, frequency and speed, and state its direction of travel.
A sound wave of frequency 400 Hz travels in air with a speed of 340 m s\(^{-1}\). Calculate the phase difference between two points on the wave separated by a distance of 0.25 m at the same instant.
Two waves of the same frequency, having amplitudes 6 cm and 8 cm, arrive at a point with a phase difference of \(90^\circ\). Calculate the amplitude of the resultant wave and the ratio \(I_{max} : I_{min}\) for these two waves.
Two interfering waves have amplitudes in the ratio 3 : 1. Calculate the ratio of the maximum intensity to the minimum intensity in the resulting interference pattern.
In a stationary wave on a string, the distance between two successive antinodes is 20 cm and the frequency of vibration is 250 Hz. Find the wavelength and the speed of the progressive waves that produced this stationary wave.
A stationary wave is represented by \(y = 0.05\cos(5\pi x)\sin(300\pi t)\), where \(x\) and \(y\) are in metre and \(t\) is in second. Find the amplitude of each component wave, the wavelength, the frequency, the speed of the component progressive waves, and the positions of the first two nodes from \(x = 0\).
A pipe closed at one end has an air column of length 0.35 m. Taking the speed of sound in air as 350 m s\(^{-1}\) and neglecting end correction, calculate the fundamental frequency and the frequencies of the first two overtones, stating which harmonic each one is.
A pipe open at both ends has a length of 60 cm. Taking the speed of sound in air as 340 m s\(^{-1}\) and neglecting end correction, calculate the fundamental frequency and the frequencies of the first two overtones.
A cylindrical pipe of inner diameter 4.0 cm and length 24.0 cm is closed at one end. Calculate the end correction, the corrected length of the air column, and the fundamental frequency of the air column if the speed of sound in air is 340 m s\(^{-1}\).
Two pipes of the same diameter, open at both ends, have lengths 30 cm and 40 cm, and their fundamental frequencies are 550 Hz and 420 Hz respectively. Calculate the end correction for these pipes.
A sonometer wire of vibrating length 0.40 m has a linear density of \(2.0 \times 10^{-3}\) kg m\(^{-1}\) and is stretched by a tension of 80 N. Calculate its fundamental frequency and the frequency of its first overtone.
The fundamental frequency of a stretched wire is 300 Hz. Keeping its length and linear density unchanged, to what value must the tension be changed so that the fundamental frequency becomes 450 Hz? Express the answer as a multiple of the original tension.
Two sonometer wires of the same material and the same length are stretched by the same tension. Their radii are in the ratio 1 : 2. Find the ratio of their fundamental frequencies.
A string 90 cm long, fixed at both ends, vibrates in three loops when driven at 210 Hz. Calculate the wavelength of the stationary wave and the speed of the progressive waves on the string.
Two tuning forks of frequencies 512 Hz and 516 Hz are sounded together. Calculate the beat frequency, the period of the beats, and the number of beats heard in 5 seconds.
Two sound waves of wavelengths 100 cm and 101 cm travelling in the same medium produce 5 beats per second. Calculate the speed of sound in that medium.
The intensity of sound at a point is \(4.0 \times 10^{-5}\) W m\(^{-2}\). Taking \(I_0 = 10^{-12}\) W m\(^{-2}\), calculate the sound level at that point in decibel. Find also the new sound level if the intensity is reduced to one thousandth of this value.
An air column of length 34 cm is set into vibration with a source of sound, the speed of sound in air being 340 m s\(^{-1}\). Neglecting end correction, calculate the frequency of the third overtone when the air column is (a) closed at one end and (b) open at both ends.
A wave of frequency 500 Hz is travelling with a speed of 350 m/s. (a) What is the phase difference between two displacements at a certain point at times 1.0 ms apart? (b) What will be the smallest distance between two points which are 45º out of phase at an instant of time?
A sound wave in a certain fluid medium is reflected at an obstacle to form a standing wave. The distance between two successive nodes is 3.75 cm. If the velocity of sound is 1500 m/s, find the frequency.
Two sources of sound are separated by a distance 4 m. They both emit sound with the same amplitude and frequency (330 Hz), but they are 180º out of phase. At what points between the two sources will the sound intensity be maximum? (Take velocity of sound to be 330 m/s)
Two sound waves travel at a speed of 330 m/s. If their frequencies are also identical and are equal to 540 Hz, what will be the phase difference between the waves at points 3.5 m from one source and 3 m from the other if the sources are in phase?
Two wires of the same material and same cross section are stretched on a sonometer. One wire is loaded with 1.5 kg and another is loaded with 6 kg. The vibrating length of the first wire is 60 cm and its fundamental frequency of vibration is the same as that of the second wire. Calculate the vibrating length of the other wire.
A pipe closed at one end can produce overtones at frequencies 640 Hz, 896 Hz and 1152 Hz. Calculate the fundamental frequency.
A standing wave is produced in a tube open at both ends. The fundamental frequency is 300 Hz. What is the length of the tube in the fundamental mode? (speed of the sound = 340 m s\(^{-1}\))
Find the fundamental, first overtone and second overtone frequencies of a pipe, open at both the ends, of length 25 cm if the speed of sound in air is 330 m/s.
A pipe open at both the ends has a fundamental frequency of 600 Hz. The first overtone of a pipe closed at one end has the same frequency as the first overtone of the open pipe. How long are the two pipes? (Take velocity of sound to be 330 m/s)
A string 1 m long is fixed at one end. Transverse vibrations of frequency 15 Hz are imposed at the free end. Due to this, a stationary wave with four complete loops is produced on the string. Find the speed of the progressive wave which produces the stationary wave. [Hint: Remember that the free end is an antinode.]
A violin string vibrates with a fundamental frequency of 440 Hz. What are the frequencies of the first and second overtones?
A set of 8 tuning forks is arranged in a series of increasing order of frequencies. Each fork gives 4 beats per second with the next one and the frequency of the last fork is twice that of the first. Calculate the frequencies of the first and the last fork.
A sonometer wire is stretched by a tension of 40 N. It vibrates in unison with a tuning fork of frequency 384 Hz. How many beats get produced in two seconds if the tension in the wire is decreased by 1.24 N?
A sonometer wire of length 0.5 m is stretched by a weight of 5 kg. The fundamental frequency of vibration is 100 Hz. Calculate the linear density of the wire.
The string of a guitar is 80 cm long and has a fundamental frequency of 112 Hz. If a guitarist wishes to produce a frequency of 160 Hz, where should the person press the string?