CBSE Class 11 Physics Wave Motion Exam Notes

Class 12 Physics: Wave Motion Exam Notes

Wave Motion

A wave motion is a disturbance in a medium which is handed over from one part of the medium to the next part due to repeated to-and-fro motion of the particles of the medium about their mean position.

Transverse wave motion is the wave motion in which the particles of the medium vibrate at right angles to the direction of propagation of the wave. The vibrations of a stretched string and electromagnetic light waves are examples of transverse waves. In longitudinal wave motion, the particles of the medium vibrate along the direction of propagation of the wave. Sound waves travel as longitudinal waves in air and require a medium of travel.

Equation of a Progressive Wave

The equation of a plane progressive wave can be represented as a wave motion in which the wave progresses along the \(x\) direction and the particles of the medium vibrate to and fro about their mean position as a function of time. The displacement \(y\) of the particles of the medium at time \(t\) in wave motion at a distance \(x\) from the origin, for a plane progressive wave travelling in the positive \(x\) direction, is given by:

\[y = a\sin\frac{2\pi}{\lambda}(vt - x)\]

where \(\lambda\) is the wavelength and \(v\) is the velocity of the wave motion. It can also be written in terms of the time period \(T\) of the wave as:

\[y = a\sin 2\pi\left(\frac tT - \frac x\lambda\right) = a\sin\left(\omega t - kx\right)\]

where \(\omega = 2\pi n\) (\(n\) is the frequency) and

\[k = \frac{2\pi}{\lambda} = \frac{\omega}{v}\]

A progressive wave travelling in the negative \(x\) direction is given by

\[y = a\sin(\omega t + kx)\quad\text{or}\quad y = a\sin\frac{2\pi}{\lambda}(vt + x)\]

As the vibrations may not start from the origin, this gives rise to a phase difference, which may be a fraction of the wavelength \(\lambda\), the time period \(T\) or the angle \(2\pi\), by which a wave may be out of step with the wave starting from the origin at time \(t = 0\). Two waves may also be out of step with each other, giving rise to a phase difference of angle \(\phi\), which may be positive or negative with respect to each other or with respect to the origin.

Therefore, the general equation for a progressive wave travelling in the positive \(x\) direction is given by

\[y = a\sin(\omega t - kx \pm \phi)\]

Velocity of a transverse wave along a string:

\[v = \sqrt{\frac{T}{\mu}}\]

where \(T\) is the tension in the string and \(\mu\) is the mass of the string per unit length.

Velocity of Sound Waves

Sound waves travel in air or in gaseous media as longitudinal waves. When these waves travel, compression and rarefaction are produced in the layers of air, such that the particles in the layers of air move to and fro about their mean position in the same direction as the direction of propagation of the sound waves. The speed \(v\) of longitudinal waves in an elastic medium of modulus of elasticity \(E\) and density \(\rho\) is given by

\[v = \sqrt{\frac{E}{\rho}}\]

For liquids and gases, \(E\) is the bulk modulus of elasticity. For a thin solid rod, \(E\) is Young's modulus. For large solids, \(E\) depends upon the bulk modulus and the shear modulus.

Newton's formula: Newton assumed that the changes produced due to the propagation of sound in gases are isothermal, which implied that a compressed layer of air at higher temperature loses heat immediately to the surroundings and a rarefied layer of air at lower temperature gains heat from the surroundings immediately, so that the temperature of the air remains constant. As the modulus of elasticity for an isothermal change is equal to the pressure \(P\), Newton's formula for the velocity of sound for isothermal changes gives

\[v = \sqrt{\frac{P}{\rho}}\]

Laplace's correction: Laplace showed that sound is propagated in air or gases under adiabatic changes, because the compressions and rarefactions produced due to the propagation of sound follow each other so rapidly that there is no time available for the compressed layer at a higher temperature and the rarefied layer at a lower temperature to equalise their temperatures with the surroundings. Thus the velocity \(v\) of sound travelling under adiabatic conditions in a gas is given by Laplace's formula:

\[v = \sqrt{\frac{E_{adiabatic}}{\rho}} = \sqrt{\frac{\gamma P}{\rho}}\]

because \(E_{adiabatic} = \gamma P\) and \(\gamma = \dfrac{C_p}{C_v}\).

Substituting \(\gamma = 1.41\) for air, density of air \(= 1.293\ \text{kg/m}^3\) and atmospheric pressure \(= 1.013\times10^5\ \text{N/m}^2\), the velocity of sound in air is \(v = 332\) m/s. In general, the velocity of sound in solids is greater than the velocity of sound in liquids, and the velocity of sound in liquids is greater than the velocity of sound in gases.

Effect of Temperature, Pressure and Humidity on Velocity of Sound

(i) Effect of temperature: As \(PV = nRT\) and \(\rho = \dfrac{M}{V}\),

\[v = \sqrt{\frac{\gamma P V}{M}} = \sqrt{\frac{\gamma R T}{M_m}}\]

where \(M_m\) is the mass of one mole of the gas. Thus the velocity of sound is directly proportional to the square root of the absolute temperature. If \(v_t\) and \(v_0\) are the velocities of sound at \(t^\circ\)C and \(0^\circ\)C respectively, then

\[\frac{v_t}{v_0} = \sqrt{\frac{T_t}{T_0}} = \sqrt{\frac{273 + t}{273}}\]

where \(T_t\) and \(T_0\) are the respective absolute temperatures.

(ii) Effect of pressure: If the temperature of the gas remains constant, the velocity of sound does not change with the change of pressure, because \(\dfrac{P}{\rho}\) is a constant quantity.

(iii) Effect of humidity: As the density of water vapour at S.T.P., \(0.8\ \text{kg/m}^3\), is lower than the density of dry air, \(1.29\ \text{kg/m}^3\), the speed of sound in air increases when the humidity increases in moist air.

Intensity of Sound

The intensity of sound \(I\) at any point is the amount of sound energy transmitted per second across a unit area normal to the direction of propagation of the sound waves. The intensity follows an inverse square law of distance, i.e. \(I \propto \dfrac{1}{r^2}\), and \(I\) is directly proportional to the square of the amplitude. The level of intensity of sound as perceived by human beings is called the loudness. The intensity level or loudness \(L\) is quantitatively measured as compared to the minimum intensity of sound audible to the human ear. The intensity level or loudness \(L\), measured in units of decibel (dB), is given as follows for a sound of intensity \(I\):

\[L = 10\log_{10}\left(\frac{I}{I_0}\right)\]

where \(I_0\) is the minimum audible intensity, which is equal to \(10^{-12}\ \text{watt/metre}^2\). Thus the intensity of sound increases by a factor of 10 when the intensity level or loudness increases by 10 decibels.

Superposition of Waves

When two or more waves travelling in the same region act on the particles simultaneously, the intensity of the resultant wave is modified due to superposition of the waves according to the following principle.

Superposition principle: If two or more waves arrive at a point simultaneously, then the resultant displacement at any point is equal to the vector sum of the displacements due to the individual waves.

\[y = y_1 + y_2 + \dots + y_n\]

where \(y\) is the resultant displacement due to the superposition of the displacements \(y_1, y_2, \dots, y_n\) of the individual waves. It can give rise to the following phenomena:

(i) Interference: When two waves of the same frequency and of constant phase difference travelling in the same direction superpose, it can give rise to a modification in intensity in the form of alternate maximum and minimum intensities, which is called the interference phenomenon.

If the waves \(y_1 = a_1\sin(\omega t - kx)\) and \(y_2 = a_2\sin(\omega t - kx + \phi)\) superpose, then by applying the principle of superposition,

\[y = y_1 + y_2 = R\sin(\omega t - kx + \theta)\]

where the resultant amplitude and phase angle are

\[R = \sqrt{a_1^2 + a_2^2 + 2a_1a_2\cos\phi},\qquad \theta = \tan^{-1}\left[\frac{a_2\sin\phi}{a_1 + a_2\cos\phi}\right]\]

When \(\phi = 2\pi n\), where \(n = 0, 1, 2, \dots\), it produces constructive interference, which gives \(R = R_{max} = a_1 + a_2\). When \(\phi = (2n+1)\pi\), where \(n = 0, 1, 2, \dots\), \(R = R_{min} = a_1 - a_2\) (in magnitude), i.e. the resultant amplitude is minimum due to destructive interference.

As intensity is proportional to the square of the amplitude, the ratio of the maximum intensity \(I_{max}\) to the minimum intensity \(I_{min}\) is given by

\[\frac{I_{max}}{I_{min}} = \frac{(a_1 + a_2)^2}{(a_1 - a_2)^2}\]

(ii) Beats: If two sources of slightly different frequencies produce sound waves in the same direction at the same point, these waves superpose to produce alternate loud and feeble sounds. Such variation in loudness is called beats. The number of times such a fluctuation in loudness from maxima to minima takes place per second is called the beat frequency.

If two waves \(y_1 = a\sin(2\pi n_1 t)\) and \(y_2 = a\sin(2\pi n_2 t)\) of respective frequencies \(n_1\) and \(n_2\) superpose at the same place \(x = 0\), then \(y = y_1 + y_2 = a\left[\sin(2\pi n_1 t) + \sin(2\pi n_2 t)\right]\), so

\[y = 2a\cos\left[\frac{2\pi(n_1 - n_2)t}{2}\right]\sin\left[\frac{2\pi(n_1 + n_2)t}{2}\right] = 2a\cos[\pi(n_1 - n_2)t]\,\sin[\pi(n_1 + n_2)t]\]

\[y = A\sin[\pi(n_1 + n_2)t],\qquad A = 2a\cos[\pi(n_1 - n_2)t]\]

The resultant wave is a harmonic wave with frequency \(\dfrac{n_1 + n_2}{2}\), but its amplitude varies harmonically as a function of time with a frequency governed by the difference of the frequencies \(n_1 - n_2\). Thus the beat frequency \(n_B\) is

\[n_B = n_1 - n_2\]

If \(n_1 - n_2\) is small, i.e. the number of times the intensity of sound fluctuates between maxima and minima per second is small (less than about 10–15 per second), the beats can be heard distinctly.

(iii) Stationary waves: When two progressive waves of the same frequency moving in opposite directions superpose, stationary waves are formed.

Consider the superposition of two such waves along a stretched string having fixed ends, where a harmonic wave travels towards the right as \(y_1 = a\sin\dfrac{2\pi}{\lambda}(vt - x)\). This wave is reflected from the second point and, due to the reflection, its amplitude becomes \(-a\) due to a change of phase \(\pi\). The reflected wave \(y_2 = a\sin\dfrac{2\pi}{\lambda}(vt + x)\) travels towards the left, and these waves superpose to give the resultant displacement \(y\):

\[y = a\sin\frac{2\pi}{\lambda}(vt - x) - a\sin\frac{2\pi}{\lambda}(vt + x) = -2a\cos\left(\frac{2\pi vt}{\lambda}\right)\sin\frac{2\pi x}{\lambda}\]

\[y = -A\cos\left(\frac{2\pi vt}{\lambda}\right),\qquad A = 2a\sin\left(\frac{2\pi x}{\lambda}\right)\]

The string executes simple harmonic motion such that the particles of the string vibrate with the same frequency but with different amplitudes. Such a resultant wave is called a standing or stationary wave. The position along the string where the amplitude is always zero is called a node, and the position where the amplitude is always maximum and equal to \(2a\) is called an antinode.

For nodes: \(A = 2a\sin\left(\dfrac{2\pi x}{\lambda}\right) = 0\), so \(\dfrac{2\pi x}{\lambda} = n\pi\) where \(n = 0, 1, 2, 3, \dots\)

\[x = \frac{n\lambda}{2}\]

This gives the position of the \(n\)th node, and the distance between successive nodes is \(\dfrac\lambda2\).

For antinodes: \(A = 2a\sin\left(\dfrac{2\pi x}{\lambda}\right) = 2a\), so \(\dfrac{2\pi x}{\lambda} = (2n-1)\dfrac\pi2\) where \(n = 1, 2, 3, \dots\)

\[x = (2n-1)\frac\lambda4,\ \text{ i.e. }\ \frac\lambda4,\ \frac{3\lambda}{4},\ \frac{5\lambda}{4},\ \dots\]

Such points are called antinodes, with maximum amplitude \(2a\). The distance between a successive node and antinode is \(\dfrac\lambda4\).

Modes of Stationary Waves in a Stretched String

In a stretched string, a transverse wave is produced by plucking it or by sending a progressive wave from a tuning fork. Such a wave is reflected from the other end to superpose with the original wave to produce a stationary wave. The fundamental or first harmonic mode will have one antinode in the middle, whereas there are nodes at the end points where the string is fixed and does not vibrate.

\[l = \frac{\lambda_1}{2},\qquad \lambda_1 = 2l\]

If a string of mass \(m\) per unit length is stretched under a tension \(T\) and a transverse wave travels along the string with a velocity \(v\), then \(v = \sqrt{T/m}\). If \(n\) is the frequency and \(\lambda\) the wavelength of the wave,

\[n\lambda = \sqrt{\frac Tm}\quad\text{or}\quad n = \frac1\lambda\sqrt{\frac Tm}\]

The fundamental frequency \(n_1\) is given by

\[n_1 = \frac{v}{\lambda_1} = \frac{1}{2l}\sqrt{\frac Tm}\]

If the string vibrates in two loops, it gives the first overtone or second harmonic mode of vibration, \(\lambda_2 = l\):

\[n_2 = \frac{v}{\lambda_2} = \frac1l\sqrt{\frac Tm} = \frac{2}{2l}\sqrt{\frac Tm}\]

If the string vibrates in three loops, it gives rise to the second overtone or third harmonic mode, \(l = \dfrac{3\lambda_3}{2}\):

\[\lambda_3 = \frac{2l}{3},\qquad n_3 = \frac{v}{\lambda_3} = \frac{3}{2l}\sqrt{\frac Tm}\]

If the string vibrates in \(p\) segments or loops, the frequency of vibration is \(n = \dfrac{p}{2l}\sqrt{\dfrac Tm}\). A sonometer, consisting of a stretched string placed on a hollow wooden box, can produce such modes of vibration in the string when the tension \(T\) is applied.

Modes of Vibration in Air Columns

Longitudinal stationary waves can be produced in open and closed pipes like organ pipes, having both ends open and one end closed respectively. If a tuning fork produces a sound wave at the open end, it is reflected from the second end such that the incident and reflected waves superpose to produce stationary waves. The closed end is always a node and the open end is always an antinode.

Open pipe: The first three modes of vibration of an open pipe are given as follows (figure).

For the fundamental or first harmonic mode:

\[l = \frac{\lambda_1}{2},\quad \lambda_1 = 2l,\quad n_1 = \frac{v}{2l}\]

where \(v\) is the velocity of sound waves. For the first overtone or second harmonic mode:

\[l = \lambda_2,\quad n_2 = \frac{v}{l}\]

For the second overtone or third harmonic mode:

\[l = \frac{3\lambda_3}{2},\quad \lambda_3 = \frac{2l}{3},\quad n_3 = \frac{3v}{2l}\]

For the \(p\)th harmonic:

\[n_p = \frac{pv}{2l}\]

Closed pipe: The first three modes of vibration of a closed pipe are given as follows (figure).

For the fundamental or first harmonic mode:

\[l = \frac{\lambda_1}{4},\quad \lambda_1 = 4l,\quad n_1 = \frac{v}{4l}\]

For the first overtone or third harmonic:

\[l = \frac{3\lambda_2}{4},\quad \lambda_2 = \frac{4l}{3},\quad n_2 = \frac{3v}{4l}\]

For the second overtone or fifth harmonic:

\[l = \frac{5\lambda_3}{4},\quad \lambda_3 = \frac{4l}{5},\quad n_3 = \frac{5v}{4l}\]

For the \(p\)th overtone or \((2p+1)\)th harmonic:

\[n = \frac{(2p+1)v}{4l}\]

End correction: At the open end of the pipe, the antinode is formed at a small distance outside the open end. Thus the corrected length of a closed pipe is \(l + e\) and that of an open pipe is \(l + 2e\), where \(e = 0.3D\) and \(D\) is the internal diameter of the pipe.

Doppler Effect

When a source of sound or an observer, or both, are moving relative to each other, there is an apparent change in the frequency of sound as heard by the observer. This is called the Doppler effect. The apparent frequency increases if the source is moving towards the observer or the observer is moving towards the source. The apparent frequency decreases if either the source is moving away from the observer or the observer is moving away from the source. This apparent change in frequency is due to the basic effect of motion of the source, which changes the effective wavelength, and the basic effect of motion of the observer, which changes the number of waves received per second by the observer.

If both the source and the observer are moving in the positive direction of the \(x\)-axis (from source towards observer), the sound of frequency \(n\) propagating in air with velocity \(v\) in still air will result in an apparent frequency \(n'\) heard by the observer as

\[n' = \left(\frac{v - v_0}{v - v_s}\right)n\]

If the direction of motion of the source or the observer is changed, the sign of \(v_0\) or \(v_s\) is changed accordingly. Thus the apparent frequency \(n'\) in still air for the different cases is as follows:

1. Both the source and the observer are moving towards the right, the source approaching a receding observer: \(n' = \left(\dfrac{v - v_0}{v - v_s}\right)n\)
2. The source and the observer are receding from each other: \(n' = \left(\dfrac{v - v_0}{v + v_s}\right)n\)
3. The source and the observer are moving towards each other: \(n' = \left(\dfrac{v + v_0}{v - v_s}\right)n\)
4. The observer is approaching a receding source: \(n' = \left(\dfrac{v + v_0}{v + v_s}\right)n\)

If the wind is blowing with velocity \(\omega\) in the direction of the sound, \(\omega\) is added to \(v\), and if the wind is blowing with velocity \(\omega\) opposite to the direction of the sound, \(\omega\) is subtracted from \(v\). The general formula for the apparent frequency \(n'\) due to the Doppler effect is

\[n' = \left(\frac{v \pm \omega \mp v_0}{v \pm \omega \mp v_s}\right)n\]