JEE Physics Waves Notes Set 03

Wave Motion and the Travelling Wave Equation

A wave is a disturbance that carries energy and momentum through a medium without carrying matter along. In a transverse wave the particles vibrate at right angles to the direction of travel; in a longitudinal wave they vibrate along it. Waves on a string are transverse, and sound in air is longitudinal.
A wave moving along +x with speed v has the form \( y=f(x-vt) \), and one moving along -x has the form \( y=f(x+vt) \). The harmonic version is \( y=A\sin(kx-\omega t+\phi) \), where \( k=\frac{2\pi}{\lambda} \) is the wave number, \( \omega=2\pi f \) the angular frequency and \( v=f\lambda=\frac{\omega}{k} \).
Reading the sign: if the kx and \( \omega t \) terms have opposite signs the wave travels towards +x; if they have the same sign it travels towards -x. A wave given by \( y=A\cos^2(\ldots) \) or any similar form should first be rewritten as a sum of simple sines and cosines.

Particle Velocity, Acceleration and Phase

Every particle of the medium performs SHM about its own mean position. Differentiating \( y=A\sin(kx-\omega t) \) with respect to t gives the particle velocity \( v_p=-A\omega\cos(kx-\omega t) \), and with respect to x gives the slope \( \frac{\partial y}{\partial x}=Ak\cos(kx-\omega t) \). Hence \( v_p=-v\frac{\partial y}{\partial x} \) for a wave moving along +x: the particle velocity is the wave speed multiplied by the negative of the slope.
Maximum particle speed is \( A\omega \) and maximum particle acceleration is \( A\omega^2 \); neither is the same as the wave speed. The ratio \( \frac{v_{p,max}}{v}=Ak \) is small for gentle waves.
Phase difference between two particles separated by a distance \( \Delta x \) is \( \frac{2\pi}{\lambda}\Delta x \); between two instants at one particle it is \( \omega\Delta t \). The wave equation \( \frac{\partial^2y}{\partial t^2}=v^2\frac{\partial^2y}{\partial x^2} \) is satisfied by every function of \( x\pm vt \).

Speed of a Transverse Wave on a String

For a taut string of tension T and mass per unit length \( \mu \), the speed is \( v=\sqrt{T/\mu} \). It depends only on the string, not on the frequency or amplitude of the wave.
A hanging rope has tension that grows with height: for a uniform rope the tension at height h above the lower end is \( \mu gh \) (plus the weight of any load), so \( v=\sqrt{gh} \) and the pulse accelerates upwards with \( a=\frac g2 \). The time to climb a rope of length L is \( 2\sqrt{L/g} \). A wire fixed between rigid supports and cooled gains tension \( T=YA\alpha\Delta T \).
The frequency of a wave is fixed by its source; when the speed changes (for instance up a hanging rope), the wavelength changes in proportion to the speed.

Energy and Intensity of a Wave on a String

A string element carries both kinetic and elastic potential energy; both are greatest at the mean position and zero at the extremes of the element's motion. The energy per unit length is \( \frac12\mu\omega^2A^2 \) (the sum of equal kinetic and potential parts).
The average power transmitted is \( P=\frac12\mu v\omega^2A^2 \), so it grows as the square of both amplitude and frequency. Intensity of any wave is proportional to \( A^2\omega^2 \), so two waves of equal intensity can have different amplitudes if their frequencies differ.

Superposition and Interference

When two waves meet, the resultant displacement is the algebraic sum of the individual displacements, and afterwards each wave continues unchanged. Two equal-frequency waves with phase difference \( \phi \) give a resultant amplitude \( A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi} \), and for equal amplitudes \( A_R=2A\cos\frac\phi2 \).
Constructive interference needs \( \phi=2n\pi \), a path difference \( n\lambda \); destructive interference needs \( \phi=(2n+1)\pi \), a path difference \( (n+\frac12)\lambda \). Intensity \( I=I_1+I_2+2\sqrt{I_1I_2}\cos\phi \), so \( I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2 \) and \( I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2 \).
Several waves can be added with phasors: draw each as an arrow of length equal to its amplitude at an angle equal to its phase, add the arrows as vectors, and read off the amplitude and phase of the resultant. Three waves of amplitude 10, 4 and 7 with successive phase steps of \( \frac\pi2 \) give a resultant of 5 units. Pulses that overlap with opposite signs give a flat string at one instant, but their energy is then entirely kinetic.

Reflection and Transmission at a Boundary

At a fixed (rigid) end a pulse is reflected inverted, a phase change of \( \pi \); at a free end it is reflected upright. Where a string joins another of different density, part of the wave is reflected and part is transmitted. With tension the same on both sides, the wave goes from speed \( v_1 \) to \( v_2 \) and the frequency does not change.
The amplitude coefficients are \( A_r=\frac{v_2-v_1}{v_2+v_1}A_i \) and \( A_t=\frac{2v_2}{v_1+v_2}A_i \), and they obey \( A_t=A_i+A_r \). If the second string is heavier (\( v_2Energy is conserved: the transmitted power fraction is \( \frac{4v_1v_2}{(v_1+v_2)^2} \) and the reflected power fraction is \( \left(\frac{v_2-v_1}{v_1+v_2}\right)^2 \). Wavelength changes in proportion to speed because frequency is fixed.

Standing Waves

Two identical waves travelling in opposite directions form a standing wave, for example \( y=2A\sin kx\cos\omega t \). Each particle vibrates in SHM of amplitude \( 2A|\sin kx| \); the nodes (zero amplitude) are at \( kx=n\pi \) and the antinodes (amplitude 2A) lie midway. Adjacent nodes, or adjacent antinodes, are \( \frac\lambda2 \) apart and a node and neighbouring antinode \( \frac\lambda4 \).
In a standing wave no energy flows past a node. All particles between two consecutive nodes (one loop) vibrate in phase, and adjacent loops are in opposite phase. All particles pass through the mean position together twice per period, and at the extremes they are all momentarily at rest together.
If the two waves have unequal amplitudes (for example a partly reflected wave of amplitude 0.8A), the result is a standing wave plus a travelling wave: the nodes are not perfectly still and the amplitudes at antinode and node are \( A_i+A_r \) and \( A_i-A_r \).

Vibrations of Strings

A string of length L fixed at both ends has nodes at both ends, so \( L=\frac{n\lambda}{2} \) and the natural frequencies are \( f_n=\frac{n}{2L}\sqrt{\frac T\mu} \), where n = 1, 2, 3, ... is the number of loops. The first harmonic is the fundamental and the nth overtone is the (n+1)th harmonic. Successive resonant frequencies differ by the fundamental.
For a given tuning fork and fixed length, the number of loops p satisfies \( p\sqrt T=\text{constant} \). Weights dipped in a liquid reduce the tension by the buoyant force. A string driven at one end, free to move there, behaves with an antinode at the driven end.
A sonometer uses these relations: \( f\propto\frac1L,\ f\propto\sqrt T,\ f\propto\frac1{\sqrt\mu} \).

Worked Examples

Worked Example 1. A wave on a string is \( y=0.02\sin(40t-2x) \) in SI units. Find the wave speed, the maximum particle speed and the tension if \( \mu=0.05 \) kg/m.
Answer:
Comparing with \( y=A\sin(\omega t-kx) \): \( \omega=40 \), \( k=2 \), so \( v=\frac{40}{2}=20\ \text{m/s} \) towards +x.
Maximum particle speed \( =A\omega=0.02\times40=0.8\ \text{m/s} \).
\( T=\mu v^2=0.05\times400=20\ \text{N} \).

Worked Example 2. A 60 cm string fixed at both ends has mass 3 g and tension 120 N. Find its fundamental frequency and the frequency of the third harmonic.
Answer:
\( \mu=\frac{0.003}{0.6}=0.005\ \text{kg/m} \); \( v=\sqrt{\frac{120}{0.005}}=155\ \text{m/s} \).
\( f_1=\frac{v}{2L}=\frac{155}{1.2}=129\ \text{Hz} \) and \( f_3=3f_1\approx387\ \text{Hz} \).

Worked Example 3. A pulse of amplitude 6 mm travels along a light string and meets a heavier string in which its speed is one third. Find the amplitudes of the reflected and transmitted pulses.
Answer:
With \( v_2=\frac{v_1}{3} \): \( \frac{A_r}{A_i}=\frac{1/3-1}{1/3+1}=-\frac12 \), so the reflected pulse is 3 mm and inverted.
\( \frac{A_t}{A_i}=\frac{2/3}{4/3}=\frac12 \), so the transmitted pulse is 3 mm.