JEE Physics Waves Notes Set 05

Equation of a Travelling Wave

A wave moving along +x with speed v is described by \( y=f(x-vt) \); for a sine wave \( y=A\sin(kx-\omega t+\phi) \). Here A is the amplitude, \( k=\frac{2\pi}{\lambda} \) the wave number, \( \omega=2\pi f=\frac{2\pi}{T} \) the angular frequency and \( \phi \) the phase constant. The wave speed is \( v=\frac{\omega}{k}=f\lambda \). A wave moving along −x has \( kx+\omega t \).
Each particle performs simple harmonic motion about its mean position with the same amplitude and frequency, lagging in phase by kx. The transverse particle velocity is \( \frac{\partial y}{\partial t}=-A\omega\cos(kx-\omega t+\phi) \), with maximum \( A\omega \) (not to be confused with the wave speed), and the maximum acceleration is \( A\omega^2 \). The particle velocity is zero at crests and troughs and maximum at the mean position. Points half a wavelength apart are in opposite phase, and particles at a distance of one wavelength vibrate in phase.
The frequency of a wave is fixed by its source and does not change when the wave enters another medium; speed and wavelength do.

Speed of a Transverse Wave on a String

For a string of tension T and mass per unit length \( \mu \): \( v=\sqrt{\frac{T}{\mu}} \), with \( \mu=\rho A \) for a wire of density \( \rho \) and cross-section A. It depends only on the string, not on the frequency or amplitude.
If the tension varies, as in a hanging rope of mass m with a block M attached, the speed varies with position, \( v=\sqrt{T(x)/\mu} \), and at the free end of an accelerating arrangement the effective gravity \( g+a \) must be used.

Energy and Power

The wave carries energy along the string. The average power transmitted is \( P_{avg}=\frac12\mu\omega^2A^2v=2\pi^2\mu vf^2A^2 \). The power is proportional to the square of the amplitude and to the square of the frequency, and to the wave speed. The intensity of a wave falls as the amplitude falls, while frequency remains the same.

Interference and Superposition

Waves add algebraically. For two waves of equal frequency with amplitudes \( A_1 \) and \( A_2 \) and phase difference \( \phi \): \( A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi} \). The maximum is \( A_1+A_2 \) at \( \phi=0,2\pi,\dots \) and the minimum is \( |A_1-A_2| \) at \( \phi=\pi,3\pi,\dots \). For equal waves \( A=2A_0\cos\frac\phi2 \). Intensity is proportional to the square of the amplitude, so the maximum intensity is four times that of a single wave.
Phase difference and path difference are related by \( \phi=\frac{2\pi}{\lambda}\Delta x \), and a time delay \( \Delta t \) gives \( \phi=\omega\Delta t \). Several waves can be added as phasors.

Reflection and Transmission

At a rigid boundary a wave is reflected with a phase change of \( \pi \) (inverted pulse), and at a free end it is reflected without a phase change. At the junction of two strings, the reflected pulse is inverted if the second string is heavier (slower wave); then the transmitted wave travels more slowly, \( v' \lt v \). If the second string is lighter, the reflected pulse is not inverted.

Standing Waves and Natural Frequencies

Two identical waves travelling in opposite directions form a stationary wave, \( y=2A\sin kx\cos\omega t \) (for a node at x = 0). Nodes are points of zero amplitude, spaced \( \lambda/2 \) apart; antinodes lie midway between nodes with amplitude 2A. All points between two adjacent nodes oscillate in phase, and neighbouring loops are in opposite phase. A stationary wave carries no net energy.
String fixed at both ends, length L: \( \lambda_n=\frac{2L}{n} \) and \( f_n=\frac{n}{2L}\sqrt{\frac{T}{\mu}}=nf_1 \), n = 1, 2, 3, ... (all harmonics). The nth harmonic has n loops, and consecutive resonant frequencies differ by the fundamental. For wires of the same length, \( f\propto\frac{1}{r}\sqrt{\frac{T}{\rho}} \).
A node at the junction of two wires of different linear density requires each wire to carry a whole number of loops at the same frequency, so \( n_1v_1=n_2v_2 \) when the lengths are equal.

Worked Examples

1. A string of mass 0.18 kg and length 6 m carries a wave of frequency 50 Hz and amplitude 7.5 cm with four cycles on it. \( \mu=0.03 \) kg/m, \( \lambda=1.5 \) m, \( v=75 \) m/s, \( P=\frac12\mu\omega^2A^2v\approx625 \) W.
2. A wire 10 m long, mass 100 g and tension 25 N: \( v=50 \) m/s and \( f_n=2.5n \) Hz.
3. If a wire resonates at 350 Hz and then at 420 Hz, its fundamental is 70 Hz.