Chapter 14 ยท Introduction

Chapter 14 โ€” Waves

A ripple on a pond, a pulse along a rope, the sound of a voice, the shaking of an earthquake: disturbances that travel, carrying energy but not matter. This chapter describes waves with one equation, finds how fast they go, and adds them together to make standing waves, musical notes and beats.

A stadium crowd does the "Mexican wave": each spectator simply stands up and sits down again, a moment after the neighbour, and yet a great hump races round the ground at some $10$โ€“$20$ metres per second. Nobody leaves their seat. That is the essence of a wave: a pattern of disturbance that travels through a medium while each part of the medium only moves a little to and fro about its own place. The same idea explains why you hear a friend across a room without any air blowing from their mouth to your ear, why a seismometer in Delhi records an earthquake in Japan minutes after it happens, and why a guitar string, fixed at both ends, sings only a particular set of notes.

This chapter builds on the oscillations of Chapter 13 โ€” every particle in a simple wave executes simple harmonic motion. You will first tell transverse waves from longitudinal ones and write the displacement of a progressive wave as $y(x,t)=a\sin(kx-\omega t+\phi)$, with its wavelength, wave number, period and frequency. Next you will find what fixes the speed of a wave: $\sqrt{T/\mu}$ for a stretched string, and $\sqrt{\gamma p/\rho}$ for sound in a gas, where Newton's famous estimate needed Laplace's correction. Then you will add waves using the principle of superposition, reflect them from fixed and free ends, and trap them between boundaries as standing waves โ€” the normal modes and harmonics of strings and organ pipes. Finally, two notes of nearly equal frequency combine to rise and fall in loudness: beats, which musicians use to tune their instruments.

Disturbance travelsโ†’Transverse / longitudinalโ†’$y=a\sin(kx-\omega t)$โ†’$v=\nu\lambda=\dfrac{\omega}{k}$โ†’$v=\sqrt{\dfrac T\mu}$โ†’$v=\sqrt{\dfrac{\gamma p}{\rho}}$โ†’Superpositionโ†’Standing wavesโ†’$\nu_n=\dfrac{nv}{2L}$โ†’Beats $\nu_1-\nu_2$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • explain how a mechanical wave carries energy through a medium without carrying the medium along, and classify waves as transverse or longitudinal (with compressions and rarefactions);
  • write and read the displacement relation $y(x,t)=a\sin(kx-\omega t+\phi)$: amplitude, phase, wavelength $\lambda$, wave number $k=\dfrac{2\pi}{\lambda}$, period, frequency and angular frequency, and tell the direction of travel from the sign in the phase;
  • use $v=\nu\lambda=\dfrac\omega k$, find the phase difference between two points of a wave, and find the velocity of a particle of the medium;
  • find the speed of a transverse wave on a stretched string, $v=\sqrt{T/\mu}$, and of sound in a fluid, $v=\sqrt{B/\rho}$, and explain why Newton's formula $\sqrt{p/\rho}$ fails and how Laplace's correction $\sqrt{\gamma p/\rho}$ fixes it, including the effects of temperature, pressure and humidity;
  • apply the principle of superposition, including interference of two waves of the same frequency, and describe the reflection of a wave at a rigid and at an open boundary;
  • describe standing waves, locate their nodes and antinodes, and find the normal modes and harmonics of a string fixed at both ends and of pipes open at both ends or closed at one end;
  • explain beats, use $\nu_\text{beat}=|\nu_1-\nu_2|$ to find unknown frequencies and tune instruments, and tackle Board, JEE Main, JEE Advanced and NEET-style problems on waves (with the Doppler effect as an exam extension).
Teaser โ€” The Stadium Wave

Thirty spectators sit $0.50\ \text{m}$ apart. Each one only stands up and sits down โ€” a simple up-and-down oscillation of period $T$ โ€” but each starts a little later than the neighbour on the left. Follow the highlighted spectator: does she ever move sideways? Then watch the crests. Change the delay and the period, and predict the speed of the wave before you read it.

0.05 s1.2 s
wave speed (spacing รท delay)โ€”
wavelength (crest to crest)โ€”
highlighted spectator's sideways motionโ€”
check: ฮป รท Tโ€”

Sections in this chapter

14.1

Transverse and Longitudinal Waves and the Displacement Relation

What a wave carries, transverse and longitudinal waves, and the progressive wave $y=a\sin(kx-\omega t+\phi)$ with its wavelength, wave number, period and frequency.

$k=\dfrac{2\pi}{\lambda},\ \ \omega=2\pi\nu$
Open section โ†’14.2

The Speed of a Travelling Wave

$v=\nu\lambda$; transverse waves on a stretched string; sound in gases, liquids and solids; Newton's formula and Laplace's correction.

$v=\sqrt{\dfrac T\mu},\ \ v=\sqrt{\dfrac{\gamma p}{\rho}}$
Open section โ†’14.3

Superposition, Reflection and Standing Waves

Adding waves, reflection at fixed and free ends, standing waves with nodes and antinodes, and the harmonics of strings and pipes.

$\nu_n=\dfrac{nv}{2L}$
Open section โ†’14.4

Beats

Two notes of nearly equal frequency, the rise and fall of loudness, the beat frequency, and tuning by beats.

$\nu_\text{beat}=|\nu_1-\nu_2|$
Open section โ†’
โœ… Before you start
You will need simple harmonic motion $x=A\cos(\omega t+\phi)$ โ€” amplitude, angular frequency $\omega=2\pi\nu=\dfrac{2\pi}T$, phase and phase difference (Section 13.1), and the velocity $\omega\sqrt{A^2-x^2}$ and energy of SHM (Section 13.2); tension in strings and Newton's second law (Section 4.3), with centripetal acceleration $\dfrac{v^2}{r}$ (Section 3.4); the bulk modulus $B$ and Young's modulus $Y$ (Section 8.2); density and pressure (Section 9.1); the ideal-gas equation (Section 12.1), and isothermal and adiabatic processes with $\gamma=\dfrac{C_p}{C_v}$ (Section 11.2). Dimensional analysis (Section 1.3) will give you the form of each wave speed before you derive it. From mathematics you need the graphs of $\sin$ and $\cos$ and the identities for $\sin A\pm\sin B$.
โ† 13.3 The Springโ€“Block Oscillator and the Simple Pendulum14.1 Transverse and Longitudinal Waves and the Displacement Relation โ†’