Propagation of Sound Waves

Why this chapter

Why this chapter exists

A voice, a drum and a temple bell all reach us in the same way. A vibrating source pushes and pulls the air, and compressions and rarefactions travel out while the air stays at home. This chapter describes the travelling pattern in two ways: by the displacement of the air and by its pressure. The two sit a quarter wavelength apart. The chapter then asks what sets the speed of sound. Newton's second law, applied to a thin slab of air, gives the answer. Newton's own value was 15 % too low, and Laplace found why. The chapter ends with the formula that predicts the speed of sound in any gas at any temperature.

You will learn

By the end of this chapter you will

Describe sound as a longitudinal wave of compressions and rarefactions, and tell amplitude, which sets loudness, from frequency, which sets pitch.
Write the displacement wave \(s = s_{\text{max}}\cos(kx - \omega t)\) and use \(v = \lambda f = \omega/k\).
Derive the pressure wave \(\Delta P = -B\,\partial s/\partial x = Bk\,s_{\text{max}}\sin(kx - \omega t)\), and explain why pressure and displacement are 90° out of phase.
Derive the wave equation for sound from Newton's second law on a thin slab of fluid, and read off \(v = \sqrt{B/\rho}\).
Derive Newton's isothermal and Laplace's adiabatic bulk moduli, \(B = P\) and \(B = \gamma P\), and explain why sound is adiabatic.
Use \(v = \sqrt{\gamma RT/M}\) to find how the speed of sound depends on temperature, molar mass and pressure.

Before you begin

From the chapter Introduction to Waves you should know transverse and longitudinal waves, the harmonic wave \(y = A\sin(kx - \omega t)\), and the wave equation \(\partial^2 y/\partial x^2 = (1/v^2)\,\partial^2 y/\partial t^2\). From the chapter Propagating Waves on a String you should know how Newton's second law on one small piece gives a wave speed. You should know the ideal gas law \(PV = nRT\) and how to differentiate \(\sin\), \(\cos\) and \(V^n\).

Already know this?

Lessons

Sound as a Longitudinal Wave

A sound starts at a vibrating source. Its amplitude sets the loudness and its frequency the pitch, independently. Seven samples to hear, the wave \(s = s_{\max}\cos(kx - \omega t)\), \(v = \lambda f = \omega/k\), and the compressions and rarefactions that travel while the air stays at home.

A sound begins at a vibrating sourceAmplitude and frequency are independentThe equation and speed of a sound waveCompressions and rarefactions travel

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Displacement and Pressure Waves

Two ways to describe one sound. The displacement wave \(s(x, t)\), the snapshot with its compressions and rarefactions, the pressure wave \(\Delta P = -B\,\partial s/\partial x = Bk s_{\max}\sin(kx - \omega t)\), and why the two are 90° out of phase.

The displacement wave of soundPressure from the slope of the displacementDisplacement and pressure are 90 degrees apart

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What Sets the Speed of Sound

The source starts a sound, but the medium sets its speed. A row of dominoes, the rule \(v = \sqrt{\text{elastic}/\text{inertial}}\), Newton's second law for a thin slab of fluid, the wave equation for sound, \(v = \sqrt{B/\rho}\), and the speed in liquids and thin solid rods.

The medium sets the speed of soundThe net force on a thin slab of fluidThe wave equation for sound and its speedSpeed of sound in liquids and thin rods

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Newton, Laplace and the Speed in a Gas

The bulk modulus of a gas, \(B = -V\,dP/dV\), depends on how the gas is squeezed. Newton's isothermal \(B = P\) gives a speed about 15 % too low; Laplace's adiabatic \(B = \gamma P\), derived step by step, agrees with experiment.

The bulk modulus of a gas depends on the processNewtons isothermal speed is too lowLaplaces adiabatic correction

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Temperature, Molar Mass and Pressure

The ideal gas law turns \(\sqrt{\gamma P/\rho}\) into \(v = \sqrt{\gamma RT/M}\). The speed of sound grows as \(\sqrt{T}\), falls as \(1/\sqrt{M}\), and ignores the pressure at a fixed temperature. The speed in air at 20 °C and at 35 °C, worked out.

The speed of sound in an ideal gasTemperature molar mass and pressureThe speed of sound at two temperatures

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

Glossary

The key words of this chapter, with the meaning this chapter gives them.

absolute temperature: The temperature \(T\) measured in kelvin from absolute zero. \(T\) in kelvin is the Celsius temperature plus 273.
adiabatic process: A change of a gas with no exchange of heat with its surroundings. For an ideal gas, \(PV^{\gamma}\) is then constant.
ambient pressure: The steady pressure \(P_0\) of the medium when no sound is present, about \(1.013 \times 10^5\,\text{Pa}\) for air at sea level.
amplitude: The largest displacement of the particles of a medium from their equilibrium positions. For sound, a larger amplitude means a louder sound.
angular frequency: The number \(\omega = 2\pi f\), in rad/s. It tells how fast the phase changes with time.
bulk modulus: A measure of how hard a material is to compress, \(B = -V\,dP/dV\), in pascals. A stiff medium has a large bulk modulus.
compression: A region of a longitudinal wave where the particles are crowded together, with a pressure and density above average.
density: The mass per unit volume of a material, \(\rho\), in kg/m³. It measures the inertia of each piece of the medium.
displacement wave: The description of sound by the shift \(s(x, t)\) of each particle from its mean position, along the direction of travel.
frequency: The number of complete oscillations in one second, measured in hertz (Hz). For sound, it sets the pitch.
ideal gas law: The relation \(PV = nRT\) for an ideal gas, which can also be written \(P = \rho RT/M\).
isothermal process: A change of a gas in which its temperature stays constant. For an ideal gas, \(PV\) is then constant.
longitudinal wave: A wave in which the particles of the medium oscillate along the direction in which the wave travels. Sound in air is longitudinal.
loudness: How strong a sound seems to the ear. A larger amplitude gives a louder sound.
medium: The material through which a mechanical wave travels, such as air, water or steel. Each part of it only oscillates about its own place.
molar mass: The mass of one mole of a substance, \(M\), in kg/mol. For air it is about \(0.029\,\text{kg/mol}\).
partial derivative: The rate of change of a quantity that depends on two variables, with one variable held fixed. For \(s(x, t)\), \(\partial s/\partial x\) holds \(t\) fixed.
phase: The argument \(kx - \omega t\) of the sine or cosine in the equation of a wave.
pitch: How high or low a sound seems to the ear. A higher frequency gives a higher pitch.
pressure amplitude: The largest pressure variation in a sound wave, \(\Delta P_{\text{max}} = Bk\,s_{\text{max}}\).
pressure variation: The small change \(\Delta P\) of pressure above or below the ambient pressure, caused by a sound wave.
rarefaction: A region of a longitudinal wave where the particles are spread apart, with a pressure and density below average.
ratio of specific heats: The number \(\gamma = C_P/C_V\), the heat capacity of a gas at constant pressure divided by that at constant volume. It is about 1.4 for air and 1.67 for helium.
wave equation: The equation \(\partial^2 s/\partial x^2 = (1/v^2)\,\partial^2 s/\partial t^2\), obeyed by every wave that travels at speed \(v\) without changing shape. For sound in a fluid, \(1/v^2 = \rho/B\).
wave number: The number \(k = 2\pi/\lambda\), in rad/m. It tells how fast the phase changes along the wave.
wavelength: The distance between two neighbouring points of a wave in the same state, such as two compressions, written \(\lambda\).
Young's modulus: A measure of how hard a solid is to stretch or squeeze along its length, the stress divided by the strain, in pascals. It sets the speed of sound along a thin rod.