Energy, Power, Intensity and Loudness

Why this chapter

Why this chapter exists

A sound carries energy from its source to your ear, yet no air travels between them. This chapter finds how much energy each cubic metre of air holds as a sound passes, and how fast it flows. The answer, the intensity, grows as the square of the pressure amplitude, which is what a microphone and the ear measure. The intensity falls with distance from a source, and the ear judges it by ratios, not by differences. So sound is measured on the decibel scale. The chapter ends at the limits of hearing, and with what makes a flute and a sitar sound different on the same note.

You will learn

By the end of this chapter you will

Derive the kinetic and the potential energy per unit volume of a sound wave, and show that they are equal for a travelling wave.
Find the average energy density \(\tfrac12\rho\omega^2 s_{max}^2\) and the intensity \(\tfrac12\rho v\omega^2 s_{max}^2 = (\Delta P_{max})^2/(2\rho v)\) of a harmonic sound.
Explain why a microphone and the ear respond to the pressure amplitude of a sound.
Use the inverse-square law \(I = P/(4\pi r^2)\) for a point source, and \(I \propto 1/r\) for a line source.
Convert between intensity and sound level in decibels, and read a difference of levels as a ratio of intensities.
State the limits of human hearing in frequency and in intensity, and place common sounds on a ladder of levels.
Name the three characteristics of a musical sound, pitch, loudness and quality, and link each to a property of the wave.

Before you begin

From the chapter Propagation of Sound Waves you should know the displacement wave \(s = s_{max}\cos(kx - \omega t)\), the pressure wave \(\Delta P = -B\,\partial s/\partial x = Bk\,s_{max}\sin(kx - \omega t)\) and the speed of sound \(v = \sqrt{B/\rho}\). From the chapter Propagating Waves on a String you should know the energy \(\tfrac12\mu\omega^2A^2\) and the power \(\tfrac12\mu v\omega^2A^2\) of a wave on a string. You should know logarithms to base 10.

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Lessons

The Energy in a Sound Wave

A sound wave carries energy in every slab of air it passes. The kinetic energy of a moving slab, the potential energy of its compression, the proof that the two are equal, and the average energy per unit volume of a harmonic sound wave.

Kinetic energy of a slab of airPotential energy of a compressed slabThe two energies of a sound wave are equalAverage energy density of a sound wave

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Power and Intensity

Intensity is the power of a sound per unit area. The energy that crosses an area in a time \(dt\), the intensity \(\tfrac12\rho v\omega^2 s_{\max}^2\) of a harmonic sound, the same intensity from the pressure amplitude, \((\Delta P_{\max})^2/(2\rho v)\), and why a microphone and the ear respond to pressure.

Intensity is power per unit areaIntensity of a harmonic sound waveWhy the ear responds to pressure amplitude

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The Inverse-Square Law

A small source spreads its power over ever larger spheres, so the intensity falls as \(1/r^2\). The sphere picture, the formula \(I = P/(4\pi r^2)\), the distance at which a 1 W source reaches the threshold of pain, and a long line source whose intensity falls only as \(1/r\).

A point source spreads over a sphereThe inverse square law for intensityA line source spreads over a cylinder

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The Decibel Scale

The ear judges loudness by ratios, so sound is measured on a logarithmic scale. The sound level \(\beta = (10\,\text{dB})\log(I/I_0)\), its benchmarks of 10, 20 and 3 dB, why 70 dB is not twice 35 dB, and how levels combine when sources are added.

The ear hears ratios of intensitySound level in decibelsDifferences of level are ratios of intensity

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The Limits of Hearing

Human hearing is limited in frequency, from 20 Hz to 20 kHz, and in intensity, from the 0 dB threshold of hearing to the 120 dB threshold of pain. A ladder of common sound levels, how the level falls with distance, and the three characteristics of a musical sound: pitch, loudness and quality.

The audible range of frequenciesThresholds of hearing and of painA ladder of common sound levelsPitch loudness and quality

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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.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

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

acoustic impedance: The product \(\rho v\) of the density of a medium and the speed of sound in it. The extra pressure of a travelling sound wave is \(\rho v\) times the particle velocity.
audible range: The range of frequencies a healthy young human ear can hear, about \(20\,\text{Hz}\) to \(20\,\text{kHz}\).
bulk modulus: The ratio of an extra pressure to the fractional decrease in volume it causes, \(B = -\Delta P/(\Delta V/V)\), in pascals. For air in a sound wave, \(B = \rho v^2 \approx 1.4 \times 10^5\,\text{Pa}\).
decibel: The unit of sound level, dB, a tenth of a bel. An increase of \(10\,\text{dB}\) means ten times the intensity.
displacement amplitude: The largest displacement \(s_{max}\) of a layer of air from its rest position in a harmonic sound wave, in metres.
energy density: The energy of a wave per unit volume of the medium, \(u\), in J/m³. For a harmonic sound wave its average is \(\tfrac12\rho\omega^2 s_{max}^2\).
harmonic: A tone whose frequency is a whole-number multiple of the lowest frequency \(f_1\) of a sound: \(f_1\), \(2f_1\), \(3f_1\) and so on.
infrasound: Sound with a frequency below about \(20\,\text{Hz}\), too low for the human ear to hear.
intensity: The power of a wave that crosses a unit area at right angles to its direction of travel, \(I = P/A\), in W/m². For a harmonic sound it is \((\Delta P_{max})^2/(2\rho v)\).
inverse-square law: The law that the intensity from a point source falls as the square of the distance, \(I = P/(4\pi r^2)\), when no energy is absorbed on the way.
line source: A long straight source of sound, such as a busy road or a row of loudspeakers. Its power spreads over cylinders, so its intensity falls as \(1/r\).
logarithmic scale: A scale on which equal steps stand for equal ratios, not equal differences. On the decibel scale each step of \(10\,\text{dB}\) is a factor of \(10\) in intensity.
loudness: How strong a sound seems to a listener. It is set mainly by the intensity and grows roughly with the sound level in decibels.
particle velocity: The velocity \(\partial s/\partial t\) with which a layer of air moves to and fro as a sound passes. It changes from moment to moment and differs from the speed of the wave.
pitch: How high or low a musical sound seems. It is set by the frequency: a higher frequency gives a higher pitch.
point source: A source of sound that is small compared with the distance to the listener. It sends its power out equally in all directions over spheres.
pressure amplitude: The largest extra pressure \(\Delta P_{max}\) of a sound wave above or below the steady pressure of the air, in pascals. For a harmonic wave \(\Delta P_{max} = Bk\,s_{max} = \rho v\,\omega\,s_{max}\).
quality: The character of a musical sound, also called timbre, that tells two instruments apart on the same note. It is set by the mix of harmonics in the sound.
sound level: A measure of a sound on a logarithmic scale, \(\beta = (10\,\text{dB})\log_{10}(I/I_0)\), with the reference \(I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2\).
threshold of hearing: The faintest intensity a healthy young ear can hear under ideal conditions, about \(1.0 \times 10^{-12}\,\text{W/m}^2\), or \(0\,\text{dB}\). It is the reference intensity \(I_0\).
threshold of pain: An intensity of about \(1\,\text{W/m}^2\), or \(120\,\text{dB}\), at which sound hurts the ear. Long exposure near it can damage hearing permanently.
ultrasound: Sound with a frequency above about \(20\,\text{kHz}\), too high for the human ear to hear.
wavefront: A surface on which every point of a wave has the same phase. Around a point source the wavefronts are spheres; around a line source they are cylinders.