David Bauer Physics & Astronomy ยท UCLA

Physics 5B Chapter 20

Sound Waves

Sound waves

Sound is a mechanical wave that travels through a medium, such as air, water, or the tissues of your body.

A source of sound, such as a loudspeaker cone or a tuning fork, vibrates back and forth. As the source moves forward, it pushes on the air next to it and squeezes that air into a smaller volume, so the pressure of that air increases. As the source moves back, the air next to it expands, and the pressure there decreases.

The squeezed air in turn pushes on the air beyond it, so the disturbance travels away from the source. In a sound wave, the disturbance is a change in the pressure (and the density) of the medium from its undisturbed value.

Transverse and longitudinal waves

In a wave on a string, each piece of the string moves perpendicular to the direction the wave travels. This is a transverse wave. In a sound wave in a fluid (a gas or a liquid), the particles of the medium move back and forth along the same direction the wave travels. This is a longitudinal wave.

Two waves traveling to the right, one above the other. Top, labeled transverse wave on a string: a sinusoidal wave on a blue string, with a black arrow labeled v pointing to the right. A red dot marks one point on the string, and a vertical double-headed arrow through it, labeled motion of the medium, shows that the point moves up and down, perpendicular to the direction of travel. Bottom, labeled longitudinal sound wave in a fluid: a horizontal tube filled with many small blue dots that are crowded together in some places and spread apart in others, with a black arrow labeled v pointing to the right. A red dot marks one particle, and a horizontal double-headed arrow below the tube, labeled motion of the medium, shows that the particle moves back and forth along the direction of travel.

Compressions and rarefactions

The figure below shows a cartoon of a fluid carrying a sinusoidal sound wave at one instant. Each dot is a small piece of the fluid. Where the dots are crowded together, the density and pressure are higher than in the undisturbed fluid. This is a compression. Where the dots are spread apart, the density and pressure are lower. This is a rarefaction.

Top: a horizontal tube filled with many small blue dots, a snapshot of a sinusoidal sound wave traveling to the right, as a black arrow labeled v shows. The dots are crowded together in evenly spaced bands, one labeled compression, and spread apart midway between them, one labeled rarefaction. Bottom: a graph of pressure p versus position x at the same instant, aligned with the tube. A dashed horizontal line marks the undisturbed pressure p sub 0. The red pressure curve is a sinusoid about p sub 0. Dashed vertical lines connect each compression in the tube to a maximum of the pressure curve and each rarefaction to a minimum. A double-headed arrow between two neighboring pressure maxima, which are also two neighboring compressions, is labeled lambda.

A particle model of a sound wave

As the wave passes, each piece of the fluid moves back and forth about its equilibrium position, in simple harmonic motion at the frequency of the source. The fluid itself does not travel with the wave.

Neighboring pieces of the fluid are slightly out of step with each other, so they crowd together in some places and spread apart in others. This pattern of compressions and rarefactions travels through the fluid at the speed of sound.

The dots are only a cartoon. Each one stands for a small parcel of fluid that contains a huge number of molecules. As we saw in the chapter on kinetic theory, the molecules themselves move in random directions at hundreds of meters per second. A sound wave adds a small, organized back-and-forth motion of each parcel as a whole.

Sound as a pressure wave

We will describe a sound wave by the pressure of the medium. If $p_0$ is the pressure with no wave present, the disturbance is the pressure variation $\Delta p = p - p_0$. It is positive in a compression and negative in a rarefaction.

A sinusoidal sound wave traveling in the $+x$ direction has the same wave function as a wave on a string, with $\Delta p$ in place of $y$:

\[\Delta p(x,t) = \Delta p_{\max}\sin(kx - \omega t).\]

The amplitude $\Delta p_{\max}$ is the pressure amplitude.

At a fixed position, the pressure oscillates in time at the frequency of the source. A microphone, or your eardrum, responds to this oscillating pressure.

The pressure variations in sound are tiny. In ordinary conversation, the pressure amplitude is a few hundredths of a pascal, less than a millionth of atmospheric pressure.

The speed of sound

Like any wave, the speed of sound is set by the medium, not by the frequency or the loudness of the sound. For two media with the same density, sound travels faster in the one that is harder to compress. For two media that are equally hard to compress, sound travels more slowly in the denser one.

Medium Speed of sound
Air ($20^\circ\mathrm{C}$) $343\ \mathrm{m/s}$
Water ($20^\circ\mathrm{C}$) $1480\ \mathrm{m/s}$
Soft tissue (typical) $1540\ \mathrm{m/s}$

Water is about 800 times as dense as air, but it is far harder to compress. As a result, sound travels about four times as fast in water as in air.

The speed of sound in air increases with temperature. We will use $v = 343\ \mathrm{m/s}$, its value at $20^\circ\mathrm{C}$, unless a problem says otherwise.

Example: Wavelengths of sound

A tuning fork vibrates at $440\ \mathrm{Hz}$, the frequency of the musical note A above middle C.

  1. What is the wavelength of the sound wave it produces in air at $20^\circ\mathrm{C}$?
  2. The sound passes from the air into a swimming pool of water at $20^\circ\mathrm{C}$. What are the frequency and the wavelength of the sound in the water?
Solution

1. In air, the speed of sound is $343\ \mathrm{m/s}$, so the wavelength is

\[\lambda = \frac{v}{f} = \frac{343\ \mathrm{m/s}}{440\ \mathrm{Hz}} = 0.780\ \mathrm{m}.\]

2. The frequency is set by the source, so it is still $f = 440\ \mathrm{Hz}$ in the water. The speed of sound in water is $1480\ \mathrm{m/s}$, so the wavelength is

\[\lambda = \frac{v}{f} = \frac{1480\ \mathrm{m/s}}{440\ \mathrm{Hz}} = 3.36\ \mathrm{m}.\]

Spherical waves

Sound in three dimensions

A wave on a string travels in one dimension, along the string. Sound spreads out through three-dimensional space.

A point source that sends out sound equally in every direction produces a spherical wave. The compressions form spheres centered on the source, called wavefronts, with rarefactions midway between them. Neighboring wavefronts are one wavelength apart, and every wavefront moves outward at the speed of sound.

A slice through a spherical sound wave, centered on a point source drawn as a black dot labeled source. The pressure is shaded in blue: darker rings are compressions and lighter rings are rarefactions. Solid blue circles centered on the source, labeled wavefronts, mark the compressions, and the rarefactions lie midway between them. Neighboring circles are equally spaced, and a double-headed arrow between two of them is labeled lambda. Black arrows point radially outward from the source in four directions, and one of them is labeled v.

Energy spreads out

A sound wave carries energy away from its source. For a point source, this energy spreads out over the wavefronts.

As a wavefront moves outward, its area increases, so the same energy is spread over a larger and larger area. In the figure, the energy that passes through one square at distance $r$ spreads over 4 squares at distance $2r$, and over 9 squares at distance $3r$.

A point source drawn as a black dot at the left, with four straight lines spreading out from it to the right, forming a square-sided cone. The cone passes through three square patches that face the source, at distances r, 2r, and 3r. The patch at distance r is one small square. The patch at distance 2r is twice as wide and twice as tall, divided into 4 small squares, and the patch at 3r is three times as wide and tall, divided into 9 small squares.

This is why a sound gets quieter as you move away from its source.

Summary

Sound in a fluid is a longitudinal wave of compressions and rarefactions. We describe it by the pressure variation, $\Delta p(x,t) = \Delta p_{\max}\sin(kx - \omega t)$. Its speed is set by the medium, about $343\ \mathrm{m/s}$ in air.