Sound Waves

Strike a tuning fork and hold it near your ear. Nothing visible travels from the fork to you, yet something does: a ripple of squeezed and stretched air, passed from layer to layer.

How a vibrating body makes sound#

Picture a prong of the fork beside a long column of air, which we imagine as thin layers. When the prong moves right, it pushes the layer next to it, which crowds against the next. That squeezed region, at higher pressure and density than normal, is a compression. The crowded layer pushes on its neighbour, which pushes on the next, so the compression moves away from the fork.

When the prong swings back left, the layers beside it follow and spread out: a region of lower pressure and density, a rarefaction. It too passes down the column. As the fork keeps vibrating, compressions and rarefactions are sent out one after another. Each layer only oscillates about its own position, while the pattern travels. That travelling pattern is a sound wave.

The pressure wave

If the prong moves in simple harmonic motion, the excess pressure (above the normal pressure) at any point varies sinusoidally with time. At the fork itself, say $p = p_0\sin\omega t$. The same disturbance reaches a point a distance x away after a time x/v, so:

The frequency ν = ω/2π is the fork’s frequency. The wavelength is the distance between neighbouring compressions, $\lambda = v/\nu$.

What we can hear#

Sound needs a medium, but it can travel through gases, liquids and solids alike. Human ears respond only to frequencies between roughly 20 Hz and 20,000 Hz (the upper limit falls with age).

A sound with a sinusoidal pressure variation contains one frequency. Most everyday sounds, such as a clap or a slammed door, are not sinusoidal. They are a mixture of many frequencies, which is why they sound like noise rather than a note.

Longitudinal, so no polarization#

In the air column the layers move back and forth along the direction in which the wave travels, so sound in a gas or liquid is a longitudinal wave. Fluids cannot resist a sideways shear, so they cannot carry transverse waves at all. (Solids can carry both kinds.) A longitudinal wave looks the same in every plane containing its direction of travel, so it cannot be polarized. A slit that blocks a transverse wave on a string has no special effect on sound.

Wavefronts#

A small source sends sound out in all directions. All the points at the same distance from it are disturbed in the same way at the same moment: they share the same phase. A surface joining points of equal phase is a wavefront.

• Around a small (point) source the wavefronts are spheres: a spherical wave.
• Far from the source, a small patch of a huge sphere is nearly flat, so the wave behaves as a plane wave. A large vibrating sheet also sends out plane waves.

A sound wave can be described in two ways: by how far the air moves, or by how much its pressure changes. Both describe the same wave, but they are not in step.

The displacement wave#

Let s be how far a layer of air, normally at position x, has moved along the direction of travel. For a sinusoidal wave:

From displacement to pressure#

Take a thin slab of air between x and x + Δx, with cross-section A. If both faces moved by the same amount, the slab would just shift and its volume would not change. A pressure change happens only when the two faces move by different amounts, s and s + Δs. Then the volume changes by $A\,\Delta s$, a fraction $\Delta V/V = \Delta s/\Delta x$.

The bulk modulus B links pressure and volume change: $p = -B\,\Delta V/V$. In the limit of a thin slab,

So the pressure is also a sinusoidal wave, with amplitude

A quarter-wavelength apart

The displacement wave has a sine; the pressure wave has a cosine. They differ in phase by $\pi/2$. Where the displacement is zero, the pressure is at its greatest or least, and where the displacement is greatest, the pressure is normal.

Why? At the middle of a compression, the layers on its left have moved right and the layers on its right have moved left. They crowd in from both sides, yet the layer at the centre itself hasn’t moved.

Which description should we use?#

Ears and microphones respond to pressure changes, not to how far the air has moved. So in this chapter we describe sound with the pressure wave, and when two sound waves meet we add their pressure changes.

Put your ear to a long steel railing while a friend taps the far end: you hear the tap twice, first through the metal and then through the air. A stiffer medium carries sound faster.

Speed in a fluid#

Take a slab of fluid of thickness Δx and cross-section A in a sound wave. The pressures on its two faces differ slightly, and this difference accelerates the slab. Newton’s second law, together with $p = -B\,\partial s/\partial x$ from the last lesson, gives

This is the wave equation, and the number in front is v². So:

In a solid rod

A thin rod can bulge sideways as it is squeezed along its length, so the relevant stiffness is Young’s modulus Y:

Speed of sound in a gas: Newton’s attempt#

For a gas, what is B? Newton assumed the temperature of each bit of air stays constant as it is squeezed and stretched. For an isothermal process $PV$ is constant, which gives $B = P$, so

The measured value at STP is about 332 m/s. Newton’s answer is about 15% too low.

Laplace’s correction#

Laplace pointed out that the compressions and rarefactions come and go far too quickly for heat to flow in or out, and air is a poor conductor anyway. A compressed region warms up; a rarefied region cools. The process is adiabatic, for which $PV^\gamma$ is constant and $B = \gamma P$, where γ = C_p/C_v ≈ 1.4 for air.

What changes the speed of sound in air?#

Pressure: no effect (at fixed temperature)

For an ideal gas $P/\rho = RT/M$. Doubling the pressure at the same temperature doubles the density too, so P/ρ and hence v are unchanged.

Temperature: v ∝ √T

Using $P/\rho = RT/M$, $v = \sqrt{\gamma RT/M}$, so v grows as the square root of the absolute temperature. For ordinary temperatures, with v₀ the speed at 0 °C and t in °C,

Humidity: moist air is faster

Water vapour (M = 18 g/mol) is lighter than dry air (about 29 g/mol). Humid air at the same pressure and temperature is therefore slightly less dense, and sound travels slightly faster in it.

We describe a sound by how loud it is, how high it is, and what it sounds like. Physics ties each of these to something measurable: intensity, frequency and waveform.

Intensity#

A sound wave carries energy. The intensity is the average energy crossing unit area (perpendicular to the direction of travel) per unit time. Its unit is W/m².

As in a wave on a string, the energy carried depends on the square of the amplitude and of the frequency. In terms of the displacement amplitude:

Using p₀ = Bks₀ and v² = B/ρ, the same thing in terms of the pressure amplitude:

The decibel scale#

The faintest sound an average ear detects at 1 kHz has an intensity of about $I_0 = 10^{-12}\ \text{W/m}^2$; sound becomes painful near 1 W/m². That is a range of a trillion. Our sense of loudness grows roughly with the logarithm of the intensity, so we use a logarithmic scale:

• Multiplying the intensity by 10 adds 10 dB; by 100 adds 20 dB.
• Doubling the intensity adds 10 log 2 ≈ 3 dB.
• A whisper is about 10 dB, a conversation 60 dB, a rock concert 110 dB; 120 dB is the threshold of pain.

Pitch, loudness and quality#

Pitch

Pitch rises with frequency. A sound containing several frequencies is heard with the pitch of its dominant one. Women’s and children’s voices usually have higher pitch than men’s because their vocal cords are shorter and lighter.

Loudness

Loudness is our sensation of intensity. It isn’t just intensity: the ear is most sensitive between about 1 and 4 kHz, so a 50 Hz hum and a 2 kHz tone of the same intensity do not sound equally loud.

Quality

A violin and a flute playing the same note have the same dominant frequency, yet we tell them apart instantly. Each instrument produces the fundamental together with a set of harmonics (whole-number multiples of it) in its own proportions. That mix gives the waveform its shape, and the shape is what we call the quality or timbre.

Two loudspeakers playing the same note don’t simply give twice the sound everywhere. Walk across the room and the note swells and fades: the two waves are adding in some places and cancelling in others.

Adding sound waves#

Sound waves obey the principle of superposition. Since detectors respond to pressure, we add the excess pressures, which are ordinary numbers, not vectors. Suppose two waves of the same frequency arrive at a point:

If they travel different distances from their sources, they arrive with a phase difference. One extra wavelength of path means one full cycle (2π) of phase, so:

The resultant is again a sinusoidal wave of the same frequency, with amplitude

Coherent and incoherent sources

A steady interference pattern needs the phase difference between the sources to stay constant. Such sources are coherent, like two speakers driven by the same amplifier. Two separate violinists are incoherent: their relative phase wanders randomly, so the cos δ term averages to zero and the intensities simply add.

Quincke’s tube#

A neat demonstration: sound enters a tube at one end and splits into two curved branches, which rejoin at the other end, where the listener is. One branch slides like a trombone, so its length can be increased. Pulling it out by d adds 2d to that path.

• When 2d is a whole number of wavelengths, the two parts arrive in phase: the sound is loud.
• Pull it a further λ/4 (adding λ/2 of path) and they arrive out of phase: the sound becomes faint.

Measuring the slide’s movement between successive minima gives λ/2, and so the wavelength.

Reflection of sound#

When sound meets a boundary, part is reflected. What happens to the pressure wave depends on the boundary:

• Rigid wall (or a closed end of a tube): the air can’t move there, so a compression comes back as a compression. The pressure wave is reflected without a phase change.
• Open end of a tube: the pressure there is held at the outside atmospheric value, so a compression comes back as a rarefaction. The pressure wave is reflected with a phase change of π.

Blow across the top of a bottle and it hums one particular note. Add water and the note rises. The air inside is a column with its own natural frequencies, just like a string.

Standing sound waves#

Two identical sound waves travelling in opposite directions along a tube add to give

Every point oscillates in step, with an amplitude $2p_0|\cos kx|$ that depends on its position: a standing wave.

• Pressure nodes, where the pressure never changes, are where cos kx = 0. Pressure antinodes, where it changes most, are where |cos kx| = 1. Neighbouring nodes (or antinodes) are λ/2 apart.
• Since pressure and displacement are π/2 out of phase, a pressure node is a displacement antinode and a pressure antinode is a displacement node.

Conditions at the ends

Strictly, the pressure node at an open end sits a little outside the tube, because the air just beyond the opening also takes part. The extra length is the end correction, about 0.6 times the tube’s radius. We ignore it unless told otherwise.

Closed organ pipe#

An organ pipe is closed at one end and open at the other. It needs a pressure antinode at the closed end and a node at the open end. Node to the nearest antinode is λ/4, so the length must be an odd number of quarter-wavelengths:

Open organ pipe#

Open at both ends, the pipe has a pressure node at each end. The length must fit a whole number of half-wavelengths:

A tuning fork, a tube and some water are enough to measure the speed of sound to within a percent. The trick is to measure the difference between two resonances, so the awkward end correction cancels.

The resonance-column method#

A long glass tube stands upright, with its bottom connected by a rubber pipe to a water reservoir. Raising or lowering the reservoir moves the water level in the tube. The air above the water is a column closed at the bottom (by the water) and open at the top.

Hold a vibrating fork over the top and slowly lower the water. At certain lengths the air column’s natural frequency matches the fork, and the sound swells loudly: resonance.

Using two resonances

The pressure node sits a small distance d (the end correction) above the open end. With the first two resonance lengths l₁ and l₂:

Subtracting removes d: λ = 2(l₂ − l₁). Then

Kundt’s tube#

Kundt’s method compares the speed of sound in two media. A long glass tube is closed at one end by an adjustable plug. A metal rod, clamped at its middle, pokes into the other end with a light disc attached. Fine powder (lycopodium or cork dust) is sprinkled thinly along the tube.

Rub the rod lengthwise with a resin-coated cloth and it rings in its fundamental longitudinal mode, which drives the air in the tube. Adjust the plug until the air column resonates. The powder is then swept away from the displacement antinodes and settles in little heaps at the displacement nodes, spaced λ/2 apart.

Speed in another gas

Fill the tube with a gas, measure the heap spacing Δl, then repeat with air (Δl′). The rod’s frequency is the same in both cases, so v ∝ λ ∝ heap spacing:

Speed in the rod

A rod of length l clamped at its middle has a node there and antinodes at both free ends, so in its fundamental $\lambda_\text{rod} = 2l$. Its frequency equals that of the air waves, whose wavelength is 2Δl. Equal frequencies give

Two guitar strings that are almost, but not quite, in tune make a sound that throbs: wah… wah… wah. As the tuner tightens one, the throbbing slows and finally stops. That throbbing is called beats.

Two close frequencies#

Take two sound waves of equal amplitude and slightly different angular frequencies ω₁ and ω₂, travelling the same way. At a fixed point, say x = 0, the total excess pressure is

where $\bar\omega = (\omega_1+\omega_2)/2$ is the average and $\Delta\omega = \omega_1 - \omega_2$. Read it as a wave at the average frequency whose amplitude slowly changes:

The beat frequency

The amplitude, and so the loudness, peaks whenever |cos(Δω t/2)| = 1. That happens every time Δω t/2 grows by π, so the time between peaks is 2π/Δω = 1/|ν₁ − ν₂|.

Using beats#

Tuning instruments

Play the instrument with a reference note and adjust it until the beats slow down and disappear: then the two match.

Finding an unknown frequency

Beats tell you the difference in frequency but not which is higher. To decide, change one source a little and see whether the beats speed up or slow down. Sticking a little wax on a fork’s prong makes it heavier, which lowers its frequency.

You can hear someone talking in the next room through an open door, even though you can’t see them. And an ambulance’s siren drops in pitch the moment it passes you. Both are everyday wave effects.

Diffraction#

When a wave meets an obstacle or passes through an opening, it spreads into the region that a straight-line path would leave in shadow. This bending is diffraction. It is strong when the obstacle or opening is about the size of the wavelength or smaller, and weak when it is much larger.

Audible sound has wavelengths from about 1.7 cm (20 kHz) to 17 m (20 Hz), similar to doors, windows, furniture and people. So sound diffracts easily around everyday objects, while light, with a wavelength under a micrometre, does not. That is why you hear around corners but can’t see around them. Low notes bend around more than high ones: from the next room, music sounds bass-heavy.

The Doppler effect#

When a source of sound and a listener move toward each other, the listener hears a higher frequency than the source emits; when they move apart, a lower one. This is the Doppler effect. All speeds below are measured relative to the air (the medium). Let v be the speed of sound and ν the frequency emitted.

Source moving, observer at rest

The source sends out a compression every period T = 1/ν. In that time the previous compression has travelled vT, but the source has moved uT after it. So the compressions ahead of it are only (v − u)T apart. They reach the observer at speed v, so the observer hears them at intervals of (v − u)T/v:

Here the wavelength itself changes. Ahead of the source:

Observer moving, source at rest

Now the wavelength is normal, λ = v/ν, but the observer runs into the compressions faster. Moving toward the source at u₀, the compressions pass at relative speed v + u₀:

Both moving

Three last stories about sound: what happens when something outruns it, why some notes sound good together, and how to build a hall where every seat hears clearly.

Faster than sound: the sonic boom#

The Doppler formula $\nu' = v\nu/(v - u_s)$ blows up when the source speed reaches the speed of sound. Beyond that, the source outruns its own sound. Each wavefront spreads from the point where it was emitted, but the source is always ahead of all of them. The fronts pile up along a cone with the source at its tip: a shock wave.

In a time t the source moves $u_s t$ while a wavefront it emitted at the start grows to radius $vt$. The cone touches all these spheres, so its half-angle θ satisfies

The cone carries a sudden, large pressure jump. When it sweeps over you, you hear a sharp bang: the sonic boom. A supersonic aircraft drags its cone along the whole time it flies faster than sound, so everyone along its path hears a boom, not just people below the point where it “broke the sound barrier”.

Musical scale#

Certain pairs of notes sound pleasant together when their frequencies are in simple ratios, such as 2 : 1, 3 : 2 or 5 : 4. A musical scale is a set of notes chosen so that many such pairs appear. The diatonic scale has eight notes; the last has twice the frequency of the first, and the gap between them is an octave.

Starting from a different first note just multiplies every frequency by the same factor, so the scale sounds the same, only higher or lower.

Acoustics of buildings#

An auditorium should let every listener hear clearly. Several problems need care:

Even distribution

Large flat or concave walls and ceilings can focus sound into some spots and leave dead spots elsewhere. Suitably curved, often convex, surfaces spread the reflected sound evenly across the audience.

Echoes

A strong reflection arriving more than about 0.1 s after the direct sound is heard as a separate echo. Distant flat walls are covered with absorbing materials or broken up so that their reflections don’t return as distinct echoes.

Reverberation

After a source stops, sound keeps bouncing around the hall and dies away gradually. This lingering is reverberation.

Long reverberation blurs speech, as each syllable overlaps the last. Curtains, carpets, soft seats and a full audience absorb sound and shorten it. But too short a reverberation time makes a hall sound “dead” and the music thin. A good hall is designed for the right value for its use: shorter for lectures, longer for concerts.

Amplification

Loudspeakers must be placed so that the sound reaching a listener from the speaker doesn’t arrive noticeably before or after the sound from the stage; electronic delays help. If a microphone picks up its own loudspeaker’s output, the sound circulates and grows into a loud howl or ringing: feedback.