Part 2 · Chapter 17

Waves — II

Sound: how compressions of the air carry a guitar note, a siren's wail, and a bat's echo — and why a jet booms

Fundamentals of Physics Prof. Mithun Mondal Reading time ≈ 55 min
i What you'll learn
  • That sound is a longitudinal wave, with speed set by an elastic and an inertial property of the medium: \(v = \sqrt{B/\rho}\).
  • The twin descriptions of a sound wave — a displacement \(s = s_{m}\cos(kx-\omega t)\) and a pressure variation \(\Delta p = \Delta p_{m}\sin(kx-\omega t)\) that are 90° out of phase, with \(\Delta p_{m} = (v\rho\omega)s_{m}\).
  • How a path-length difference sets the interference at a point: \(\phi = (\Delta L/\lambda)\,2\pi\), fully constructive at \(\Delta L/\lambda = 0,1,2,\dots\)
  • Sound intensity \(I = \tfrac{1}{2}\rho v\omega^{2}s_{m}^{2}\), its \(1/r^{2}\) falloff, and the decibel scale \(\beta = (10\,\mathrm{dB})\log(I/I_{0})\).
  • Resonance in pipes (\(f = nv/2L\) open–open; \(f = nv/4L\) open–closed), beats \(f_{\text{beat}} = |f_{1}-f_{2}|\), the Doppler effect \(f' = f\,\frac{v \pm v_{D}}{v \pm v_{S}}\), and the Mach cone \(\sin\theta = v/v_{S}\).
Section 17-1

What Is Physics?

The physics of sound runs through a surprising number of fields. Physiologists study how speech is produced and how hearing loss can be eased; acoustic engineers tune concert halls and quiet freeways; aviation engineers wrestle with the shock waves of supersonic jets; medical researchers listen to the heart and lungs for signs of trouble; and biologists ask how a cat purrs or how a penguin finds its mate in a crowd. All of it begins with one question: what are sound waves?

Section 17-2

Sound Waves

As in Chapter 16, a mechanical wave needs a material medium. Mechanical waves come in two kinds: transverse waves, with oscillations perpendicular to the travel direction, and longitudinal waves, with oscillations parallel to it. In this book a sound wave is defined, roughly, as any longitudinal wave — the kind that seismic teams use to probe the crust, that sonar uses to find obstacles, and that ultrasound uses to image the body.

Near a small point source, sound spreads outward as spherical wavefronts — surfaces of equal oscillation — while rays, drawn perpendicular to the wavefronts, point in the direction of travel. Close to the source the wavefronts are spherical; far away, their curvature is so slight that we treat them as flat planar wavefronts.

Section 17-3

The Speed of Sound

The speed of any mechanical wave depends on an inertial property of the medium (to store kinetic energy) and an elastic property (to store potential energy). For a string those were the linear density and the tension. For sound in a fluid the inertial property is the density \(\rho\), and the elastic property is the bulk modulus \(B = -\Delta p/(\Delta V/V)\), which measures how strongly the medium resists being compressed.

Speed of sound in a medium
\[ v = \sqrt{\frac{B}{\rho}} \]
A stiffer medium (larger \(B\)) carries sound faster; a denser one (larger \(\rho\)) slows it. In air at 20 °C, \(v = 343\) m/s.
Table 17-1 · The speed of sound (m/s)
MediumSpeedMediumSpeed
Air (0 °C)331Water (20 °C)1482
Air (20 °C)343Seawater1522
Helium965Aluminum6420
Hydrogen1284Steel5941
Water (0 °C)1402Granite6000
Why sound is faster in water than in air. Water is about 1000 times denser than air, which alone would slow sound. Yet sound travels several times faster in water. The reason is that water is far less compressible — its bulk modulus is more than 1000 times that of air, and the stiffness wins. Stiffness, not lightness, is the dominant factor.
Section 17-4

Traveling Sound Waves

As a sinusoidal sound wave travels along a tube of air, each thin element of air oscillates back and forth in simple harmonic motion about its equilibrium position — longitudinally, along the travel direction. Because the motion is along \(x\) (not the \(y\) of a string), we write the displacement as \(s(x,t)\) and use a cosine:

Displacement and pressure forms of a sound wave
\[ s(x,t) = s_{m}\cos(kx - \omega t) \qquad\qquad \Delta p(x,t) = \Delta p_{m}\sin(kx - \omega t) \]
\(s_m\) is the displacement amplitude (a few tens of nanometres for ordinary sound); \(\Delta p_m\) is the pressure amplitude. As before \(k = 2\pi/\lambda\) and \(\omega = 2\pi f\). A positive \(\Delta p\) is a compression, a negative one an expansion.

The two descriptions are tied together. A short derivation from the bulk modulus relates the pressure amplitude to the displacement amplitude:

Pressure amplitude in terms of displacement amplitude
\[ \Delta p_{m} = (v\rho\omega)\,s_{m} \]
The pressure amplitude is normally a tiny fraction of atmospheric pressure (~10⁵ Pa). Even a sound at the threshold of pain has \(\Delta p_m \approx 28\) Pa.
Displacement and pressure are 90° out of phase. The cosine of the displacement and the sine of the pressure differ by \(\pi/2\) rad. So where an air element is at maximum displacement (a node of pressure), the pressure variation is momentarily zero; where the displacement is zero, the pressure swing is greatest. This quarter-cycle offset is the key to reading standing-sound-wave patterns in pipes.
Section 17-5

Interference

Like all waves, sound waves obey superposition, so two identical waves reaching a common point can reinforce or cancel. If two in-phase sources send waves to a point \(P\) along paths of length \(L_{1}\) and \(L_{2}\), the only thing that matters is the path-length difference \(\Delta L = |L_{2} - L_{1}|\). Since one wavelength of path corresponds to \(2\pi\) of phase:

Phase difference from path-length difference
\[ \phi = \frac{\Delta L}{\lambda}\,2\pi \]
Fully constructive interference when \(\Delta L/\lambda = 0, 1, 2, \dots\) (waves in phase). Fully destructive when \(\Delta L/\lambda = 0.5, 1.5, 2.5, \dots\) (waves a half-wavelength out of step).
🔊
Constructive vs destructive interference
ΔL = mλ → loud  ·  ΔL = (m + ½)λ → quiet

An integer number of wavelengths of extra path puts the waves back in step (a loud spot); an odd number of half-wavelengths puts them exactly out of step (a quiet spot). Moving a detector around two in-phase sources therefore sweeps through alternating maxima and minima of loudness.

Section 17-6

Intensity and Sound Level

Beyond pitch and speed, a sound has intensity \(I\) — the average rate per unit area at which it transports energy across a surface, \(I = P/A\). As with a string wave, the intensity grows with the squares of the frequency and the displacement amplitude:

Sound intensity, and the inverse-square law for a point source
\[ I = \tfrac{1}{2}\rho v\,\omega^{2}s_{m}^{2} \qquad\qquad I = \frac{P_{s}}{4\pi r^{2}} \]
An isotropic point source of power \(P_s\) spreads its energy over a sphere of area \(4\pi r^2\), so intensity falls off as \(1/r^2\): double the distance, quarter the intensity.

The ear handles an enormous range — the loudest tolerable intensity is about \(10^{12}\) times the faintest detectable one. To tame that span we use a logarithmic measure, the sound level in decibels:

🎚️
Sound level (decibels)
β = (10 dB) log(I / I₀),   I₀ = 10⁻¹² W/m²

The reference \(I_{0}\) sits near the threshold of hearing, so it reads 0 dB. Every factor-of-10 jump in intensity adds 10 dB. Conversation is ~60 dB, a rock concert ~110 dB, the pain threshold ~120 dB.

Section 17-7

Sources of Musical Sound

Musical sound comes from standing waves on strings, membranes, bars, and — for wind instruments — columns of air in pipes. As in Chapter 16, reflections at the ends send waves back, and when the wavelength is matched to the pipe length the superposition builds a large, sustained standing wave that radiates a clear note. The boundary rule is simple: a closed end forces a displacement node (the air can't move there), and an open end forces a displacement antinode.

Resonant frequencies of pipes
\[ f = \frac{nv}{2L}\;(n = 1,2,3,\dots)\ \text{[two open ends]} \qquad f = \frac{nv}{4L}\;(n = 1,3,5,\dots)\ \text{[one open end]} \]
A pipe open at both ends supports every harmonic. A pipe open at one end and closed at the other supports only the odd harmonics — the second harmonic simply cannot exist in it.
Same note, different sound. A flute and an oboe playing the same note share a fundamental frequency but generate different mixes of higher harmonics. Your ear hears the harmonics superimposed as one net wave, and the differing harmonic recipes are exactly what give each instrument its distinctive timbre.
Section 17-8

Beats

Sound two tones of nearly equal frequency together and you hear a single tone at their average frequency, but its loudness swells and fades in a slow throb. These are beats. Adding \(s_{1} = s_{m}\cos\omega_{1}t\) and \(s_{2} = s_{m}\cos\omega_{2}t\) and using a product-to-sum identity gives an amplitude that itself oscillates slowly, producing one loudness peak per half-cycle of that envelope:

Beat frequency
\[ f_{\text{beat}} = |f_{1} - f_{2}| \]
Two tones at 552 and 564 Hz are heard as a 558 Hz tone throbbing 12 times a second. Musicians tune by ear this way — adjusting an instrument against a reference until the beats slow to a stop.
Section 17-9

The Doppler Effect

Drive toward a parked police siren and its pitch rises; drive away and it falls. Any relative motion between a source and a detector through the air shifts the detected frequency — the Doppler effect. Measuring all speeds relative to the air, the detected frequency \(f'\) relates to the emitted frequency \(f\) by:

🚓
General Doppler effect (sound)
f′ = f · (v ± v_D) / (v ± v_S)

Here \(v\) is the speed of sound, \(v_{D}\) the detector's speed, and \(v_{S}\) the source's speed, all relative to the air. The sign rule is one phrase: toward means shift up, away means shift down. Choose each sign so that motion toward the other raises \(f'\) and motion away lowers it.

It works for light too. The same up-shift/down-shift logic governs electromagnetic waves — the "redshift" of receding galaxies and the blue-shift of approaching ones — though the exact formula differs because light needs no medium. For sound, always anchor your reasoning in the body of air and the words toward and away.
Section 17-10

Supersonic Speeds, Shock Waves

When the source moves as fast as sound it keeps pace with its own wavefronts, and the Doppler formula predicts an infinite frequency. When it moves faster than sound — at supersonic speed — the wavefronts pile up along a cone trailing the source, the Mach cone. The bunched wavefronts form a shock wave, heard as the sharp crack of a sonic boom.

Mach cone half-angle
\[ \sin\theta = \frac{v}{v_{S}} \]
The ratio \(v_S/v\) is the Mach number; "Mach 2.3" means 2.3 times the speed of sound. The faster the source, the narrower (smaller \(\theta\)) the cone. The crack of a whip is a miniature sonic boom — its tip briefly breaks the sound barrier.
Worked Examples

Putting It to Work

1 From pressure amplitude to displacement amplitude

Problem. The loudest sound the ear tolerates has a pressure amplitude of about \(\Delta p_{m} = 28\,\mathrm{Pa}\). For a 1000 Hz tone in air (\(\rho = 1.21\,\mathrm{kg/m^{3}}\), \(v = 343\,\mathrm{m/s}\)), what is the displacement amplitude \(s_{m}\)?

Solution. Invert \(\Delta p_{m} = (v\rho\omega)s_{m}\) with \(\omega = 2\pi f\).

s_m = Δp_m / (vρ·2πf)
\[ s_{m} = \frac{\Delta p_{m}}{v\rho(2\pi f)} = \frac{28}{(343)(1.21)(2\pi)(1000)} \approx 1.1 \times 10^{-5}\,\mathrm{m} = 11\,\mu\mathrm{m} \]

About one-seventh the thickness of a book page — and the faintest audible sound moves the air by only ~11 pm, a tenth of an atom's radius. The ear is an extraordinarily sensitive detector.

2 Speed of sound from bulk modulus

Problem. Water at 20 °C has bulk modulus \(B = 2.2 \times 10^{9}\,\mathrm{Pa}\) and density \(\rho = 998\,\mathrm{kg/m^{3}}\). Find the speed of sound in water.

Solution. Apply \(v = \sqrt{B/\rho}\).

v = √(B/ρ)
\[ v = \sqrt{\frac{2.2 \times 10^{9}}{998}} = \sqrt{2.20 \times 10^{6}} \approx 1.48 \times 10^{3}\,\mathrm{m/s} \]

About 1480 m/s — matching the tabulated value, and more than four times the speed in air, thanks to water's huge stiffness.

3 Intensity and sound level of a point source

Problem. A small speaker emits \(P_{s} = 1.0\,\mathrm{W}\) of sound isotropically. Find the intensity and the sound level at \(r = 2.5\,\mathrm{m}\).

Solution. Use \(I = P_{s}/4\pi r^{2}\), then \(\beta = (10\,\mathrm{dB})\log(I/I_{0})\) with \(I_{0} = 10^{-12}\,\mathrm{W/m^{2}}\).

I = P_s/4πr², then β = 10 log(I/I₀)
\[\begin{gathered} I = \frac{1.0}{4\pi(2.5)^{2}} \approx 1.27 \times 10^{-2}\,\mathrm{W/m^{2}} \\ \beta = (10)\log\!\frac{1.27 \times 10^{-2}}{10^{-12}} \approx 101\,\mathrm{dB} \end{gathered}\]

About 101 dB — louder than a busy street and well on the way to the pain threshold. Step back to 5.0 m and the intensity drops fourfold, lowering the level by about 6 dB.

4 Resonance in a tube — both ends vs one end open

Problem. A cardboard tube of length \(L = 67.0\,\mathrm{cm}\) resonates in its fundamental mode (\(v = 343\,\mathrm{m/s}\)). What fundamental do you hear (a) with both ends open, and (b) with one end closed by your ear?

Solution. Open–open uses \(f = nv/2L\); open–closed uses \(f = nv/4L\), both with \(n = 1\).

f₁ = v/2L (open–open); f₁ = v/4L (open–closed)
\[\begin{gathered} f_{1}^{\text{open}} = \frac{343}{2(0.670)} \approx 256\,\mathrm{Hz} \\ f_{1}^{\text{closed}} = \frac{343}{4(0.670)} \approx 128\,\mathrm{Hz} \end{gathered}\]

Closing one end halves the fundamental. And because only odd harmonics survive there, the 256 Hz tone (an even multiple of 128 Hz) can no longer sound.

5 Doppler shift of a siren

Problem. A siren emits \(f = 1000\,\mathrm{Hz}\). Take \(v = 343\,\mathrm{m/s}\). What do you hear if (a) you drive toward the stationary siren at \(v_{D} = 33.3\,\mathrm{m/s}\), and (b) the siren moves away from stationary you at \(v_{S} = 33.3\,\mathrm{m/s}\)?

Solution. Use \(f' = f\,\frac{v \pm v_{D}}{v \pm v_{S}}\), choosing signs so "toward" shifts up and "away" shifts down.

Detector toward (numerator +); source away (denominator +)
\[\begin{gathered} \text{(a)}\quad f' = 1000\,\frac{343 + 33.3}{343} \approx 1097\,\mathrm{Hz} \\ \text{(b)}\quad f' = 1000\,\frac{343}{343 + 33.3} \approx 912\,\mathrm{Hz} \end{gathered}\]

Approaching raises the pitch by ~97 Hz; receding lowers it. The up-shift and down-shift are not equal in size because the source and detector speeds enter the formula in different places.

Review

Chapter Summary

✓ Speed of sound

\(v = \sqrt{B/\rho}\) — set by stiffness (\(B\)) and inertia (\(\rho\)). In air at 20 °C, 343 m/s.

✓ Traveling sound wave

\(s = s_{m}\cos(kx-\omega t)\) and \(\Delta p = \Delta p_{m}\sin(kx-\omega t)\), 90° out of phase.

✓ Pressure amplitude

\(\Delta p_{m} = (v\rho\omega)s_{m}\) — usually a minute fraction of atmospheric pressure.

✓ Interference

\(\phi = (\Delta L/\lambda)2\pi\). Loud at \(\Delta L/\lambda = 0,1,2,\dots\); quiet at \(0.5,1.5,\dots\)

✓ Intensity & decibels

\(I = \tfrac{1}{2}\rho v\omega^{2}s_{m}^{2}\), \(I = P_{s}/4\pi r^{2}\), \(\beta = (10\,\mathrm{dB})\log(I/I_{0})\).

✓ Pipe resonance

Open–open: \(f = nv/2L\), all \(n\). Open–closed: \(f = nv/4L\), odd \(n\) only.

✓ Beats

Two close frequencies throb at \(f_{\text{beat}} = |f_{1} - f_{2}|\).

✓ Doppler & shock

\(f' = f\,\frac{v \pm v_{D}}{v \pm v_{S}}\) (toward = up). Supersonic: \(\sin\theta = v/v_{S}\).

Practice

Problems

Unless told otherwise, take the speed of sound in air as \(343\,\mathrm{m/s}\) and the density of air as \(1.21\,\mathrm{kg/m^{3}}\). Interference problems hinge on the path-length difference; intensity problems on \(1/r^{2}\) and the decibel definition; pipe problems on the open/closed boundary rules; Doppler problems on "toward = up, away = down."

  1. What is the bulk modulus of oxygen if 32.0 g occupies 22.4 L and the speed of sound in it is 317 m/s?
  2. A stone is dropped into a well and the splash is heard 3.00 s later. How deep is the well? (Account for both the fall and the sound's travel time.)
  3. Diagnostic ultrasound at 4.50 MHz is used to image soft tissue. Find its wavelength (a) in air and (b) in tissue, where the sound speed is 1500 m/s.
  4. A traveling sound wave has pressure \(\Delta p = (1.50\,\mathrm{Pa})\sin\pi[(0.900\,\mathrm{m^{-1}})x - (315\,\mathrm{s^{-1}})t]\). Find its (a) pressure amplitude, (b) frequency, (c) wavelength, and (d) speed.
  5. Two sources, in phase at 540 Hz, send sound (330 m/s) in the same direction to a point 4.40 m from one and 4.00 m from the other. What is the phase difference there?
  6. A 1.0 W point source emits sound isotropically. Find the intensity (a) 1.0 m and (b) 2.5 m from the source.
  7. A certain sound source is raised in level by 30.0 dB. By what factor does (a) its intensity and (b) its pressure amplitude increase?
  8. A source emits isotropically; the intensity 2.50 m away is \(1.91 \times 10^{-4}\,\mathrm{W/m^{2}}\). What is the source's power?
  9. A glass tube 1.00 m long, open at the top and closed by adjustable water below, resonates with a 686 Hz fork. (a) For how many water levels does resonance occur? Find the (b) least and (c) second-least water heights.
  10. Organ pipe A (both ends open) has a fundamental of 300 Hz. The third harmonic of pipe B (one end open) equals the second harmonic of A. Find the lengths of (a) pipe A and (b) pipe B.
  11. Two identical piano wires have a fundamental of 600 Hz at equal tension. What fractional increase in one wire's tension produces 6.0 beats/s when both are sounded?
  12. A 540 Hz whistle moves in a circle of radius 60.0 cm at 15.0 rad/s. What are the (a) lowest and (b) highest frequencies heard by a distant listener at rest relative to the circle's center?
  13. A police car (siren 500 Hz) chases a speeder; both travel at 160 km/h on a straight road. What Doppler shift does the speeder hear?
  14. A jet flies at Mach 1.5 at a height of 5000 m (sound speed 331 m/s). Find (a) the Mach cone angle and (b) how long after the jet passes overhead the shock wave reaches you.
Tip: three habits cover most of this chapter. For interference, convert path difference to wavelengths first — integers are loud, half-integers are quiet. For decibels, remember each ×10 in intensity is +10 dB, and intensity itself falls as \(1/r^{2}\). For Doppler, write the general formula once and decide each sign by asking whether that motion is toward (shift up) or away (shift down) — never memorize four separate cases.