Waves & Oscillations · Chapter 10

Sound

Sound is a longitudinal pressure wave in a medium. Its physics spans the piano and the sonic boom, the bat's sonar and the medical ultrasound scan.

PrerequisitesWave properties (Ch. 8) — wavelength, frequency, speed, superposition

Learning Goals

$Calculate the speed of sound in an ideal gas from \gamma, R, T, and M$ .
$Convert between intensity in W/m^{2} and decibels using the logarithmic scale$ .
Apply the Doppler formula to find observed frequency for moving sources and observers.
Determine resonant frequencies of open-open and open-closed tubes.
Predict beat frequency when two nearly equal frequencies are superposed.

10.1 The Nature of Sound

Sound is a longitudinal mechanical wave: the displacement of the medium (air molecules) is parallel to the direction of wave propagation. A vibrating source creates alternating regions of compression (high pressure) and rarefaction (low pressure) that propagate outward as a traveling wave.

The speed of sound depends on the medium's elastic and inertial properties. In an ideal gas it is:

v_s=\sqrt{\frac{\gamma RT}{M}}

(10.1)

where γ is the adiabatic index (≈1.4 for air), R = 8.314 J/(mol·K), T is the absolute temperature, and M is the molar mass. At 20°C, v_s ≈ 343 m/s in air. Sound travels faster in denser solids — about 5100 m/s in steel — because the restoring force (bulk modulus) increases faster than the density.

10.2 Intensity and the Decibel Scale

The intensity I of a sound wave is the power transported per unit area, measured in W/m². For a point source radiating isotropically at power P in free space, intensity falls as the inverse square of distance: I = P/(4πr²).

Because human hearing spans twelve orders of magnitude in intensity — from 10⁻¹² W/m² (threshold of hearing) to 1 W/m² (painful) — we use a logarithmic scale:

\beta=10\log_{10}\left(\frac{I}{I_0}\right) \qquad \text{(decibels, dB)}

(10.2)

where I₀ = 10⁻¹² W/m² is the reference intensity. Ordinary conversation ≈ 60 dB, a rock concert ≈ 110 dB, a jet engine at 30 m ≈ 140 dB (threshold of pain). Every 10 dB increase is a factor of 10 in intensity and roughly a factor of 2 in perceived loudness.

Example 10.1 — Sound Level at Different Distances

A speaker outputs 1 W of acoustic power. Find the intensity and decibel level at 1 m and 10 m.

At 1 m:

I_{1} = P/(4\pir^{2}) = 1/(4\pi) = 0.0796 W/m^{2}

dB at 1 m:

\beta_{1} = 10 log_{10}(0.0796 / 10^{-12}) = 10 \times 10.9 = 109 dB

At 10 m:

I_{10} = 0.0796/100 = 7.96\times10^{-4} W/m^{2}

dB at 10 m:

\beta_{10} = 10 log_{10}(7.96\times10^{-4} / 10^{-12}) = 10 \times 8.9 = 89 dB

Note:

10\times distance \to intensity drops 100\times \to sound level drops 20 dB. ✓ (inverse square

10.3 The Doppler Effect

When a source of sound moves relative to an observer, the observed frequency differs from the emitted frequency. This is the Doppler effect: motion toward the observer compresses the wavefronts, raising the perceived pitch; motion away stretches them, lowering it.

Definition 10.1 — Doppler Frequency Formula

For a source moving at speed v_{s} and observer moving at speed v_{o}, both measured relative

to the medium (positive when moving toward each other):

f_{obs} = f_{0} \times (v + v_{o}) / (v - v_{s}

where v = 343 m/s is the speed of sound. If source approaches: v_{s}

0 \to f_{obs}

f_{0}. If source recedes: v_{s}

0 \to f_{obs}

f_{0}

Figure 10.1. Doppler effect simulation. At rest, the wavefronts are evenly spaced. As the source moves right, wavefronts bunch toward the observer on the right (higher pitch) and spread out to the left (lower pitch). Push the source toward Mach 1 to see the Mach cone form.

10.4 Resonance in Tubes

A sound wave reflecting inside a tube creates a standing wave. The frequencies at which standing waves form are the natural frequencies or resonant modes. These are the fundamentals and harmonics heard from organ pipes, clarinets, and trumpets.

Theorem 10.1 — Resonant Frequencies in Tubes

Open–open tube(

both ends open, pressure nodes at ends): f_{n} = nv/(2L), n = 1, 2, 3, \cdots \qquad (all harmonics

Open–closed tube(

one open, one closed end): f_{n} = nv/(4L), n = 1, 3, 5, \cdots \qquad (odd harmonics only

A closed end is a displacement node (pressure antinode); an open end is a displacement antinode (pressure node).

Example 10.2 — Fundamental Frequency of an Organ Pipe

An open o $rgan pipe is 2.0 m long. Find its fundamental frequency and the first three harmonics. (v_$ /s)

Fundamental (n=1):

f_{1} = v/(2L) = 343/(2\times2.0) = 343/4 = 85.75 Hz

Second harmonic:

f_{2} = 2f_{1} = 171.5 Hz

Third harmonic:

f_{3} = 3f_{1} = 257.3 Hz

Note:

If the pipe were closed at one end (length same), f_{1} = v/(4L) = 42.9 Hz — an octave lower

, with only odd harmonics.

10.5 Beats

When two sound waves of slightly different frequencies f₁ and f₂ overlap, their superposition produces a slowly varying amplitude modulation called beats. The beat frequency — the rate at which the amplitude pulsates — is:

f_\mathrm{beat}=|f_1-f_2|

(10.3)

Musicians use beats to tune instruments: when f_beat → 0, the two instruments are in tune. A guitar string slightly sharp relative to a 440 Hz tuning fork produces a 3 Hz beat — three loud pulses per second — which slows to zero as the string is loosened to pitch.

Definition 10.2 — Common Traps

Sound needs a medium: unlike light, sound does not propagate through vacuum.
Decibels are logarithmic: $adding 10 dB multiplies intensity by 10; it does not add a fixed W/m^{2} amount$
Doppler signs follow relative motion: approach raises observed frequency; recession lowers it.
Open and closed pipe ends swap node types: open ends are pressure nodes, closed ends are pressure antinodes.

Exercises — 8.1–8.5 Sound

A 400

Hz source moves toward a stationary observer at 30 m/s. Find the observed frequency. (v =

/s)

Straightforward

An open organ pipe is 2.0 m long. Find its fundamental frequency. (v = 343 m/s

Straightforward

A sound measures 109 dB at 1 m from a speaker. What does it measure at 10 m?

Intermediate

4.Two violin strings sound at 445 Hz and 441 Hz. How many beats per second are heard, and which string must be adjusted to bring them into tune?

Intermediate

5.Derive the Mach cone half-angle as a function of Mach number. At Mach 2, what is the cone angle? Explain why an observer on the ground hears silence until the boom arrives.

Challenging

Key Takeaways

$Sound is a longitudinal pressure wave; speed in air \approx 343 m/s at 20°C, faster in denser so$ lids.
$Intensity I = P/(4\pir^{2}); decibel scale \beta = 10 log(I/I_{0}) compresses the 10^{12} dynamic range$ .
$Doppler effect: f_{obs} = f_{0}(v+v_{o})/(v-v_{s}) — approach raises pitch, recession lowers it$ .
Resonance: open–open pipe has all harmonics; open–closed pipe has only odd harmonics.
$Beats: f_{beat} = |f_{1}-f_{2}|; used by musicians to tune by ear$ .
$Sonic boom at Mach \ge 1: shock cone half-angle = arcsin(1/M$ .