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Sound Waves

Source lecture(s): SC133 Lec 24

Intuition

Sound is a longitudinal pressure wave: air molecules shuffle back and forth along the travel direction, piling into compressions and thinning into rarefactions that race outward at ~343 m/s. Your eardrum doesn't receive matter from the source — it receives the pattern, a few pascals of pressure wiggle riding on the full atmosphere.

Speed of sound

The universal medium formula, \(v = \sqrt{\text{stiffness}/\text{inertia}}\):

\[v = \sqrt{\frac{B}{\rho}} \qquad\text{(air, ideal-gas form): } v = \sqrt{\frac{\gamma R T}{M}} \approx 343\,\text{m/s at }20°\text{C}\]
  • Rises with temperature (~0.6 m/s per °C) — instruments sharpen as halls warm.
  • Faster in water (~1480 m/s) and steel (~5900 m/s): stiffness wins over density.
  • Depends on the gas, not the pressure: helium's tiny \(M\) nearly triples \(v\), raising your vocal-tract resonances — the duck voice.

Intensity and the decibel

Sound intensity (power per area) from a point source spreads over spheres:

\[I = \frac{P}{4\pi r^2} \qquad\text{— the inverse-square law}\]

Ears operate across 12 orders of magnitude, so we use a log scale:

\[\beta = 10\log_{10}\frac{I}{I_0}\ \text{dB}, \qquad I_0 = 10^{-12}\,\text{W/m}^2\]

+10 dB = ×10 intensity (≈ "twice as loud" perceptually); +3 dB = ×2 intensity. Whisper 30 dB, conversation 60 dB, rock concert 110 dB, pain ~130 dB.

The Doppler effect

Relative motion between source and listener shifts the received frequency:

\[f' = f\,\frac{v \pm v_\text{listener}}{v \mp v_\text{source}} \qquad\text{(upper signs: approaching)}\]

Approach ⇒ higher pitch (wavefronts bunched); recede ⇒ lower. The ambulance-siren drop, radar speed guns, ultrasound blood-flow imaging, and — with light — the redshift that revealed the expanding universe. A source at the sound speed piles its wavefronts into a shock cone.

Worked example: concert loudness

You measure 80 dB at 10 m from a speaker. What at 40 m?

Inverse square: \(I\) drops ×16 ⇒ \(\Delta\beta = 10\log 16 \approx 12\) dB: 68 dB. (Real rooms add reverberation; outdoors this rule is good.)

Worked example: passing train

A train horn at 400 Hz passes you at 30 m/s. Pitch heard on approach and recession?

\[f_\text{approach} = 400\,\frac{343}{343 - 30} \approx 438\,\text{Hz} \qquad f_\text{recede} = 400\,\frac{343}{343 + 30} \approx 368\,\text{Hz}\]

— a drop of nearly two semitones as it passes: the classic neee-ooow.

Common mistakes

  • Doppler depends only on relative speed? For sound, no — the medium matters; moving source and moving listener give (slightly) different formulas. (For light, only relative motion counts.)
  • Confusing loudness (dB, logarithmic, perceptual) with intensity (W/m², physical) — "twice the watts" is +3 dB, barely noticeable.
  • Thinking molecules travel to your ear. Displacement amplitudes are micrometres; it's the pattern that propagates.
  • Applying the approach formula while the source passes abeam — at closest approach the shift is momentarily zero; only the radial velocity component counts.

Knowledge graph position

Prerequisites: Traveling waves. Leads to: Beats, Shock waves, acoustics and audiology.

Quiz

Q1 (computational). Thunder arrives 4 s after the lightning flash. How far away was the strike?

Answer

\(d = vt \approx 343\times4 \approx 1.4\,\text{km}\) — the "3 seconds per kilometre" rule.

Q2 (conceptual). Why does inhaling helium raise the pitch of your voice but not the frequency of a tuning fork struck in the same room?

Answer

Your vocal tract is a resonant cavity: its formant frequencies scale with the gas sound speed (\(f \sim v/L\)). The tuning fork's frequency is set by its own metal elasticity, not the surrounding gas — the gas only carries the sound.

Q3 (multiple choice). Moving toward a stationary siren at high speed, you hear: (a) higher pitch and it drops as you pass (b) the true pitch (c) lower pitch rising

Answer

(a). Approach raises \(f'\); after passing, recession lowers it below the true value — the shift flips sign at closest approach.