Displacement & Pressure Waves
Where the air moves most, the pressure changes least.
Lesson at a glanceShow the key points
- Displacement wave: s = s₀ sin ω(t − x/v).
- The volume strain of a thin slab gives the excess pressure p = (Bωs₀/v) cos ω(t − x/v), so p₀ = Bks₀ (16.2).
- Pressure and displacement differ in phase by π/2: pressure extremes are at zero displacement and vice versa.
- Ears and microphones respond to pressure, so describe sound (and add sound waves) using pressure.
- Midway between two facing speakers driven together, displacement cancels but pressure doubles.
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 fraction .
The bulk modulus B links pressure and volume change: . 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 . 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.
A sound wave of wavelength 50 cm has a pressure amplitude of 0.020 Pa in air (bulk modulus 1.4 × 10⁵ N/m²). Find the displacement amplitude.
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.
- 1Displacement wave s = s₀ sin ω(t − x/v); pressure wave p = p₀ cos ω(t − x/v).
- 2p₀ = Bks₀ = (Bω/v)s₀.
- 3Pressure and displacement differ in phase by π/2: pressure extremes sit where the displacement is zero.
- 4Detectors respond to pressure, so we add sound waves as pressure changes.
Check yourself
Keep exploring
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