5.1
What is sound?
§5 · Energy, momentum, impedance

5.1 Plane harmonic waves

The simplest solution to the wave equation is a plane harmonic wave:

p′(r,t)  =  P0 cos⁡(ωt−k⋅r),p'(\mathbf{r}, t) \;=\; P_0\, \cos(\omega t - \mathbf{k} \cdot \mathbf{r}),

or, in complex form,

p~′(r,t)  =  P0 ei(ωt−k⋅r).\tilde p'(\mathbf{r}, t) \;=\; P_0\, e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}.
where
p′(r,t)p'(\mathbf{r}, t)
acoustic pressure perturbation (departure from ambient) Pa
P0P_0
peak pressure amplitude Pa
r\mathbf{r}
position vector of the field point m
tt
time s
ω\omega
angular frequency rad/s
k\mathbf{k}
wavevector: points along propagation, magnitude k=∣k∣=2π/λk = |\mathbf{k}| = 2\pi/\lambda rad/m
cc
speed of sound in the medium m/s

The vector k\mathbf{k} is the wavevector: its direction is the direction of propagation, and its magnitude k=∣k∣k = |\mathbf{k}| is the wavenumber. The relation ω=ck\omega = c k is the dispersion relation of acoustic waves — linear, the same at all frequencies. Substituting the plane wave into ∂t2p′=c2∇2p′\partial_t^2 p' = c^2 \nabla^2 p':

−ω2P0 ei(⋯ )  =  c2(−∣k∣2)P0 ei(⋯ ),-\omega^2 P_0\, e^{i(\cdots)} \;=\; c^2 (-|\mathbf{k}|^2) P_0\, e^{i(\cdots)},

i.e. ω2=c2∣k∣2\omega^2 = c^2 |\mathbf{k}|^2 — confirmed.

k

Cream regions are at equilibrium pressure; red is compression; blue is rarefaction. Slide the direction to see the wavevector rotate. Increase the wavelength to spread the bands and shorten the arrow; decrease it to pack the bands in and grow the arrow, since |k| = 2π/λ.

The interactive above renders a snapshot of the plane-wave pressure field over a 2-D region. Cream is equilibrium pressure; red is compression; blue is rarefaction. The black arrow is the wavevector k\mathbf{k} pointing in the direction of propagation, with length proportional to its magnitude ∣k∣=2π/λ|\mathbf{k}| = 2\pi/\lambda. Slide the direction to rotate k\mathbf{k} and watch the wavefronts (loci of constant pressure) align perpendicular to it. Slide the wavelength to widen or compress the bands — and watch the arrow shorten as the bands spread and lengthen as they pack in, since a shorter wavelength means a larger wavenumber. With the animation playing, the whole pattern slides bodily at speed cc along the direction of k\mathbf{k} — the entire content of the wave equation’s plane-wave solution.

Wavelength and frequency, for air

The kinematics of λ\lambda, ff, TT, and the relation c=λfc = \lambda f are general to any harmonic wave and were developed — with an interactive — in 3.3 (harmonic traveling waves). Here we only instantiate them for air, where c≈343c \approx 343 m/s, because the resulting wavelengths set the scale for everything in the chapters that follow:

These wavelengths are what decide whether a wave reflects, diffracts, or wraps around an object: diffraction and the room (chapter 7) turn on the ratio of λ\lambda to obstacle size, and the radiating efficiency of a source (chapter 6) turns on the ratio of λ\lambda to source size.

Velocity, density, pressure — all in phase

The plane wave also fixes the relationship between the three perturbation fields. From the linearised Euler equation, ρ0∂tv′=−∇p′\rho_0 \partial_t \mathbf{v}' = -\nabla p', applied to a wave going in the +k^+\hat{\mathbf{k}} direction:

ρ0(iω)v~′  =  −(−ik)p~′  =  ik p~′,\rho_0 (i\omega) \tilde{\mathbf{v}}' \;=\; -(-i\mathbf{k}) \tilde p' \;=\; i\mathbf{k}\, \tilde p',

so

v~′  =  kρ0ω p~′  =  k^ρ0c p~′.\tilde{\mathbf{v}}' \;=\; \frac{\mathbf{k}}{\rho_0 \omega}\, \tilde p' \;=\; \frac{\hat{\mathbf{k}}}{\rho_0 c}\, \tilde p'.
where
v′\mathbf{v}'
particle velocity (motion of the fluid itself, not the wave) m/s
ρ′\rho'
density perturbation kg/m³
ρ0\rho_0
ambient (equilibrium) density of the medium kg/m³
k^\hat{\mathbf{k}}
unit vector along the propagation direction —

Velocity is in phase with pressure (no ii in the relation), pointing in the direction of propagation, with magnitude ∣v~′∣=∣p~′∣/(ρ0c)|\tilde v'| = |\tilde p'| / (\rho_0 c). Similarly, ρ′=p′/c2\rho' = p' / c^2. All three fields oscillate together — coherence in time at every point and coherence in space at every instant.

distance along k →
pressure p′ particle velocity v′

All three fields are proportional to cos(kx − ωt), so their crests, zeros, and troughs line up on the dashed guide: they move in phase. The curves are scaled to equal height to show that phase agreement — physically p′, v′, and ρ′ have different magnitudes, related by v′ = p′/ρ₀c and ρ′ = p′/c². Pause to scrub the phase by hand.

The three curves share every crest, zero, and trough: that is what “in phase” means. Their magnitudes differ — pressure and velocity are tied by the factor ρ0c\rho_0 c, the specific acoustic impedance that lesson 5.4 is named for — but their timing is identical. This is special to a single travelling plane wave; at a wall, or in a standing wave, pressure and velocity fall a quarter-cycle out of phase, and that phase shift is exactly what stores energy without radiating it.

The “shape” of a sound

A real sound — speech, music, applause — is not a single plane wave. But because the wave equation is linear, any sound can be written as a sum (or integral) of plane waves with different ω\omega, k\mathbf{k}, amplitudes, and phases. The plane wave is the basis element for the rest of the book. Master what a plane wave does (which we will, over the next 5 lessons), and you understand what any linear sound does — just sum the contributions.

This is the entire program of Fourier analysis applied to acoustic fields. The frequency picture (chapter 8) makes it operational. For chapter 5 we work with one plane wave at a time, since everything we are about to compute is linear in the field (linear adds to linear; we can sum at the end).

What we will compute, in this chapter

For a plane wave of pressure amplitude P0P_0:

  1. The energy density E\mathcal{E} — energy per unit volume in the field (lesson 5.2).
  2. The intensity II — energy flowing per unit area per unit time (lesson 5.3).
  3. The specific acoustic impedance Z=p′/v′Z = p' / v', the ratio of pressure to particle velocity (lesson 5.4).
  4. The decibel, the engineer’s logarithmic scale for intensity (lesson 5.5).
  5. The momentum the wave carries and the radiation pressure it exerts on obstacles (lesson 5.6).

We have the wave; now let us see what it carries.