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})}.

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; decrease it to pack them in.

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. 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. 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

The wavelength is λ=2π/k\lambda = 2\pi/k. The temporal period is T=2π/ωT = 2\pi/\omega. The dispersion relation ω=ck\omega = c k becomes the familiar λf=c\lambda f = c (with f=ω/2πf = \omega/2\pi). For air at 343 m/s:

These wavelengths matter for diffraction (chapter 7) and for the antenna properties of acoustic sources (chapter 6).

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'.

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.

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.