6.1 A pulsating sphere — the monopole

The simplest acoustic source is a sphere of radius $a$ whose surface oscillates radially in and out:

r_\text{surface}(t) \;=\; a + \delta\, \cos(\omega t),

with $\delta \ll a$ . As the sphere expands, it pushes against the surrounding air, creating a pressure pulse that propagates outward. As it contracts, it draws air back in, creating a pressure trough. Repeat indefinitely. This is the acoustic monopole — and despite its idealised geometry, it is the cleanest model for any source small compared to the wavelength.

The radial wave equation

For a spherically symmetric field $p(r, t)$ , the wave equation reduces (after applying $\nabla^2$ in spherical coordinates and using symmetry):

\frac{\partial^2 (r p)}{\partial t^2} \;=\; c^2\, \frac{\partial^2 (r p)}{\partial r^2}.

The combination $r p$ satisfies a 1-D wave equation. So its solutions are precisely d’Alembert’s:

r\, p(r, t) \;=\; f(r - c t) + g(r + c t),

i.e.

p(r, t) \;=\; \frac{f(r - c t)}{r} + \frac{g(r + c t)}{r}.

The first term is an outgoing wave; the second is incoming. For radiation from a source, we keep only the outgoing piece:

p(r, t) \;=\; \frac{f(r - c t)}{r}.

The amplitude decays as $1/r$ with distance, an automatic consequence of energy conservation as the wave spreads over an expanding spherical surface.

A harmonic monopole

For a sinusoidally pulsating sphere with surface velocity $U_0 \cos(\omega t)$ (amplitude $U_0 = \omega \delta$ ), the outgoing pressure field is

p(r, t) \;=\; \frac{A}{r}\, \cos(\omega t - k(r - a)),

with the amplitude $A$ fixed by matching the boundary condition at $r = a$ (the radial velocity of the air must match the radial velocity of the sphere). For a small source — much smaller than the wavelength, $ka \ll 1$ — the matching gives

A \;\approx\; \frac{i \omega \rho_0\, Q_0}{4\pi},

where $Q_0 = 4\pi a^2 U_0$ is the volume velocity of the source — the rate at which volume is being displaced per unit time. This is the canonical quantity for small monopole sources: not the surface velocity, not the surface area, but their product (volume per unit time).

What a monopole sounds like at distance

Far from the source ( $k r \gg 1$ ), the field is a spherical outgoing wave with pressure amplitude proportional to $Q_0 / r$ . The intensity falls as $1/r^2$ — the inverse-square law. The far-field pressure carries phase information about the source motion through the time delay $r/c$ : a kick at the source at time $t = 0$ produces a kick at radius $r$ at time $t = r/c$ . This is the basis of time-of-arrival localisation in audition (chapter 3 of the Hearing book) and of every sonar / radar system.

What makes a source a monopole

The key feature: a monopole changes the local volume of the medium. A breathing balloon, an exploding firework, a popping cork, a bursting bubble — all are monopole sources. The sound radiates equally in all directions ( $p$ depends only on $r$ , not on angle).

A vibrating object that doesn’t change its volume — a violin string moving sideways, a tuning fork prong moving back-and-forth — is not a monopole. It is at least a dipole (lesson 6.4), with a quite different radiation pattern.

The volume-velocity $Q_0$ is the strength of a monopole. Two monopoles with the same $Q_0$ at the same frequency radiate identically — regardless of their size or shape — provided both are small compared to the wavelength. This is why a small loudspeaker and a popping balloon sound similar when blown up to the same dB level: the wavelength-scale source structure doesn’t matter, only $Q_0$ .

Looking ahead

⏳ The history — Rayleigh and the Theory of Sound

John William Strutt, third Baron Rayleigh, published The Theory of Sound in two volumes (1877, 1894) — the first comprehensive mathematical treatment of acoustics. The work covered vibrating strings, membranes, and plates; the propagation of sound in tubes and in the open air; diffraction, scattering, and radiation from sources of various geometries. Rayleigh’s formulation of the monopole, dipole, and quadrupole radiation patterns — the subject of this chapter — remains standard.

Rayleigh wrote the book while managing his family estate at Terling Place in Essex, before taking up the Cavendish Professorship at Cambridge. The Theory of Sound was unusual for its time in treating acoustics as a branch of mathematical physics rather than of music or physiology. It set the vocabulary and the methods for the field for the next century; nearly every derivation in the Sound book traces a lineage to Rayleigh’s two volumes.

Read the original: The Theory of Sound, Vol. I (Lord Rayleigh (J.W. Strutt), 1877)

The next two lessons take the monopole field apart in more detail: how the pressure varies with distance (lesson 6.2 — the inverse-square law and its corrections near the source), how cylindrical sources differ from spherical ones (lesson 6.3), and then on to dipole and piston sources whose radiation patterns are directional rather than spherically symmetric.