6.4 The dipole as two opposing monopoles

A monopole changes the local volume of the medium. But many vibrating objects don’t: a tuning fork’s prong moves left, then right; a violin string vibrates transversely; a loudspeaker cone moves forward, the back of the cabinet moves backward by an equal amount. These sources don’t create volume; they displace it.

The cleanest model for such a source is the acoustic dipole: two equal-and-opposite monopoles separated by a small distance $d$ .

The construction

Place a positive monopole (radiating with $+Q_0$ volume velocity) at $+d/2$ along the $z$ -axis, and a negative monopole ( $-Q_0$ ) at $-d/2$ . Each radiates the spherical-wave field of lesson 6.1. The total field is the sum:

p(\mathbf{r}, t) \;=\; \frac{i \omega \rho_0 Q_0}{4\pi}\!\left[\frac{e^{-ik r_+}}{r_+} - \frac{e^{-ik r_-}}{r_-}\right],

where $r_\pm$ are the distances to the two sources.

The far-field pattern

For $d \ll \lambda$ and $r \gg d$ , expand $r_\pm \approx r \mp (d/2) \cos\theta$ , where $\theta$ is the angle between $\mathbf{r}$ and the dipole axis. The two contributions almost cancel; the residual is the dipole’s far-field pattern:

p(\mathbf{r}, t) \;\approx\; \frac{\omega^2 \rho_0\, Q_0\, d}{4\pi c\, r}\, \cos\theta\, \cos\!\big(\omega t - k r\big).

Two things to notice:

Directionality. The pressure goes as $\cos\theta$ . It is zero perpendicular to the dipole axis (the two monopoles cancel exactly), and maximum along the axis. A figure-eight pattern in 2-D, a “dumbbell” in 3-D.
An extra factor of $\omega/c = k$ . Compared to a monopole of the same volume velocity, the dipole radiates an amplitude smaller by $\sim k d$ . Since $k d \ll 1$ for a small dipole, the dipole is a much weaker radiator than a monopole of the same volume velocity at the same frequency. (Conversely, at higher frequencies — where $k d$ approaches $1$ — the dipole becomes more efficient.)

The intensity radiated by a dipole:

\langle I \rangle \;\propto\; \frac{\omega^4\, Q_0^2\, d^2}{c^2\, r^2}\, \cos^2\theta.

The $\omega^4$ is the same scaling as Rayleigh scattering of light (and for the same reason — both are dipole radiation in 3-D). It explains why high-frequency dipole sources (tuning forks at audible pitches) radiate much more efficiently than low-frequency ones.

Concrete examples

Unbaffled loudspeaker. A bare loudspeaker cone is a dipole: when the cone moves forward, it compresses the air in front and creates a rarefaction behind. The forward and backward radiation are out of phase, partially cancelling. This is why speakers are mounted in cabinets (which seal off the back, converting the source from dipole to monopole) or in infinite baffles (which similarly block the back radiation).
Tuning fork. The two prongs each move outward, then inward, in opposite directions. The fork is a quadrupole at high order, but its dominant radiating mode is a dipole. This is why tuning forks are quiet (poor radiators of their fundamental), and why you press them against a resonator (the resonator’s vibrating surface, with the same total volume displacement, is a far better monopole radiator).
Animal vocalisations. The mouth opens and closes: a monopole. The chest expands and contracts: also a monopole (much smaller). But a wing beat or a fin flick: a dipole, often. Different species’ acoustic signatures reflect what kind of source mechanism they have.

Dipole moment

The dipole’s strength is captured by the dipole moment $\mathbf{p}_d = Q_0\, \mathbf{d}$ . Two dipoles with the same $\mathbf{p}_d$ are interchangeable at distances large compared to either size. This is analogous to the electric-dipole moment of electromagnetism (and the analogy goes much deeper — radiation patterns, frequency dependence, even the language).

Higher multipoles

Pairs of opposing dipoles make quadrupoles; pairs of opposing quadrupoles make octupoles; etc. Each higher multipole is more directional but radiates less efficiently at the same volume velocity. The multipole expansion of an arbitrary source is the acoustic analogue of the spherical-harmonic expansion of an arbitrary electromagnetic source, with the same Bessel-function radial profiles.

For the most part in this book we will care only about monopoles and dipoles. The exception is the piston in a baffle (next lesson), which is genuinely a directional radiator and which the multipole expansion treats poorly.