6.5 The piston in a baffle

A flat circular disk of radius $a$ , vibrating along its normal in a rigid infinite plane (the baffle), is the canonical model for a loudspeaker. It is also the cleanest example of a directional acoustic source — one whose radiation pattern depends strongly on the angle to the source axis. The piston-in-baffle problem is in every acoustics textbook and is worth working through in outline because the directivity it exhibits is universal: any source large compared to the wavelength radiates more in some directions than others.

The setup

A rigid circular piston of radius $a$ vibrates along its normal axis (call it the $z$ -axis) with uniform surface velocity $U_0 \cos(\omega t)$ . The piston sits flush with an infinite rigid plane wall — the baffle — at $z = 0$ . We want the radiated pressure field for $z > 0$ .

The boundary condition is: the normal velocity at $z = 0$ is $U_0 \cos(\omega t)$ over the piston ( $\rho < a$ ) and zero everywhere else.

The radiation integral

Each point on the piston surface radiates as an infinitesimal monopole with volume velocity $U_0\, dA$ . Integrating over the piston gives the pressure at any field point. After the (long) algebra, the far-field pressure at angle $\theta$ from the axis is

p(r, \theta, t) \;=\; \frac{i \omega \rho_0\, U_0\, A}{2\pi r}\, D(\theta)\, e^{i(\omega t - k r)},

with $A = \pi a^2$ the piston area and the directivity function

D(\theta) \;=\; \frac{2 J_1(k a \sin\theta)}{k a \sin\theta},

where $J_1$ is the Bessel function of the first kind, order 1. The combination $J_1(x)/x$ is normalised to $D = 1$ at $\theta = 0$ .

The beam

$D(\theta)$ has structure:

$D(0) = 1$ — maximum on-axis.
$D(\theta) = 0$ at angles where $k a \sin\theta = 3.83$ (the first zero of $J_1$ ) — the first null of the beam.
Further zeros at $7.02, 10.17, \ldots$ — secondary nulls.
Between nulls, side lobes at decreasing amplitude.

The angular width of the main lobe depends on $k a = 2\pi a / \lambda$ :

Small piston / low frequency ( $k a \ll 1$ ): $D(\theta) \approx 1$ everywhere. The piston radiates nearly isotropically, like a monopole.
Large piston / high frequency ( $k a \gg 1$ ): the main lobe is narrow, with half-width $\sin^{-1}(1.22 \lambda / D)$ where $D = 2a$ is the piston diameter — the same formula as the Airy disk in optics.

This is Fraunhofer diffraction. The piston’s directional radiation pattern is the Fourier transform of its aperture function. The connection to chapter 8 (Fourier domain) and chapter 7 (diffraction) is not a coincidence.

Crossover with the loudspeaker

For a 30-cm-diameter woofer cone ( $a = 0.15$ m), the half-angle of the main lobe is approximately:

At 100 Hz ( $\lambda = 3.4$ m): $\sin^{-1}(1.22 \cdot 3.4 / 0.3) \approx$ 90° (nearly omnidirectional).
At 1 kHz ( $\lambda = 0.34$ m): $\sin^{-1}(1.22 \cdot 0.34 / 0.3) \approx$ 88° (still nearly omnidirectional).
At 10 kHz ( $\lambda = 3.4$ cm): $\sin^{-1}(1.22 \cdot 0.034 / 0.3) \approx$ 8° (sharply beamed!).

This is why a single loudspeaker’s high-frequency response sounds coloured off-axis: the beam pattern aims most of the high-frequency energy along the axis, and off-axis listeners hear a roll-off. Multi-driver speakers use small dome tweeters for high frequencies precisely to broaden the beam at the wavelengths where directionality would otherwise dominate.

Radiation impedance

The piston also has a frequency-dependent radiation impedance (its loading on the surrounding air). For $k a \ll 1$ , the radiation resistance is $\sim (k a)^2$ — small, like a monopole. For $k a \gg 1$ , it asymptotes to $\rho_0 c$ — the piston is impedance-matched to the surrounding air and radiates with maximum efficiency.

This is why subwoofers are big: at 50 Hz, $\lambda = 7$ m, and a 0.5-m woofer has $k a = 0.22$ , giving radiation resistance $\sim 0.05 \cdot \rho_0 c$ — most of the cone’s motion is reactive, sloshing air without radiating. Larger woofers, or horn-loaded enclosures (which artificially increase the effective $a$ ), recover the lost efficiency.

Looking ahead

We have surveyed isotropic and directional sources. The radiated wave then propagates outward — and may meet boundaries: walls, fluid interfaces, tube ends, slits. The next chapter is about that interaction. Reflection, refraction, diffraction, modes — all the things the wave does when it bumps into something.