SPRDSenior Physics Research DatabaseDickinson College

All projectsDirectional speaker

Theory

Why ultrasound beams are narrow

The directivity of a piston and of an array, and why the same panel that beams ultrasound cannot beam 1 kHz.

Week 1 of 4; the paper is due at the end of week 4

Goals

  • Derive or carefully follow the far-field directivity of a baffled circular piston.
  • Compute the beam half-angle and Rayleigh distance of the department speaker at its carrier frequency and at 1 kHz.
  • Explain why an ordinary loudspeaker cannot make a narrow audio beam.

Why this matters

The whole point of the project is a beam of sound you can aim. This step shows why the size of the source compared with the wavelength sets the beam width, and why you need ultrasound to get a narrow beam from a panel you can carry.

The physics

A circular piston of radius aa in a rigid baffle radiates a far-field pressure with angular dependence

H(θ)=2J1(kasin⁡θ)kasin⁡θ,H(\theta) = \frac{2 J_1(ka\sin\theta)}{ka\sin\theta},

with the first null at kasin⁡θ=3.832ka\sin\theta = 3.832. The far field starts roughly at the Rayleigh distance R0=ka2/2=πa2/λR_0 = ka^2/2 = \pi a^2/\lambda.

At 60 kHz in air, λ≈5.7\lambda \approx 5.7 mm. A panel with a=5a = 5 cm has ka≈55ka \approx 55, so the first null sits about 4° off axis and R0≈1.4R_0 \approx 1.4 m. The same panel playing 1 kHz has ka<1ka < 1: it radiates almost equally in every direction.

An array of NN small transducers with spacing dd behaves like the piston multiplied by an array factor. When dd is larger than λ\lambda, extra main lobes called grating lobes appear at sin⁡θ=mλ/d\sin\theta = m\lambda/d.

Hints

Hint 1Plot it before you trust it

Plot ∣H(θ)∣|H(\theta)| in dB for ka=1ka = 1, 5 and 55 on the same axes. The change from omnidirectional to a pencil beam is one of the most convincing figures you can put in your paper.

Hint 2Look at the real speaker

Measure the radiating face of the department’s speaker and estimate kaka at its carrier frequency. If you can see individual transducers, measure their spacing: is it more than a wavelength? Where would the first grating lobe be, and how much does the directivity of a single transducer suppress it?

Hint 3Near field versus far field

Your listener will probably stand 1–3 m away, which is not far beyond R0R_0. Keep a note of this. It is one reason the simple far-field formulas in the next step do not tell the whole story and the KZK model is worth the effort.

What past teams did

2026–27 team

First team on the project, so there are no notes yet. Add the beam angle you predicted and how it compared with what you measured.