Nonlinear acoustics and the parametric array (Directional speaker)

Why this matters

This is the central idea of the project and the heart of your paper’s introduction. If you can explain it to a classmate in two minutes, the rest of the project is simulation and measurement.

The physics

Sound speed depends slightly on the local pressure: the peaks of a strong wave travel a little faster than the troughs. To second order this is captured by the coefficient of nonlinearity $\beta = 1 + B/2A$ , about 1.2 for air. Including it, plus thermoviscous loss, gives the Westervelt equation

\nabla^2 p - \frac{1}{c_0^2}\frac{\partial^2 p}{\partial t^2} + \frac{\delta}{c_0^4}\frac{\partial^3 p}{\partial t^3} = -\frac{\beta}{\rho_0 c_0^4}\frac{\partial^2 p^2}{\partial t^2}.

The right-hand side acts as a source. If the primary beam contains frequencies $f_1$ and $f_2$ , then $p^2$ contains $f_1 - f_2$ , so every bit of air along the beam radiates a little difference-frequency sound. Those sources are spread along the axis and phased like an end-fire array, which is why the audible sound comes out as a narrow beam with almost no side lobes.

For an amplitude-modulated primary $p_1 = P_0 E(t)\sin\omega_0 t$ , Berktay’s far-field solution on the axis is

p_\text{audio}(z,t) \propto \frac{\beta P_0^2 a^2}{\rho_0 c_0^4\, \alpha_0\, z}\;\frac{\partial^2}{\partial t^2} E^2\!\left(t - \frac{z}{c_0}\right),

where $\alpha_0$ is the absorption at the carrier frequency.

Hints

Hint 1Read the formula term by term

What does $P_0^2$ tell you about how loud the audio can be? What does the second time derivative do to low frequencies compared with high ones? Translate that into dB per octave.

Hint 2Put in plain AM

Take $E(t) = 1 + m\cos\omega_m t$ . Expand $E^2$ and differentiate twice. Compare the amplitude at $2\omega_m$ with the amplitude at $\omega_m$ . What does that mean for a listener when $m = 1$ ?

Hint 3Absorption helps and hurts

Notice $\alpha_0$ in the denominator. Strong absorption of the carrier shortens the virtual array, which is bad for directivity, but it also removes the ultrasound before it reaches the listener. Keep both effects in mind when you explain why the manufacturer chose the speaker’s carrier frequency.

What past teams did

2026–27 team

First team on the project. Add the explanation or analogy that finally made the parametric array click for you.