The Acoustic Lens: Focusing Sound the Way Glass Focuses Light

Sound is a wave, and waves can be focused

Light bends when it crosses from air into glass because it travels at a different speed in each medium; a shaped piece of glass, a lens, uses that bending to focus light to a point. Sound obeys exactly the same geometric logic. Sound speed depends on the medium it travels through — roughly 343 m/s in air, faster in denser or stiffer materials, slower in others — and a region shaped to have a different sound speed than its surroundings bends passing wavefronts the same way a glass lens bends light. That region is an acoustic lens.

Solving the wave equation instead of tracing rays

The most direct way to simulate this is to solve the acoustic wave equation itself on a grid, rather than tracing individual rays. In two dimensions, pressure u(x, y, t) obeys

∂²u/∂t² = c(x, y)² ∇²u
c(x, y) = local sound speed — constant in the background medium, different inside the lens region

and the simulation advances that equation directly with a finite-difference time-domain (FDTD) scheme: at every grid point, estimate the spatial second derivative (the Laplacian) from the four or eight nearest neighbours, use it to update the point's acceleration, then step the pressure field forward in time with a simple leapfrog update. Run it long enough from a point source and the field looks exactly like ripples on a pond — because ripples on a pond and sound in air satisfy the same class of wave equation, just with a different c.

Why a variable c(x, y) is enough to build a lens

Nothing in the simulation needs to know about rays or Snell's law explicitly — focusing emerges automatically from the wave equation once c(x, y) varies in space. A wavefront crossing into a region of lower sound speed slows down on the side that enters first, which rotates the wavefront the same way a marching band rotates when one end steps into mud; a region shaped like a converging lens (thicker in the middle, where the speed change is largest) bends the wavefronts from a distant source so they all converge on a single focal point, exactly where the ray picture built on Snell’s law, n₁ sinθ₁ = n₂ sinθ₂ with the acoustic refractive index n = c_background / c_lens, predicts they should. Watching the two pictures agree — wavefronts visibly bending in the FDTD simulation, converging exactly where the ray-optics formula says the focal point should be — is a direct demonstration that ray optics is the short-wavelength limit of the full wave equation.

Where this shows up: medical ultrasound

This is not just a classroom analogy — shaped acoustic lenses, usually silicone or a similar low-speed polymer bonded to the front of a piezoelectric transducer array, are how medical ultrasound probes focus their beam at a chosen depth inside the body to get a sharp image at that depth. Phased-array probes go a step further and focus electronically, delaying the drive signal to each element of the array so that the many small wavefronts they emit arrive in phase at the focal point — a lens made of timing rather than shape, but governed by the identical wave equation and the identical idea of engineering where wavefronts add up constructively.

What to look for in the simulation

Watch the wavefronts as straight or circular lines approaching the lens region, note how they visibly kink and curve as they cross into the different-speed material, and follow them to the point downstream where they pile up into a bright focus. Changing the lens shape or its sound-speed contrast moves that focal point exactly the way changing a glass lens's curvature or refractive index moves an optical focus.

Frequently asked questions

Try it live

Everything above runs in your browser — open Acoustic Lens and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.