It should be possible to reconstruct the shape, size and make up of an object by studying how sound propagates in it. We use the word “sound” quite loosely to mean vibrations, or wave-like signals. For example if we have a block of material and we knock it, it should tremble and shake for a bit before it settles down and becomes still. It settles down because the energy we impart it, and sets it vibrating, slowly dissipates and heats the block up. Let us now consider a simple, two dimensional, surface: the surface of a drum. The edges are securely fastened to a rim which can have any size or shape. We are used to round drums, but there are oval drums or even square ones. If we bang the drum, the membrane will vibrate and emit sound. The sound we hear is simply the motion that the vibration of the membrane imparts on the air around it. We could, for example, put a pencil between our teeth and hold it to the rim. Our head would pick up the vibrations and we would hear the sound of the drum.
The sound that each drum makes depends on a number of things. It depends on its size for example. The smaller the drum, the higher the pitch it emits because the quicker it vibrates and the shorter the size of the vibrations. It will also depend on its shape. For example, a circular drum is symmetric in that it we can turn it round its centre by any amount, and it always looks the same. This severely restricts the type of sound it can make. A square drum also has some symmetry. We can turn it around by 90 degrees and it will look unchanged. But we can’t rotate it by an arbitrary amount, unlike the circular drum. The types of vibrations that these two drums can host are different. Finally, the constitution of the membrane will greatly affect what it sounds like. If it is thick and tightly stretched it will tremble more quickly than if it is thin and flaccid. So by listening to its sound, we can figure out what a drum looks like and what it is made of. Of course it is never that simple because the different properties (shape, size and constitution) may affect the sound in similar ways and it becomes difficult to disentangle what is responsible for making the drum sound the way it does. For example a small, loosely stretched may sound vaguely similar to a large, taut one. They won’t sound exactly the same, and the more precise our hearing, the more accurately we can distinguish between them.