0.1 Mathematical foundations

Sound waves travel at definite velocities characteristic of the propagation medium. Reflection occurs when the waves hit a change in medium and experience a change in velocity. Fermat’s principle of least time states that a wave, in going between two points S and R, must traverse a path length that is stationary with respect to variations of that path. While this principle was formulated for light beams, it also holds for other waves.

Fermat’s principle can be understood from a phasor perspective. Waves that traverse paths close to that of a maximum or minimum (stationary) path will arrive by routes that only differ slightly in path length. Hence, they will arrive from S to R nearly in-phase, and they will add constructively. Waves taking other paths far away from the stationary one will arrive mainly out of phase with each other and cancel.

For an ellipse, the sum of the distances from the two focal points to a point on the ellipse is a constant. Thus the paths SQR for all points Q on an ellipse are stationary. If a pulse is fired from the source S and arrives at receiver R a time t later, we can then place the source and receive at the focal points of an ellipse. The sum of the distances from each focal point to a point on the ellipse is given by:

where v is the speed of sound in the given medium. The ellipse represents the locus of all possible image points for one source/receiver pair.

This way, we can draw such an ellipse for each source-receiver pair. Each image exhibits elliptical wavefronts. The intensity of an oscillatory integral is given by:

where a(x) is the amplitude function, ω is the frequency, and φ(x) is the phase of the signal. Because we are sending a pulse, we can make a large ω approximation and look at the case where . For a small section dx, the high number of oscillations due to large ω causes the integral to equal zero except where. This point is a point of stationary phase, and occurs where the wavefront is. Therefore, after summing up all the ellipses generated by each source-receiver pair, only the outline of the reflecting surface is visible.

Below are images showing the summing of increasing numbers of ellipses. The reflecting surface is mostly horizontal, with a mountain barely visible on the right.