Waves of sound travel with a velocity of approximately c = 345 meters / second in air.
The propagation of sound over a distance d thus involves a time delay of d/c seconds.
A digital delay line of length M.
A digital delay line (Fig. 9) defined by the difference equation
(where the input signal
for n < 0 and y[n] is the output signal) can be interpreted as a physical model of traveling-wave propagation over a distance d, where
M = fsd / c is the number of discrete-time samples corresponding to a distance d at a sample rate fs.
Assuming the input to the delay line is a traveling-wave variable at time n T, where T is the sampling period, the output of the delay line represents a time-delayed version of that wave value.
The delay-line implements lossless, plane-wave propagation. If we wish to simulate spherical-wave propagation of pressure, the output would be scaled by a factor 1/d.