The lossless one-dimensional wave equation was previously derived for a stretched string as:
where
is the speed of wave motion on the string, is the string tension, and is the mass density of the string.
The traveling-wave solution of the wave equation was published by d'Alembert in 1747. It has the general form
for arbitrary functions and . A function of can be interpreted as a fixed waveshape traveling to the right (positive direction) and a function can be interpreted as a fixed waveshape traveling to the left (negative direction), both with speed .
To develop a discrete-time model or simulation of traveling wave motion, it is necessary to sample the traveling-wave amplitudes in both time and space.
The temporal sampling interval is seconds, which corresponds to a sample rate
samples per second.
The spatial sampling interval is given most naturally by
meters, or the distance traveled by sound in one temporal sampling interval. In this way, each traveling-wave component moves left or right one spatial sample for each time sample.
Using the change of variables
the traveling-wave solution becomes
This representation can be further simplified by suppressing , with a resulting expression for physical displacement at time and location given as the sum of the two traveling-wave components