The one-dimensional wave equation was previously derived for a stretched string as:
is the speed of wave motion on the string, T 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
y(t, x) = y+(t - x/c) + y-(t + x/c),
for arbitrary functions and . A function of (t - x/c) can be interpreted as a fixed waveshape traveling to the right (positive x direction) and a function (t + x/c) can be interpreted as a fixed waveshape traveling to the left (negative x direction), both with speed c.
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 Ts 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 Ts, with a resulting expression for physical displacement at time n and location m given as the sum of the two traveling-wave components
y+[(n - m)Ts] = y+(n - m) can be interpreted as the output of an m-sample delay line of input y+(n). Similarly, the term
y-[(n + m)Ts] = y-(n + m) can be interpreted as the input to an m-sample delay line with output y-(n).
Physical wave variables are given by the superposition of traveling waves. In a one-dimensional system, we can use two systems of unit delay elements to model left- and right-going traveling waves and sum delay-line values at corresponding ``spatial'' locations to obtain physical outputs, as depicted below.
Discrete-time simulation of ideal, lossless wave propagation with observation points at x = 0 and
x = 3 X = 3 cTs.
Any ideal, lossless, one-dimensional waveguide can be simulated in this way. The model is exact at the sampling instants to within the numerical precision of the processing system.
To avoid aliasing, the traveling waveshapes must be initially bandlimited to less than half the sampling frequency.
In many modeling contexts, the calculation of physical output values can be limited to just one or two discrete spatial locations. Individual unit delay elements are more typically combined and represented by digital delay lines, as shown below.
Digital waveguide simulation of ideal, lossless wave propagation using delay lines.
The delay lines can be initialized with displacement data corresponding any bandlimited, arbitrary waveshape.
Real wave propagation is never lossless. Sound waves in air lose energy via molecular frictional forces. Mechanical vibrations in strings are dissipated through yielding terminations, the viscosity of the surrounding air, and via internal frictional forces. In general, these losses vary with frequency.
Losses are often well approximated by the addition of a small number of terms to the wave equation.
In the simplest case, we can add a frequency-independent force term that is proportional to the transverse string velocity. Using the wave equation derived for the string,
where is the resistive proportionality constant. Assuming the friction coefficient is relatively small, the following general class of solutions to this equation can be found:
When this solution is sampled, we get
y(tn, xm) = gmy+(n - m) + g-my-(n + m),
Discrete-time simulation of ideal, lossy wave propagation.
Because the system is linear and time-invariant, the loss terms can be commuted and implemented at discrete points for efficiency.
In the more realistic situation where losses are frequency dependent (and typically of ``lowpass'' characteristic), the g factors are replaced with frequency responses of the form . These responses can likewise be commuted and implemented at discrete spatial locations within the system.
Thus far, we have only considered wave propagation along or within a uniform, one-dimensional medium of seemingly infinite length. In an anechoic or non-reflecting waveguide, waves traveling in only one directon may exist and can thus be simulated with just a single delay line.
In most situations, however, the media in which waves travel are of finite length and reflections occur at the boundaries that give rise to waves traveling in two directions per dimension.
The simplest case is an ideal termination that is completely rigid.
If we consider a string to be fixed at a position L, the boundary condition at that point is y(t, L) = 0 for all time. From the traveling-wave solution to the wave equation, we then have
y+(t - L/c) = -y-(t + L/c), which indicates that displacement traveling waves reflect from a fixed end with an inversion (or a reflection coefficient of -1). The simulation of displacement wave motion in a string rigidly terminated at both its ends (and without losses) is shown in the figure below.
Digital waveguide simulation of wave propagation on a string fixed at both ends.