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.
Frequency-independent losses can be incorporated into the digital waveguide scheme with solutions of the form
y(tn, xm) = gmy+(n - m) + gmy-(n + m),
as diagrammed below, where the g terms can be related to physical damping properties of the propagation medium.
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 ``lowpass'' characteristics), the g factors are replaced with frequency responses of the form . These responses can likewise be commuted and implemented with digital filters at discrete spatial locations within the system.