Up to now, we have been concerned with the simulation of traveling waves in linear, time-invariant (LTI) media. The main example considered was wave propagation in air, but waves on vibrating strings behave analogously. We saw that the point-to-point propagation of a traveling plane wave in an LTI medium can be simulated simply using only a delay line and an LTI filter. The delay line simulates propagation delay, while the filter further simulates (1) an independent attenuation factor at each frequency by means of its amplitude response (e.g., to simulate air absorption), and (2) a frequency-dependent propagation speed using its phase response (to simulate dispersion). If there is additionally spherical spreading loss, the amplitude may be further attenuated by , where is the distance from the source. For more details about the acoustics of plane waves and spherical waves, see, e.g., [321,352]. Appendix B contains a bit more about elementary acoustics,
So far we have considered only traveling waves going in one direction. The next simplest case is 1D acoustic systems, such as vibrating strings and acoustic tubes, in which traveling waves may propagate in two directions. Such systems are simulated using a pair of delay lines called a digital waveguide.