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 cases are ideal terminations that are either completely rigid or completely free.
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
yr(t - L/c) = -yl(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, without losses, and initialized with a plucked string shape is diagrammed in the figure below.
Digital waveguide simulation of ideal lossless wave propagation on a string fixed at both ends (with pluck initialization).