Boundary Conditions

• Thus far, we have only considered wave propagation along or within a 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 ideal displacement wave motion in a string rigidly terminated at both its ends (and without losses) is shown in the figure below.

  Figure 11: Digital waveguide simulation of wave propagation on a string fixed at both ends.