DW Non-Displacement Inputs

Since a displacement input at position $m$ corresponds to symmetrically exciting the right- and left-going traveling-wave components $y^{+}_m$ and $y^{-}_m$ , it is of interest to understand what it means to excite these components antisymmetrically. As discussed in §E.3.3, an antisymmetric excitation of traveling-wave components can be interpreted as a velocity excitation. It was noted that localized velocity excitations in the FDTD generally correspond to non-localized velocity excitations in the DW, and that velocity in the DW is proportional to the spatial derivative of the difference between the left-going and right-going traveling displacement-wave components (see Eq.(E.13)). More generally, the antisymmetric component of displacement-wave excitation can be expressed in terms of any wave variable which is linearly independent relative to displacement, such as acceleration, slope, force, momentum, and so on. Since the state space of a vibrating string (and other mechanical systems) is traditionally taken to be position and velocity, it is perhaps most natural to relate the antisymmetric excitation component to velocity.

In practice, the simplest way to handle a velocity input $v_m(n)$ in a DW simulation is to first pass it through a first-order integrator of the form

$$H(z) = \frac{1}{1-z^{-1}} = 1 + z^{-1}+ z^{-2} + \cdots \protect$$ (E.37)

to convert it to a displacement input. By the equivalence of the DW and FDTD models, this works equally well for the FDTD model. However, in view of §E.3.3, this approach does not take full advantage of the ability of the FDTD scheme to provide localized velocity inputs for applications such as simulating a piano hammer strike. The FDTD provides such velocity inputs for ``free'' while the DW requires the external integrator Eq.(E.37).

Note, by the way, that these ``integrals'' (both that done internally by the FDTD and that done by Eq.(E.37)) are merely sums over discrete time--not true integrals. As a result, they are exact only at dc (and also trivially at $f_s/2$ , where the output amplitude is zero). Discrete sums can also be considered exact integrators for impulse-train inputs--a point of view sometimes useful when interpreting simulation results. For normal bandlimited signals, discrete sums most accurately approximate integrals in a neighborhood of dc. The KW-converter filter $H(z)=1-z^{-2}$ has analogous properties.