### Digital Waveguide (DW) Scheme

We now derive the digital waveguide formulation by *sampling* the
*traveling-wave* solution to the wave equation. It is easily
checked that the lossless 1D wave equation
is solved
by any string shape which travels to the left or right with speed
[100]. Denote
*right-going* traveling waves in general by
and
*left-going* traveling waves by
, where and
are assumed twice-differentiable. Then, as is well known, the
general class of solutions to the lossless, one-dimensional,
second-order wave equation can be expressed as

Sampling these traveling-wave solutions yields

where a ``'' superscript denotes a ``right-going'' traveling-wave component, and ``'' denotes propagation to the ``left''. This notation is similar to that used for acoustic-tube modeling of speech [297].

Figure E.1 (initially given as Fig.C.3)
shows a signal flow diagram for the computational model of
Eq.(E.5), termed a *digital waveguide model* (developed
in detail in Appendix C). Recall that, by the sampling theorem, it
is an exact model so long as the initial conditions and any ongoing
additive excitations are bandlimited to less than half the temporal
sampling rate [451, Appendix G]. Recall also that
the position along the string,
meters, is laid
out from left to right in the diagram, giving a physical
interpretation to the horizontal direction in the diagram, even though
spatial samples have been eliminated from explicit consideration. (The
arguments of and have physical units of time.)

The left- and right-going traveling wave components are summed to produce a physical output according to

In Fig.E.1, ``transverse displacement outputs'' have been arbitrarily placed at and .

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Finite Difference Time Domain (FDTD) Scheme