The complete, linear, time-invariant generalization of the lossy, stiff
string is described by the differential equation
 |
(C.33) |
which, on setting

, (or taking the 2D
Laplace transform
with zero
initial conditions), yields the algebraic equation,
 |
(C.34) |
Solving for

in terms of

is, of course, nontrivial in general.
However, in specific cases, we can determine the appropriate
attenuation per sample

and wave
propagation speed

by numerical means. For example, starting at

, we
normally also have

(corresponding to the absence of static
deformation in the medium). Stepping

forward by a small
differential

, the left-hand side can be approximated by

. Requiring the generalized wave
velocity

to be continuous, a physically reasonable assumption, the
right-hand side can be approximated by

, and
the solution is easy. As

steps forward, higher order terms become
important one by one on both sides of the equation. Each new term in

spawns a new solution for

in terms of

, since the order of
the polynomial in

is incremented. It appears possible that
homotopy continuation methods [
316] can be used to
keep track of the branching solutions of

as a function of

.
For each solution

, let

denote the real part of

and let

denote the imaginary part. Then the
eigensolution family can be seen in the form

. Defining

, and
sampling according to

and

, with

as before, (the spatial
sampling
period is taken to be frequency invariant, while the temporal
sampling
interval is modulated versus frequency using
allpass filters), the
left- and right-going sampled eigensolutions become
where

. Thus, a general map of

versus

, corresponding to a
partial differential equation of any
order in the form (
C.33), can be translated, in principle, into an
accurate, local, linear, time-invariant, discrete-time simulation.
The
boundary conditions and initial state determine the initial
mixture of the various solution branches as usual.
We see that a large class of wave equations with constant
coefficients, of any order, admits a decaying, dispersive,
traveling-wave type solution. Even-order time derivatives give rise
to frequency-dependent dispersion and odd-order time derivatives
correspond to frequency-dependent losses. The corresponding digital
simulation of an arbitrarily long (undriven and unobserved) section of
medium can be simplified via commutativity to at most two pure delays
and at most two linear, time-invariant filters.
Every linear, time-invariant filter can be expressed as a zero-phase
filter in series with an allpass filter. The zero-phase part can be
interpreted as implementing a frequency-dependent gain (damping in a
digital waveguide), and the allpass part can be seen as
frequency-dependent delay (dispersion in a digital waveguide).
Next Section: Spatial DerivativesPrevious Section: Lossy
Finite Difference Recursion