In any real
vibrating string, there are energy losses due to yielding
terminations, drag by the surrounding air, and internal
friction within the
string. While losses in solids generally vary in a complicated way with
frequency, they can usually be well approximated by a small number of
odd-order terms added to the
wave equation. In the simplest case,
force is
directly proportional to
transverse string
velocity, independent of
frequency. If this proportionality constant is

, we obtain the
modified
wave equation

 |
(C.21) |
Thus, the
wave equation has been extended by a ``first-order'' term,
i.e.,
a term proportional to the first derivative of

with respect to time.
More realistic loss approximations would append terms proportional to

,

, and so on, giving
frequency-dependent losses.
Setting

in the
wave equation to find the relationship
between temporal and spatial frequencies in the eigensolution, the wave
equation becomes
where

is the wave velocity in the lossless case.
At high frequencies (large

), or when the friction coefficient

is small relative to the
mass density

at the lowest frequency of interest,
we have the approximation
 |
(C.22) |
by the
binomial theorem. For this small-loss approximation, we obtain the
following relationship between temporal and
spatial frequency:
 |
(C.23) |
The eigensolution is then
![$\displaystyle e^{st+vx} = \exp{\left[st\pm \left({s + \frac{\mu}{2\epsilon }}\r...
...c{x}{c}\right)\right]} \exp{\left(\pm\frac{\mu}{2\epsilon }\frac{x}{c}\right)}.$](http://www.dsprelated.com/josimages_new/pasp/img3335.png) |
(C.24) |
The right-going part of the eigensolution is
 |
(C.25) |
which
decays exponentially in the direction of
propagation,
and the left-going solution is
 |
(C.26) |
which also decays exponentially in the direction of travel.
Setting

and using superposition to build up arbitrary
traveling
wave shapes, we obtain the general class of solutions
 |
(C.27) |
Sampling these exponentially decaying
traveling waves at intervals of

seconds (or
meters) gives
The
simulation diagram for the lossy
digital waveguide is shown in
Fig.
C.5.
Figure C.5:
Discrete simulation of the ideal, lossy waveguide.
![\includegraphics[scale=0.9]{eps/floss}](http://www.dsprelated.com/josimages_new/pasp/img3340.png) |
Again the discrete-time simulation of the decaying
traveling-wave solution
is an
exact implementation of the continuous-time solution at the
sampling positions and instants, even though losses are admitted in the
wave equation. Note also that the losses which are
distributed in
the continuous solution have been consolidated, or
lumped, at
discrete intervals of

meters in the simulation. The loss factor
summarizes the distributed loss incurred in one
sampling interval. The lumping of distributed losses does not introduce
an approximation error at the sampling points. Furthermore,
bandlimited
interpolation can yield arbitrarily accurate reconstruction between
samples. The only restriction is again that all
initial conditions and
excitations be bandlimited to below half the
sampling rate.
In many applications, it is possible to realize vast computational savings
in
digital waveguide models by
commuting losses out of unobserved and
undriven sections of the medium and consolidating them at a minimum number
of points. Because the digital simulation is
linear and time invariant
(given constant medium parameters

), and because linear,
time-invariant elements commute, the diagram in Fig.
C.6 is
exactly equivalent (to within numerical precision) to the previous diagram
in Fig.
C.5.
Figure C.6:
Discrete simulation of the ideal, lossy waveguide.
Each per-sample loss factor
may be ``pushed through'' delay
elements and combined with other loss factors until an input or output
is encountered which inhibits further migration. If further
consolidation is possible on the other side of a branching node, a
loss factor can be pushed through the node by pushing a copy
into each departing branch. If there are other inputs to the
node, the inverse of the loss factor must appear on each of
them. Similar remarks apply to pushing backwards through a node.
![\includegraphics[scale=0.9]{eps/flloss}](http://www.dsprelated.com/josimages_new/pasp/img3343.png) |
Frequency-Dependent Losses
In nearly all natural wave phenomena, losses increase with frequency.
Distributed losses due to air drag and internal bulk losses in the
string tend to increase monotonically with frequency. Similarly, air
absorption increases with frequency, adding loss for sound waves in
acoustic tubes or open air [
318].
Perhaps the apparently simplest modification to Eq.

(
C.21) yielding
frequency-dependent damping is to add a
third-order
time-derivative term [
392]:
 |
(C.28) |
While this model has been successful in practice
[
77], it turns out to go
unstable at very high
sampling rates. The technical term for
this problem is that the
PDE is
ill posed
[
45].
A well posed replacement for Eq.

(
C.28) is given by
 |
(C.29) |
in which the third-order partial derivative with respect to time,

, has been replaced by a third-order
mixed partial derivative--twice with respect to

and
once with respect to

.
The solution of a lossy
wave equation containing higher odd-order
derivatives with respect to time yields
traveling waves which
propagate with frequency-dependent attenuation. Instead of
scalar
factors

distributed throughout the diagram as in Fig.
C.5,
each

factor becomes a
lowpass filter having some
frequency-response per sample denoted by

. Because
propagation is passive, we will always have

.
More specically, As shown in [
392], odd-order partial
derivatives with respect to time in the
wave equation of the form
correspond to attenuation of
traveling waves on the string. (The
even-order time derivatives can be associated with variations in
dispersion as a function of frequency, which is considered in
§
C.6 below.) For

, the losses are
frequency
dependent, and the per-sample
amplitude-response ``rolls off''
proportional to
In particular, if the
wave equation (
C.21) is modified by adding
terms proportional to

and

, for instance, then the
per-sample propagation gain

has the form
where the

are constants depending on the constants

and

in the
wave equation. Since these per-sample loss filters are
linear and time-invariant [
449], they may also be consolidated
at a minimum number of points in the
waveguide without introducing any
approximation error, just like the constant gains

in Fig.
C.5.
This result does not extend precisely to the
waveguide mesh
(§
C.14).
In view of the above, we see that we can add odd-order time
derivatives to the wave equation to approximate experimentally
observed frequency-dependent damping characteristics in
vibrating
strings [
73]. However, we then have the problem that
such wave equations are ill posed, leading to possible
stability
failures at high
sampling rates. As a result, it is generally
preferable to use mixed derivatives, as in Eq.

(
C.29), and try to
achieve realistic damping using higher order
spatial derivatives
instead.
A large class of well posed
PDEs is given by [
45]
 |
(C.30) |
Thus, to the ideal
string wave equation Eq.

(
C.1) we add any number
of even-order partial derivatives in

, plus any number of mixed
odd-order partial derivatives in

and

, where differentiation
with respect to

occurs only once. Because every member of this
class of
PDEs is only second-order in time, it is guaranteed to be
well posed, as shown in §
D.2.2.
In an efficient digital simulation, lumped loss factors of the form

are approximated by a rational
frequency response

. In general, the coefficients of the optimal rational
loss
filter are obtained by minimizing

with respect to the filter coefficients or the
poles
and zeros of the filter. To avoid introducing frequency-dependent
delay, the loss filter should be a
zero-phase,
finite-impulse-response (FIR) filter [
362].
Restriction to zero phase requires the
impulse response

to
be finite in length (
i.e., an
FIR filter) and it must be symmetric
about time zero,
i.e.,

. In most implementations,
the zero-phase FIR filter can be converted into a
causal,
linear
phase filter by reducing an adjacent
delay line by half of the
impulse-response duration.
We will now derive a finite-difference model in terms of string
displacement samples which correspond to the lossy
digital waveguide
model of Fig.
C.5. This derivation generalizes the lossless case
considered in §
C.4.3.
Figure
C.7 depicts a
digital waveguide section once again in
``physical canonical form,'' as shown earlier in Fig.
C.5, and
introduces a doubly indexed notation for greater clarity in the
derivation below
[
442,
222,
124,
123].
Figure C.7:
Lossy digital waveguide--frequency-independent loss-factors
.
 |
Referring to Fig.
C.7, we have the following time-update
relations:
Adding these equations gives
This is now in the form of the
finite-difference time-domain (FDTD)
scheme analyzed in [
222]:
with

, and

. In
[
124], it was shown by
von Neumann analysis
(§
D.4) that these parameter choices give rise to a stable
finite-difference scheme (§
D.2.3), provided

. In the
present context, we expect
stability to follow naturally from starting
with a passive digital waveguide model.
The preceding derivation generalizes immediately to
frequency-dependent losses. First imagine each

in Fig.
C.7
to be replaced by

, where for passivity we require
In the time domain, we interpret

as the
impulse response
corresponding to

. We may now derive the frequency-dependent
counterpart of Eq.

(
C.31) as follows:
where

denotes
convolution (in the time dimension only).
Define
filtered node variables by
Then the frequency-dependent FDTD scheme is simply
We see that generalizing the FDTD scheme to frequency-dependent losses
requires a simple filtering of each node variable

by the
per-sample
propagation filter

. For computational efficiency,
two spatial lines should be stored in memory at time

:

and

, for all

. These
state variables enable computation of

, after which each sample of

(

) is filtered
by

to produce

for the next iteration, and

is filtered by

to produce

for the next iteration.
The frequency-dependent generalization of the FDTD scheme described in
this section extends readily to the
digital waveguide mesh. See
§
C.14.5 for the outline of the derivation.
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