Without damping, the
ideal plucked string sounds more like a cheap
electronic organ than a string because the sound is perfectly
periodic
and never decays. Static
spectra are very boring to the ear. The
discrete
Fourier transform (
DFT) of the initial ``string loop''
contents gives the
Fourier series coefficients for the periodic tone
produced.

The simplest change to the ideal
wave equation of Eq.

(
6.1) that
provides damping is to add a term proportional to
velocity:
 |
(7.14) |
Here,

can be thought of as a very simple
friction coefficient,
or resistance. As derived in §
C.5, solutions to this
wave
equation can be expressed as sums of left- and right-going
exponentially decaying traveling waves. When

, we get
non-decaying
traveling waves as before. As discussed in
§
2.2.2,
propagation losses may be introduced by the substitution
in each delay element (or wherever one sample of delay models one
spatial sample of wave propagation). By commutativity of
LTI systems,
making the above substitution in a
delay line of length

is
equivalent to simply scaling the output of the
delay line by

.
This
lumping of propagation loss at one point along the
waveguide serves to minimize both computational cost and round-off
error. In general
finite difference schemes, such a simplification is
usually either not possible or nonobvious.
To illustrate how significant the computational savings can be,
consider the simulation of a ``damped guitar string'' model in
Fig.
6.11. For simplicity, the length

string is
rigidly terminated on both ends. Let the string be ``plucked'' by
initial conditions so that we need not couple an input mechanism to
the string. Also, let the output be simply the
signal passing through
a particular delay element rather than the more realistic summation of
opposite elements in the bidirectional
delay line. (A
comb filter
corresponding to pluck position can be added in series later.)
Figure 6.11:
Discrete simulation of the rigidly terminated
string with distributed resistive losses. The
loss factors
are embedded between the delay-line elements.
![\includegraphics[width=\twidth]{eps/fstring}](http://www.dsprelated.com/josimages_new/pasp/img1418.png) |
In this string simulator, there is a loop of delay containing

samples where

is the desired
pitch of the string. Because
there is no input/output coupling, we may lump
all of the losses at
a single point in the delay loop. Furthermore, the two reflecting
terminations (gain factors of

) may be commuted so as to cancel them.
Finally, the right-going delay may be combined with the left-going delay to
give a single, length

, delay line. The result of these inaudible
simplifications is shown in Fig.
6.12.
Figure 6.12:
Discrete simulation of the rigidly terminated
string with consolidated losses (frequency-independent). All
loss factors
have been ``pushed'' through delay elements and
combined at a single point.
![\includegraphics[width=\twidth]{eps/fsstring}](http://www.dsprelated.com/josimages_new/pasp/img1421.png) |
If the
sampling rate is

kHz and the desired pitch is

Hz, the loop delay equals

samples. Since delay lines are
efficiently implemented as circular buffers, the cost of implementation is
normally dominated by the loss factors, each one requiring a multiply
every sample, in general. (Losses of the form

,

, etc., can be efficiently implemented using shifts and
adds.) Thus, the consolidation of loss factors has reduced computational
complexity by
three orders of magnitude, i.e., by a factor of

in this case. However, the physical accuracy of the simulation has
not been compromised. In fact, the
accuracy is improved because
the

round-off errors per
period arising from repeated multiplication
by

have been replaced by a single round-off error per period
in the multiplication by

.
Next Section: Frequency-Dependent DampingPrevious Section: The Ideal Struck String