110
D. Rocchesso: Sound Processing
waveguide models
traveling waves
Replacing the second derivatives by central second-order differences
plicit updating scheme for the i-th spatial sample of displacement (or pressure)
is:
p(i, n + 1) = 2 1 -
c
2
t
2
x
2
p(i, n) - p(i, n - 1)
+
c
2
t
2
x
2
[p(i + 1, n) + p(i - 1, n)] ,
(41)
where t and x are the time and space grid steps. The von Neumann analysis
assumes that the equation parameters are locally constant and checks the time
evolution of a spatial Fourier transform of (41). In this way a spectral amplifi-
cation factor is found whose deviations from unit magnitude and linear phase
give respectively the numerical loss (or amplification) and dispersion errors. For
the scheme (41) it can be shown that a unit-magnitude amplification factor is
ensured as long as the Courant-Friedrichs-Lewy condition [70]
ct
x
1
(42)
is satisfied, and that no numerical dispersion is found if equality applies in (42).
A first consequence of (42) is that only strings having length which is an integer
number of ct are exactly simulated. Moreover, when the string deviates from
ideality and higher spatial derivatives appear (physical dispersion), the simula-
tion becomes always approximate. In these cases, the resort to implicit schemes
can allow the tuning of the discrete algorithm to the amount of physical dis-
persion, in such a way that as many partials as possible are reproduced in the
band of interest [22].
It is worth noting that if c in equation (40) is a function of time and space,
the finite difference method retains its validity because it is based on a local (in
time and space) discretization of the wave equation. Another advantage of finite
differencing over other modeling techniques is that the medium is accessible
at all the points of the time-space grid, thus maximizing the possibilities of
interaction with other objects.
As opposed to finite differencing, which discretize the wave equation (see
eqs. (40) and (41)), waveguide models come from discretization of the solution
of the wave equation. The solution to the one-dimensional wave equation (40)
was found by D'Alembert in 1747 in terms of traveling waves
p(x, t) = p
+
(t - x/c) + p
-
(t + x/c) .
(43)
Eq. (43) shows that the physical quantity p (e.g. string displacement or acous-
tic pressure) can be expressed as the sum of two wave quantities traveling
in opposite directions. In waveguide models waves are sampled in space and
time in such a way that equality holds in (42). If propagation along a one-
dimensional medium, such as a cylinder, is ideal, i.e. linear, non-dissipative and
non-dispersive, wave propagation is represented in the discrete-time domain by
a couple of digital delay lines (Fig. 14), which propagates the wave variables p
+
and p
-
.
6
The reader is invited to derive (41) by substituting in (40) the first-order spatial deriva-