Sound Modelling
109
partial differential equations
cellular models
finite difference methods
waveguide models
ordinary differential
equations
one-dimensional distributed
resonators
Kirchhoff variables
wave equation
I - interaction: a system that connects E and R in such a way that the physical
variables at the two ends are compatible.
Although in our example the resonator is a lumped mechanical oscillator, usually
the resonator is a medium where waves propagate. Therefore the resonator is
a distributed system, described by partial differential equations (PDE). Among
the different ways of discretizing it, we mention
· Network of elementary coupled oscillators (cellular models);
· Numerical integration of the PDE (for instance, finite difference methods);
· Discretization of the solutions of the PDE (waveguide models).
The exciter is usually a lumped system described by ordinary differential
equations (ODE) that can be integrated using numerical methods, the bilin-
ear transformation, or the impulse invariance method. Often the exciter ex-
hibits strong nonlinearities, such as the pressure-flow characteristic of a clarinet
reed [31].
The interaction block is the place where the different discretizations of the
exciter and resonator blocks talk to each other. Moreover, this is the right place
to insert sound component that are difficult to capture with a physical model,
either because the physics is too complicated or because we just don't know
to model some phenomena. For instance, where the clarinet reed (exciter) is
connected to the bore (resonator), small flow-dependent noise bursts can be
injected to increase the simulation realism.
In a system such as the one of figure 13, if each block is separately discretized
a computability problem may arise when the blocks are connected to each other.
Namely, if the realization of each block has a delay-free input-output path then
a non-computable delay-free loop will appear in the model. There are techniques
to cope with these delay-free loops (implicit solvers) or to eliminate them [16].
5.4.3
One-dimensional distributed resonators
Physical systems such as strings or acoustic tubes can be idealized as one-
dimensional distributed resonators, described by a couple of dual variables, here
called Kirchhoff variables, which are functions of time and longitudinal space.
For a string, the Kirchhoff variables are force and velocity. For the acoustic tube,
these variables are pressure and air flow. In any case, each of these variables is
governed by the wave equation [63]
2
p(x, t)
t
2
= c
2
2
p(x, t)
x
2
,
(40)
where c is the wave speed in the medium. The symbol p in (40) can be thought
of as the instantaneous and local air pressure inside a tube.
One of the most popular ways of solving PDEs such as (40) is finite dif-
ferencing, where a grid is constructed in the spatial and time variables, and
derivatives are replaced by linear combinations of the values on this grid. Two
are the main problems to be faced when designing a finite-difference scheme for
a partial differential equation: numerical losses and numerical dispersion. There
is a standard technique [70], [103] for evaluating the performance of a finite-
difference scheme in contrasting these problems: the von Neumann analysis.