Physical Modelling for Sound Synthesis - Game Sound Technology and Player Interaction: Concepts and Developments - page 344

turies of tradition, learned which design subtleties
are important in particular instruments and what
is necessary to make an instrument sound “good”.
In more recent times, the exact way in which in-
struments produce and radiate sound has been an
area of research for physicists and acousticians.
Mathematical equations have been devised to
describe the physical behaviour that causes sound
producing vibrations for many instruments and
this has increased our understanding of them.
The interested reader is directed to The Physics of
Musical Instruments (Fletcher & Rossing, 1991)
which gives the derivation of such equations for
many popular instruments. Methods have been
developed to solve these equations numerically
enabling computers or signalling processing hard-
ware to simulate the instruments thereby revealing
more about why they sound as they do. This has
not only increased our understanding of instru-
ments and sound-producing objects but it has also
given musicians new tools with which to create
sounds and music. For a more thorough explana-
tion of the methods described below, and others,
one may wish to read Discrete Time Modelling of
Musical Instruments (Välimäki, Pakarinen, Erkut,
& Karjalainen, 2006).
Physical modelling techniques were first de-
veloped in the 1960s based on the causal systems
that produce sound in reality (Kelly & Lochbaum,
1962; Ruiz, 1969). Back then, the limited process-
ing power and availability of computers was pro-
hibitive but, as computers became more powerful
and accessible, the algorithms being developed
could afford to be more computationally demand-
ing and this allowed researchers to develop many
different physical modelling techniques.
The finite element method (FEM) has been
employed in many areas of scientific research from
car crash simulations to complex weather systems.
It involves dividing a distributed physical system,
which is complex, into discrete elements, which are
simple. The advantage of FEM over other physi-
cal modelling techniques is the ease with which
it can be applied to complex geometries that do

not behave uniformly throughout. As well as this,
more accuracy can be gained in important parts of
a structure, for example the part of a car which will
be impacted upon during a crash simulation, by
using a finer grid in that area. However difficulties
arise when one attempts to extract an audio signal
from the surface vibrations of an FEM simulation
due to the irregularity of the grid and in general
it is a computationally expensive technique. For
this reason FEM is quite uncommon in acoustic
applications, and while sometimes used in a pre-
processing stage it is less suitable for direct sound
synthesis in real-time.
Methods such as finite difference models and
mass-spring networks also discretise an object in
space and time. The current position of a point
on an object is calculated from its own posi-
tion at previous temporal intervals and that of
neighbouring points. The mass-spring networks
method treats an object as a mesh-like network
of point masses connected by springs and damp-
ers. The positions of points in the network are
calculated using Newton's second law of motion
and Hooke's law. The finite difference method
is based on partial differential equations (PDEs),
also derived using Newton's second law of mo-
tion and Hooke's law, that describe the vibration
of an object. To enable these equations to be
solved numerically, they are discretised in space
and time using finite approximations. With the
finite difference method, the same equations
are used to describe all parts of the object being
modelled and so the method is suited to objects
that behave uniformly throughout and that can be
fitted to a regular grid. Both these methods can
be used in real-time and offer some advantages,
for example, it is easy to extract an audio signal
from the surface vibrations of an object modelled
using these techniques. The techniques allow for
real-time interaction, at one or many points on
the object, which could be external excitation or
coupling with another object. They can also model
distributed nonlinear effects, that is, effects due to
larger or smaller vibrations beyond the amplitude