Introduction to Acoustics

722 Part E Music, Speech, Electroacoustics
Part E 17.4
17.4 Modal (Damped Sinusoidal) Synthesis
The simplest physical system that does something
acoustically (and musically) interesting is the
mass/spring/damper (Fig. 17.10 [17.12]. The differential
equation describing that system has a solution
that is a single exponentially decaying cosine wave.
The Helmholtz resonator (large, contained air cavity
with a small long-necked opening, like a pop bottle,
Fig. 17.11 behaves like a mass/spring/damper system,
with the same exponentially damped cosine behavior.
The equations describing the behavior of these systems
is:
d 2 y/dt 2 + (r/m)dy/dt + (k/m)y = 0 , (17.10)
y(t) = y0 e (−rt/2m) � �
�k/m � �
cos t − (r/2m) 2 .
y +
0
–
m
Mass
Spring
k
(r)
Damping
Fig. 17.10 Mass/spring/damper system
PA
PV
Fig. 17.11 Helmholtz resonator
(17.11)
Of course, most systems that produce sound are
more complex than the ideal mass/spring/damper system,
or a pop bottle. And of course most sounds are
more complex than a simple damped exponential sinusoid.
Mathematical expressions of the physical forces
(thus the accelerations) can be written for nearly any
system, but solving such equations is often difficult or
impossible. Some systems have simple enough properties
and geometries to allow an exact solution to be
written out for their vibrational behavior. An ideal string
under tension is one such system.
Here we will resort to some graphical arguments and
our prior discussion of the Fourier transform to motivate
further the notion of sinusoids in real physical systems.
The top of Fig. 17.12 shows a string, lifted from a point
in the center (halfway along its length). Below that is
shown a set of sinusoidal mode that the center-plucked
String plucked
in center
Odd modes
Even (forbidden) modes
2
6
1
5 3
4
Fig. 17.12 Plucked string (top). The center shows sinusoidal
modes of vibration of a center-plucked string. The bottom
shows the even modes, which would not be excited by the
center-plucked condition
(1,1) (1,2) (1,3)
(2,2) (2,3)
Fig. 17.13 Square-membrane modes
(3,3)