Perceptual Coherence : Hearing and Seeing

336 Perceptual Coherence
plucking, or bowing a string) can change the vibration pattern of the
source. Second, there may be large timing differences between the amplitudes
of the vibrations at different frequencies throughout the sound.
We still multiply the source by the filter, but due to the temporal evolution
of the amplitudes of the component vibration, that multiplication must be
done separately at every time point across the duration of the sound (i.e.,
frequency and time are nonseparable). It is this change in the vibration pattern
that can be critical for object identification.
Vibratory Modes
If we excite a material by striking it, vibrating it, blowing across it, bending
or twisting it, and then stop the excitation, the material may begin to vibrate
at one or more of its natural resonant frequencies. There are two physical
properties that yield continuing vibrations:
1. The material must possess a stiffness or springlike property that provides
a restoring force when the material is displaced from equilibrium. At
least for small displacements, nearly all materials obey Hooke’s law that
the restoring force is proportional to displacement x, F =−Kx. Hooke’s law
is correct only at small displacements. At longer displacements, springlike
materials tend to become harder so that the restoring force increases. This
nonlinearity in the restoring force at those longer displacements can change
the natural frequencies of the vibration modes.
2. The material must possess sufficient inertia so that the return motion
overshoots the equilibrium point. The overshoot displacement in turn creates
a restoring force in the direction of the original displacement that recharges
the motion. The overshoots create a continuing vibration that eventually dies
out due to friction (although the vibration frequency remains the same
throughout the decay, an important property).
For materials that satisfy properties (1) and (2), the amplitude of the
movement across time for every single vibratory pattern will resemble the
motion of a sinusoidal wave and is termed simple harmonic motion. In
nearly all vibratory systems, any source excitation will create several vibratory
modes, each at a different frequency. Each such vibratory mode will
have a distinct motion and can be characterized by its resonant frequency
and by its damping or quality factor. The resonant frequency of each mode
is simply the frequency at which the amplitude of vibration is highest. The
material itself will determine the possible frequencies of vibration. For
strings under tension, the resonant frequency of the first vibration mode
(termed the fundamental frequency) is equal to:
F = 1/(2π)[K(tension)/M(mass of the string)] 1/2 . (8.1)