Perceptual Coherence : Hearing and Seeing

340 Perceptual Coherence
frequencies found for strings. Nevertheless, each of the vibration modes
of a plate has great similarity to those found for vibrating strings. The
two-dimensionality of plates merely allows new spatial patterns of vibrations
to occur.
For a uniform string, the motion for the lowest resonant frequency is
an up-down motion of the string as a whole. Similarly, the motion for the
first mode of a uniform plate is an up-down motion of the plate as a whole
(figure 8.1A). For a string, the motion for the second vibration mode
consists of the out-of-phase motion of the two halves, and the maximum
amplitude occurs at the midpoint of each half; there is a null point at the
center of the string. This, again, is the same motion found for the second
mode of a plate: The plate vibrates in two out-of-phase sections at the same
frequency; there is a single null line (figure 8.1B). Because the plate has
two dimensions, there is one vibratory mode that divides the plate into two
vibratory regions along the length and one that divides the plate along the
width. It is quite possible to have the length and width vibratory mode
simultaneously. The third string mode consists of three out-of-phase vibrations;
the third plate modes consist of three out-of-phase motions in regions
of the plate. The nodal lines (zero amplitude) can run along either the width
or length (figure 8.1C). Standing waves for each vibratory mode along one
dimension of the plate are independent of the standing waves along the
other dimension. Given the two dimensions of the plate, there also are vibration
modes that are twisting motions along the diagonals (figure 8.1D).
If the membrane is not uniform (e.g., wooden plates of a violin then some
of the vibration modes can extend for long distance. An example of such a
third vibration mode is shown in figure 8.1E.
As found for vibrating strings, the position and manner of excitation determine
the relative amplitudes of the vibration modes. Any vibratory mode
can be stilled by either (1) exciting the plate, say by hitting it with a narrow
hammer, at a null region; or (2) mechanically restricting the displacement
at a region of maximum amplitude (antinode).
Source (String Vibration)-Filter (Sound Body) Model
Strings are not good sound radiators; the periodic physical motion creates a
compression wave in front of the string and a rarefaction wave in back, and
they effectively cancel each other. The string must transfer its vibration
energy to the sound body and the shape of the violin body has evolved to
increase that transfer to maximize the sound output power. As described
above, each of the string vibrations can excite one or more of the sound
body vibration modes. Depending on the match between the vibration
frequencies of the string and sound body, some of the string vibrations are
effectively amplified and others are effectively nulled, and the spectral