Perceptual Coherence : Hearing and Seeing

338 Perceptual Coherence
of sound quality between different pitches and loudness that can be used to
identify objects.
String and Percussion Instruments: Vibrations in Strings
and Plates
Vibrations of Strings
Suppose we pluck a string at the midway point. When we release the string
there will be two traveling waves, one moving toward each end of the string.
At the ends, each wave will be reflected back toward the middle of the string.
If the reflections continue, the wave pattern along the string can become stationary.
If we view the motion from the side, the individual points of the
string oscillate up and down, but the wave profile does not move laterally.
The original traveling waves have summed to produce a standing wave. The
overall motion of the string is complex (although each point simply moves up
and down in sinusoidal motion), because the motion is actually the sum of
several simple harmonic motions. The simplest harmonic motion is the upand-down
movement of the string as a whole. The second harmonic motion
occurs when the string effectively splits into two parts, and each half undergoes
an up-and-down motion out of phase with each other. The frequency of
each of these vibrations is twice the frequency of the motion of the string as a
whole. The motion of all of the higher harmonics can be understood in exactly
the same fashion. The string breaks up into more and more subunits; the
vibratory frequency of each subunit is the number of subunits times the frequency
of the fundamental vibration (F 0 , 2F 0 , 3F 0 , 4F 0 , 5F 0 ...). 3
The fundamental frequency of the string is determined by its physical
construction and the amount of tension (equation 8.1). It is not determined
by how the string is excited. However, the position and the manner of excitation
do determine the relative amplitudes of the vibration modes. The
maximum amplitude of a resonance mode occurs when the point of excitation
occurs at an antinode (a point of maximum displacement), and the
minimum amplitude occurs when the point of excitation occurs at a node
(a stationary point). Thus, if the string is excited at the center point, the
first up-and-down resonance will be maximized because that is the point
of maximum movement, but the second resonance will be minimized because
that is the point of zero movement separating the vibration of the
two halves of the string. In general, the overall amplitude of the vibration
3. By convention, F 0 is termed the fundamental, 2F 0 the second harmonic (F 0 being the
first harmonic), 3F 0 the third harmonic, and so on. If the vibration modes are not related by integers
(e.g., the vibration modes found for 2-dimensional wooden violin plates), the vibration
modes are termed partials.