Vibrating Bodies and Acoustic Spectra

Longitudinal Vibration

This is where the vibrational forces and displacements occur in the direction of the wave propigation, eg. in a column of air such as a woodwind instrument or along the string of a guitar or violin (where the tension forces act in the direction of the axis of the string). Perhaps the most important of all longitudinal vibrating systems is the human voice.

	An example of longitudinal vibration is a long spiral string with a compression pulse travelling along it. More pulses need to be applied with the right timing to set up a standing wave because friction quickly damps the first pulse.
	A representation of the harmonic waveforms of a string. The tension waves in a string are easy to see. When a string is plucked reflections of the pulse reflect from both ends and create standing waves which are as long as the whole string or integer fractions of the string length (ie. 1/2, 1/3, 1/4 length etc.).

The string doesn't move at the points where the waveform crosses the straight line. These are called node points. Since the frequency of vibration of such waves is inversely proportional to their wavelength, overtones or partial frequencies occur at integer multiples of the fundamental (lowest) frequency of the string (ie. 2, 3, 4 times etc). When overtones are integer multiples of the fundamental they are known as harmonics.

The acoustic spectrum of a vibrating object is the recorded amplitudes (pressures of radiated air waves) for the various partial frequencies over a certain time. These amplitudes may be effected by the type of excitation of the vibrations, the efficiency with which the vibrational energy of the object is transferred to the surrounding air, and the internal damping of vibration in the object itself.

The timbre of a sound is closely related to its acoustic spectrum. For example different speech vowels often contain the same overtones with different amplitudes.

This graph shows the harmonics of my voice speaking the words "Australian Bell". Frequency (Hz) is the vertical axis and time is the horizontal axis. The amplitude of each harmonic is shown by the darkness of the line. Notice the differences between the vowels AU, A, I, and E in this sentence (for an Australian accent!).

Flexural Vibration

This is where the vibrational displacements are not in the direction of propigation, eg. when a xylophone bar is struck in its center the initial displacement is in the direction of the mallet head but the wave propigation is along the length of the bar.

The physics of flexural vibrations are such that the overtones do not naturally lie in harmonic relationships. For the more technically minded this is because they occur in dispersive media where the velocity of the wave varies with its frequency (so the reflected impulses don't occur twice as quickly, even if the wavelength is halved). Most flexural vibrations are too complex to describe by a single mathematical formula, so complex computer programs involving thousands of mathematic equations are now used to describe them. In a paper soon to be published, Dr Mclachlan of Australian Bell has used such a program to detail the waveform shapes and frequencies of vibration for the types of vibrations that contribute to the sound of bells over a wide range of shapes.

The following images show three of the types of vibration found in bells, the 2,0, 3,0 and 2,1 modes. The first number refers to the the number of vertical nodal lines (lines along the surface of the object that don't move in this mode of vibration), and the second number refers to the number of horizontal nodal lines. The blue color is the part of the bell that vibrates with the least amplitude and the red vibrates with the greatest amplitude. We call modes with only vertical nodal lines (eg. 2,0 mode) circumferential modes, modes with only horizontal nodal lines axial modes, and modes with both horizontal and vertical nodal lines (eg. 2,1 mode), mixed modes.

2,0 mode	3,0 mode	2,1 mode

Sounds where the frequencies of the overtones are not in harmonic relationships are described as inharmonic.

This graph shows partial frequencies in the sound of a gong.

The following sections describe the musical implications of flexural vibrations with respect to human perception and the evolution of the bell.