Physical acoustics - modes of vibration

This page presents some animations and movies of various modes of vibration that play a prominent role in music. To view the animations you will need to have a Quicktime 3.0 plug-in for your browser.


Basics

We characterise the vibrating system by its dimensions (x1, x2, x3) and the displacement from equilibrium position (y1, y2, y3). For a one dimensional vibrator such as a string or rod, we drop the subscripts and refer to displacement y(x) and look for its time development. If this time development has a periodic repetitiveness then it can generate a sound that propagates through the surrounding medium (usually air). Whether we can hear this sound will also depend on the frequency response of our ears and the sound level.

 

The vibrating string

A real string as is found on a violin, guitar or piano is a complicated system to analyse (and hence synthesise) exactly, but its essence may be understood simply in an idealised model. For this we assume that the end connections are "perfect" and that the thickness of the string and amplitude dependence may be ignored. Under these circumstances the resonant modes of vibration are simple sinusoidal functions with the restriction that the ends be stationary.

A general solution for the displacement at point x at time is found to be

y(x) = A sin(kx)

where A is the amplitude and k is the wave number. The amplitude also varies sinusoidally with time at frequency f. The wavenumber and frequency are related by the wavelength l according to

The requirement of fixed ends leads directly to the allowed frequencies of vibration denoted by an index "n".

where c is the speed of sound in the string (which depends on the tension and density) and L is the length of the string.

The following animations show the time development for the lowest frequency string modes.

n = 1 n = 2 n = 3 n = 4

 
 

The vibrating beam

The elastic properties of a solid bar lead to similar vibrations as for the string, but with a slightly different aspect. Mathematically it is due to the hyperbolic functions sinh and cosh which are also permitted. The general solution may be written at a particular time as

y(x) = A cos(bx) + B sin(bx) - A cosh(bx) - B sinh(bx)

where A and B represent the amplitude, and b is determined by the length and end conditions. Variation of the amplitude in time is sinusoidal with frequencies given by

where k is the "radius of gyration" which depends on the relative thickness of the bar. The mode specific parameter b may be found from end considerations. The first few values for a cantilever are

bL = 1.875,   4.694,   7.855,   10.996

 
 

Cantilever: fixed - free ends

n = 1 n = 2 n = 3 n = 4

 
 

The vibrating circular membrane

A drum plays an important role in many musical styles but this is usually for its rythmic rather than harmonious qualities. Nevertheless the sound does originate in the resonant vibrations of the drum skin. The two dimensional nature requires non-sinusoidal spatial functions called (after the mathematician) Bessell functions. As for the previous examples these oscillate sinusoidally in time at a number of frequencies determined by the size, material, tension and edge connection of the membrane. An animation of one the the lower frequency modes is given here.