Sound is a longitudinal wave. The motion of air (about an equilibrium position) in the direction of the wave produces a pressure and rarefaction wave called sound. If the motion of the air about this equilibrium position is described by

then the pressure change is

with

the bulk modulus of air related to the speed of the wave by

The intensity of sound waves is described in terms of decibels with

Many musical instruments rely on vibrating columns of air to produce musical sounds. Organ pipes are a simple example. The possible resonant wavelengths (and frequency) correspond to maximum displacement (motion in the direction of the wave) of the air at the open end of the pipe and no displacement at the closed end of the pipe. Considering the various configurations of pipes the various allowed wavelengths can be determined using the following depictions.

First consider a closed at one end, open at the other end pipe.  The first wavelength that can be supported in this pipe is one where there is no displacement at the closed end and there is displacement at the open end. This is depicted below.

The pipe length corresponds to a quarter of a wavelength.

The next possible wave is depicted below and in this case the length of the pipe corresponds to three-quarters of a wavelength

This process can be carried on producing the fundamental, the first harmonic, the second harmonic and so on.

Now consider an open at both ends pipe.  In this situation there is desplacement at both ends of the pipe and this situation is depicted below.

The pipe length corresponds to a half wavelength.

The next possible wave is depicted below and in this case the length of the pipe corresponds to a full wavelength,

Again, the fundamental wave is followed by the first harmonic, second harmonic and so on.

The Doppler effect describes the apparent shift in frequency for a source or observer moving according to

Copyright © Robert M. Oman 2002