Mathematical Description of Sound

Mathematical description

Waves can be described mathematically using a series of parameters. The amplitude of a wave (commonly notated as A, or another letter) is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium). In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.

The units of the amplitude depend on the type of wave — waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter). The amplitude may be constant (in which case the wave is a c.w. or continuous wave), or may vary with time and/or position. The form of the variation of amplitude is called the envelope of the wave.


The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum.


Waves can be represented by simple harmonic motion.
The period T is the time for one complete cycle for an oscillation of a wave. The frequency f (also frequently denoted as ν) is how many periods per unit time (for example one second) and is measured in hertz. In other words, the frequency and period of a wave are reciprocals of each other.


The angular frequency ω represents the frequency in terms of radians per second.

There are two velocities that are associated with waves. The first is the phase velocity, which gives the rate at which the wave propagates.


The second is the group velocity, which gives the velocity at which variations in the shape of the wave's amplitude propagate through space. This is the rate at which information can be transmitted by the wave. It is given by