The medium that carries a wave is called the transmission medium. It can be classified into one or more of the following categories:
- A linear medium, if the amplitudes of different waves at any particular point in the medium can be added.
- A bounded medium, if the medium is finite in extent; otherwise, the medium is called an unbounded medium.
- A uniform medium, if the physical properties of the medium are the same in different parts of the medium.
- An isotropic medium, if the physical properties of the medium are the same in different directions.
Mathematics of specific cases
Propagation through strings
The speed (v) of a wave traveling along a string is directly proportional to the square root of the tension (T) over the linear density (ρ):
Traveling waves have a disturbance (amplitude ) that varies with both time () and distance (). This can be expressed mathematically as:
where is the amplitude envelope of the wave, is the wave number, and is the phase of the wave.
The wave equation
The wave equation is a differential equation that describes how a harmonic wave changes over time. The equation has slightly different forms, depending on how the wave is transmitted and the medium it is traveling through. For a one-dimensional wave traveling down a rope along the -axis with velocity () and amplitude () (which generally depends on both x and t), the wave equation is:
In three dimensions, the equation becomes:.
It should be noted that the velocity () depends on both the type of wave and the medium through which it is being transmitted.
A general solution for the wave equation in one dimension was given by French physicist-mathematician Jean Le Rond d'Alembert (1717-1783). It is
This can be viewed as two pulses travelling down a taut rope in opposite directions; F in the +x direction, and G in the -x direction. If we substitute for x above, replacing it with directions x, y, z, we then can describe a wave propagating in three dimensions.
In quantum mechanics, the Schrödinger equation describes the wavelike behavior of subatomic particles. Solutions of this equation are wave functions that can be used to describe the probability density of a particle. Quantum mechanics also describes particle properties that other waves (such as light and sound) have on the atomic and subatomic scales.
- Electromagnetic spectrum
- Doppler effect
- French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 074874479.
All links retrieved August 10, 2013.
- Vibrations and Waves - An online textbook
- Sounds Amazing - AS and A-Level learning resource for sound and waves