A wave is a disturbance that propagates in a periodically repeating fashion, often transferring energy. A mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces) through which they travel and can transfer energy from one place to another without any of the particles of the medium being displaced permanently; there is no associated mass transport. Instead, any particular point oscillates around a fixed position. However, Electromagnetic radiation, and probably Gravitational radiation are not mechanical waves, and can travel through a Vacuum, without a medium.
Waves are characterised by crests (highs) and troughs (lows), either perpendicular (in the case of transverse waves) or parallel (in the case of longitudinal waves) to wave motion.
Transverse waves are those with vibrations perpendicular to the wave's direction of travel; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations along the wave's direction of travel; examples include most sound waves.
Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow elliptical paths.
Waves can be described using a number of standard variables including: Frequency, Wavelength, Amplitude and period. The Amplitude of a wave is the measure of the magnitude of the maximum disturbance in the medium during one wave cycle, and is measured in units depending on the type of wave. For examples, 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 period (T) is the time for one complete cycle for an oscillation of a wave. The frequency (F) is how many periods per unit time (for example one second) and is measured in Hertz. These are related by:
| f = | 1 –– T |
When waves are expressed mathematically, the Angular frequency (ω, radians/second) is often used; it is related to the frequency f by:
| f = | ω ––––– 2 π |
where A(z, t) is the amplitude envelope of the wave, k is the wave number and φ is the phase. The velocity v of this wave is given by:
| v = | ω –– k | = λ f, |
where λ is the Wavelength of the wave.
| v = | √ | –––––––– T/( ρ) | . |
This equation can be found using Dimensional analysis
Not all waves are sinusoidal. One example of a non-sinusoidal wave is a pulse that travels down a rope resting on the ground, extending in direction x, travelling at velocity c. The height of the pulse above the ground is φ. The distance the pulse travels between some time t and time 0 is ct.
In one dimension the wave equation has the form
| 1 ––– c 2 | ∂ 2 φ –––––––– ∂ t 2 | = | ∂ 2 φ –––––––– ∂ x 2 | . |
A general solution, given by d'Alembert is
F and E can be considered to be the shapes of two pulses travelling down the rope, one in the +x direction, and one in the -x direction. If we substitute for x above, instead directions x, y, z, we then can describe a wave propagating in three dimensions.
A non-linear wave-equation can cause mass transport.
The Schrödinger equation describes the wave-like behaviour of particles in Quantum mechanics. Solutions of this equation are wave functions which 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 scale and below.