# Wave : Waves

A wave is a disturbance that propagates. Apart from electromagnetic radiation, which can travel through vacuum, waves have a medium through which they travel and can transfer energy from one place to another without any of the particles of the medium being displaced permanently. Instead, any particular point oscillates around a fixed position.

## Characteristic properties

All waves have common behaviour under a number of standard situations. All waves can experience the following:

• Reflection - when a wave turns back from the direction it was travelling, due to hitting a reflective material.
• Refraction - the change of direction of waves due to them entering a new medium.
• Diffraction - the spreading out of waves, for example when they travel through a small slit.
• Interference - the addition of two waves that come in to contact with each other.
• Dispersion - the splitting up of a wave up depending on frequency.

## Transverse and longitudinal waves

Transverse waves are those with vibrations perpendicular to the wave's direction of travel; examples include ripples on the surface of a pond, waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations along the wave's direction of travel; examples include sound waves.

### Polarisation

Transverse waves can be polarised. Normally transverse waves can oscillate in any angle on the plane perpendicular to the direction of travel - these are described as unpolarised waves. Polarisation means to create light which has oscillations in only one line perpendicular to the line of travel.

## Physical description of a wave

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:

$f=\frac{1}{T}$.

When waves are expressed mathematically, the angular frequency (ω, radians/second) is often used; it is related to the frequency f by:

$f=\frac{\omega}{2 \pi}$.

### Travelling waves

Waves that remain in one place are called standing waves - eg vibrations on a violin string. Waves that are moving are called travelling waves, and have a disturbance that varies both with time t and distance z. This can be expressed mathematically as:

$y=A(z,t) \cos (\omega t - kz + \phi)$,

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:

$v=\frac{\omega}{k}= \lambda f$,

where λ is the wavelength of the wave.

### The wave equation

In the most general sense, not all waves are sinusoidal. One example of a non-sinusoidal wave is a pulse that travels down a rope resting on the ground. In the most general case, any function of x, y, z, and t that is a non-trivial solution to the wave equation is a wave. The wave equation is a differential equation which describes a harmonic wave passing through a certain medium. The equation has different forms depending on how the wave is transmitted, and on what medium.

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.