# Wave

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.

## The medium which carries a wave

A medium that can carry a wave is classified by one or more of the following properties:

• A linear medium if the amplitudes of different waves at any particular point in the medium can be added.
• A bounded medium if it is finite in extent, otherwise unbounded.
• A uniform medium if its physical properties are unchanged at different locations in space.
• An isotropic medium if its physical properties are the same in different directions.

## Examples of waves

File:Waves crashing.jpg
A wave crashing against the shore

## Characteristic properties

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

• Reflection – the change of direction of waves, due to hitting a reflective surface.
• Refraction – the change of direction of a wave due to them entering a new medium.
• Diffraction – the spreading out of waves, for example when they travel through a small slit.
• Interference – the superposition of two waves that come into contact with each other.
• Dispersion – the splitting up of waves by frequency.
• Rectilinear propagation – the movement of waves in straight lines.

## Transverse and longitudinal waves

File:Elliptical trajectory on ripples.png
When an object bobs up and down on a ripple in a pond, it experiences an elliptical trajectory because ripples are not simple transverse sinusoidal waves.

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.

### Polarization

Transverse waves can be polarized. Unpolarised waves can oscillate in any direction in the plane perpendicular to the direction of travel, while polarized waves oscillate in only one direction 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:

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

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

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

### Travelling waves

Waves that remain in one place are called standing waves - e.g. 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:

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

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:

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

where λ is the wavelength of the wave.

### Propagation through strings

The speed of a wave travelling along a string (v) is directly proportional to the square root of the tension (T) over the linear density (ρ):

$\displaystyle v=\sqrt{\frac{T}{\rho}}.$

This equation can be found using dimensional analysis

### The wave equation

The wave equation is a differential equation which describes a harmonic wave passing through a medium, discussed above. The equation has different forms depending on how the wave is transmitted, and on what medium.

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

$\displaystyle \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=\frac{\partial^2\phi}{\partial x^2}.$

A general solution, given by d'Alembert is

$\displaystyle \phi(x,t)=F(x-ct)+E(x+ct).$

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.