Wave

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Surface waves in water

This article is about waves in the most general scientific sense. For waves in the ocean, see ocean surface wave. For other uses, see Wave (disambiguation).

A wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions. Thus, while mechanical waves require a medium to transverse the distance, electromagnetic waves can travel through a vacuum.

1 Introduction / Definitions
2 Characteristics
- 2.1 Polarization
- 2.2 Examples
3 Mathematical description
- 3.1 The wave equation
- 3.2 Traveling waves
- 3.3 Standing wave
- 3.4 Propagation through strings
4 Transmission medium
5 See also
6 Sources
7 External links

[edit] Introduction / Definitions

Agreeing on a single, all-encompassing definition for the term wave is non-trivial. A vibration can be defined as a back-and-forth motion around a point of rest (e.g. Campbell & Greated, 1987: 5) or, more generally, as a variation of any physical property of a system around a reference value. However, defining the necessary and sufficient characteristics that qualify a phenomenon to be called a wave is, at least, flexible. The term is often understood intuitively as the transport of disturbances in space, not associated with motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall, 1980: 8). However, this notion is problematic for a standing wave (e.g. a wave on a string), where energy is moving in both directions equally, or for electromagnetic / light waves in a vacuum, where the concept of medium does not apply.

For such reasons, wave theory represents a peculiar branch of physics that is concerned with the properties of wave processes independently from their physical origin (Ostrovsky and Potapov, 1999). The peculiarity lies in the fact that this independence from physical origin is accompanied by a heavy reliance on origin when describing any specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave-like transfer / transformation of vibratory energy. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as opposed to optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved (e.g. in the case of air: vortices, radiation pressure, shock waves, etc., in the case of solids: Rayleigh waves, dispersion, etc., and so on).

Other properties, however, although they are usually described in an origin-specific manner, may be generalized to all waves. For example, based on the mechanical origin of acoustic waves there can be a moving disturbance in space-time if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion (or rather infinitely fast wave motion). On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion (or rather infinitely slow wave motion). Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (i.e. its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times. Similarly, wave processes revealed from the study of wave phenomena with origins different from that of sound waves can be equally significant to the understanding of sound phenomena. A relevant example is Young's principle of interference (Young, 1802, in Hunt, 1978: 132). This principle was first introduced in Young's study of light and, within some specific contexts (e.g. scattering of sound by sound), is still a researched area in the study of sound. As another example, the phenomenon of dispersion demonstrates that wave modulations behave as regular waves. When modulations propagate in media where the speed of wave propagation depends on frequency, they separate from the complex wave they belonged to and travel independently carrying energy, similarly to the rest of the frequency components of the complex wave. It is true that this separation will never happen in a non-dispersive medium such as air, where all frequencies move with the same speed. Nonetheless, the important point is that the dispersive case serves to illustrate that modulations in general and amplitude fluctuations in particular behave as waves. Dispersion provides a case where modulations are isolated from the waves that carry them and can therefore be studied easier (assuming that the only characteristic that changes during dispersion is the modulations' velocity). In addition, systems with dispersion provide better cases for the mathematical analysis of the kinematic properties of waves (i.e. frequency, wavelength, phase and group velocities). In the case of sound waves, diffraction, absorption, reverberation, and interference are examples of phenomena that have been better understood with the aid of dispersion theory.

To summarize, the term wave implies three general notions: vibrations in time, disturbances in space, and moving disturbances in space-time associated with the transfer/transformation of energy. Based on these notions, the following origin-specific definition may be adopted for sound waves in air (Vassilakis, 2001): "Sound-waves in air represent a transfer of vibratory energy characterized by: i) rate (frequency), ii) starting position (phase), and iii) magnitude (amplitude) of vibration. In general, amplitude can be expressed equivalently in terms of maximum displacement, velocity, or pressure relative to a reference value. Sound waves in air are manifested as alternating air-condensations and rarefactions that spread away from the vibrating source with a velocity usually not related to the velocity amplitude of the vibration. They result in pressure/density disturbance patterns in the surrounding medium, which, in general, correspond to the signal that plots the vibration of the source over time." This definition will serve as an initial operational definition of sound waves in air to which further qualifications may be added as needed.

[edit] Characteristics

Periodic waves are characterized by crests (highs) and troughs (lows), and may usually be categorized as either longitudinal or transverse. Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.

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

A = At deep water.
B = At shallow water. The circular movement of a surface particle becomes elliptical with decreasing depth.
1 = Progression of wave
2 = Crest
3 = Trough

Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.

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

Reflection - wave direction change from hitting a reflective surface
Refraction - wave direction change from entering a new medium
Diffraction - wave circular spreading from entering a hole of comparable size to their wavelengths
Interference - superposition of two waves that come into contact with each other (collide)
Dispersion - wave splitting up by frequency
Rectilinear propagation - wave movement in straight lines

[edit] Polarization

Main article: Polarization

A wave is polarized if it can only oscillate in one direction. The polarization of a transverse wave describes the direction of oscillation, in the plane perpendicular to the direction of travel. Longitudinal waves such as sound waves do not exhibit polarization, because for these waves the direction of oscillation is along the direction of travel. A wave can be polarized by using a polarizing filter.

[edit] Examples

An ocean surface wave crashing into rocks

Examples of waves include:

Ocean surface waves, which are perturbations that propagate through water.
Radio waves, microwaves, infrared rays, visible light, ultraviolet rays, x-rays, and gamma rays make up electromagnetic radiation. In this case, propagation is possible without a medium, through vacuum. These electromagnetic waves travel at 299,792,458 m/s in a vacuum.
Sound — a mechanical wave that propagates through air, liquid or solids.
waves of traffic (i.e. propagation of different densities of motor vehicles, etc.) — these can be modelled as kinematic waves, as first presented by Sir M. J. Lighthill
Seismic waves in earthquakes, of which there are three types, called S, P, and L.
Gravitational waves, which are fluctuations in the gravitational field predicted by general Relativity. These waves are nonlinear, and have yet to be observed empirically.
Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect.

[edit] Mathematical description

From mathematical point of view most primitive (or fundamental) wave is harmonic (sinusoidal) wave which is described by the equation f(x,t) = Asin(wt-kx)), where A is the amplitude of a wave - 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.

A wavenumber $k$ can be associated with the wavelength by the relation

$k = \frac{2 \pi}{\lambda}. \,$

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. These are related by:

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

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. It is related to the frequency by

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

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

$v_p = \frac{\omega}{k} = {\lambda}f.$

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

$v_g = \frac{\partial \omega}{\partial k}. \,$

[edit] The wave equation

Main article: Wave equation

The wave equation is a differential equation that describes the evolution of a harmonic wave over time. The equation has slightly different forms depending on how the wave is transmitted, and the medium it is traveling through. Considering a one-dimensional wave that is travelling down a rope along the x-axis with velocity $v$ and amplitude $u$ (which generally depends on both x and t), the wave equation is

$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,$

In three dimensions, this becomes

$\frac{1}{v^2}\frac{\partial^2 u}{\partial t^2} = \nabla^2 u. \,$

where $\nabla^2$ is the Laplacian.

The velocity v will depend 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 d'Alembert. It is

$u(x,t)=F(x-vt)+G(x+vt). \,$

This can be viewed as two pulses travelling down the 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.

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.

[edit] Traveling waves

Simple wave or traveling wave, also sometimes called progressive wave is a disturbance that varies both with time $t$ and distance $z$ in the following way:

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

where $A (z, t)$ is the amplitude envelope of the wave, $k$ is the wave number and $φ$ is the phase. The phase velocity v_p of this wave is given by

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

where $λ$ is the wavelength of the wave.

[edit] Standing wave

Main article: standing wave

Standing wave in stationary medium. The red dots represent the wave nodes

A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, longitudinal waves propagate out to where the string is held in place at the bridge and the "nut", where upon the waves are reflected back. The two opposed waves each cancel the wave propagation of the other. This effect is known as interference. There is no net propagation of energy.

Also see: Acoustic resonance, Helmholtz resonator, and organ pipe

[edit] Propagation through strings

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

$v=\sqrt{\frac{T}{\mu}} \,$

[edit] Transmission medium

Main article: Transmission medium

The medium that carries a wave is called a 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 it is finite in extent, otherwise an unbounded medium.
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.