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wave equations (Topic)

Wave Equations

Describe the propagation of a wide variety of periodic signals, 'perturbations', or waves through either a medium or vacuum in terms of certain types of differential equations. There are a large number of such wave equations ranging from classical to quantum, and then to relativistic QFT and gravitational ones. Examples of waves are extremely numerous, and include: elastic waves (for example, vibrations, sound, ultrasound, sea waves, ground waves – as in an earthquake, etc.), electromagnetic waves (including: radio waves, light, X-rays, $\gamma$-rays, and so on), plasma waves, spin waves, gravitational waves, and so on.

Furthermore, every quantum `particle' has an associated, 'de Broglie wave', which is called the wave-particle duality in quantum theory; for example, electrons, protons, neutrons, quarks, neutrinos, and all other sub-atomic or elementary `particles' have their 'own' associated waves whose wavelength is inversely proportional to their energy (viz. de Broglie). The waves are represented as solutions of such differential equations, often with specified boundary conditons; thus a wave has both an amplitude and a phase.

Thus, any oscillation has a resulting, or corresponding wave that propagates; alternatively, a wave can also be represented as a continuous sequence of local oscillations of a propagating field. The intensity of the propagated signal or field is proportional to the square of the amplitude of the wave, whereas the phase can be thought of as the time interval that has elapsed from the beginning of the wave propagation to the point in space where its phase is determined or measured. There are two basic types of waves: longitudinal (for example, in an elastic medium such as sounds and sea waves)–that are propagating via longitudinal oscillations of particles in the elastic medium which occur along the direction of propagation of the wave front, or transversal waves (for example, electromagnetic) that can also propagate in vacuum.

Because of the periodic nature of the waves and of their propagation, wave superposition is readily analyzed in terms of either Fourier series or integrals. General solutions of the wave equations are thus usually expressed in terms of Fourier series whose components are `monochromatic' (single- frequency) waves.

Examples of Differential Wave Equations

  1. Transverse electrical wave, with varying vector field $\mathbf{E}$, travelling at a constant speed $c$:

    $\displaystyle \nabla^2 \mathbf{E} = c^{-2} \frac{ \partial^2 \mathbf{E}}{\partial t^2}$
  2. The wave equation in one dimension is

    $\displaystyle \frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}. $
  3. “The wave equation in $n$ dimensions is

    $\displaystyle \frac{\partial^2 u}{\partial t^2}=c^2\nabla^2 u. $
    where $u$ is a function of the location variables $x_1,x_2,\ldots,x_n$, and time $t$. Here, $\nabla^2$ is the Laplacian with respect to the location variables, which in Cartesian coordinates is given by $\nabla^2=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\cdots+\frac{\partial^2}{\partial x_n^2}$”, on line
  4. The Schrödinger equation for a quantum system has the general form:

    $\displaystyle \textbf{H} \psi= (-i) \partial \psi / \partial t$



"wave equations" is owned by bci1.

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See Also: wave equation, telegraph equation, velocity of wave, wave function space

Also defines:  wave-particle duality, wave amplitude, wave phase, longitudinal wave, transversal wave
Keywords:  wave-particle duality, wave amplitude, wave phase, longitudinal wave, transversal wave, field, propagation

Attachments:
derivation of wave equation (Derivation) by pahio

Cross-references: system, speed, vector field, square, field, boundary, energy, quarks, neutrons, quantum theory, quantum particle, spin, plasma, QFT, differential equations, types, waves
There are 14 references to this object.

This is version 7 of wave equations, born on 2008-10-19, modified 2009-02-18.
Object id is 315, canonical name is WaveEquation2.
Accessed 2195 times total.

Classification:
Physics Classification: 03.75.-b (Matter waves )
 41.20.Jb (Electromagnetic wave propagation; radiowave propagation )
 03.65.Ge (Solutions of wave equations: bound states)
 03.65.Pm (Relativistic wave equations)
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