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Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck constant is the canonical momentum.
When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the electric field. [1]
More, all α are in an n-dimensional set A = A 1 × A 2 × ... × A n where each A i is the set of allowed values for α i; all ω are in an m-dimensional "volume" Ω ⊆ ℝ m where Ω = Ω 1 × Ω 2 × ... × Ω m and each Ω i ⊆ R is the set of allowed values for ω i, a subset of the real numbers R. For generality n and m are not ...
For long waves (λ ≫ h) with small Ursell number, U ≪ 32 π 2 / 3 ≈ 100, [3] linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves ( λ > 7 h ) [ 4 ] – like the Korteweg–de Vries equation or Boussinesq equations – has to be used.
wave vector: radian per meter (m −1) Boltzmann constant: joule per kelvin (J/K) wavenumber: radian per meter (m −1) stiffness: newton per meter (N⋅m −1) ^ Cartesian z-axis basis unit vector unitless angular momentum: newton meter second (N⋅m⋅s or kg⋅m 2 ⋅s −1)
Other examples of mechanical waves are seismic waves, gravity waves, surface waves and string vibrations. In an electromagnetic wave (such as light), coupling between the electric and magnetic fields sustains propagation of waves involving these fields according to Maxwell's equations .
The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile () is a sinusoidal function. That is, (,) = (() +) The parameter , which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient is its "spatial frequency"; and the scalar is its "phase shift".
A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization. Real-world examples include sound waves (vibrations in pressure, a particle of displacement, and particle velocity propagated in an elastic medium) and seismic P waves (created by earthquakes and explosions).