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By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves ...
The phase velocity varies with frequency. The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.
The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves. From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear ...
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] The ...
The equation says the matter wave frequency in vacuum varies with wavenumber (= /) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass ( ℏ ω 0 = m 0 c 2 {\displaystyle \hbar \omega _{0}=m_{0}c^{2}} ) and a quadratic part due to kinetic energy.
The following equations have solutions which satisfy the superposition principle, that is, the wave functions are additive. Throughout, the standard conventions of tensor index notation and Feynman slash notation are used, including Greek indices which take the values 1, 2, 3 for the spatial components and 0 for the timelike component of the ...
In the physical sciences, the wavenumber (or wave number), also known as repetency, [1] is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length , expressed in SI units of cycles per metre or reciprocal metre (m −1 ).
Panels (E–F) show two different wave functions that are solutions of the Schrödinger equation but not standing waves. The wave function of an initially very localized free particle. In quantum physics , a wave function (or wavefunction ) is a mathematical description of the quantum state of an isolated quantum system .