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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 ...
Linearity. The Schrödinger equation is a linear differential equation, meaning that if two state vectors and are solutions, then so is any linear combination of the two state vectors where a and b are any complex numbers. [13]: 25 Moreover, the sum can be extended for any number of state vectors.
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.
Electromagnetic wave equation. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or ...
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction ...
The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y ...
Helmholtz equation. In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: where ∇2 is the Laplace operator, k2 is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number.
Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. [1] Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇ 2 X (The last term in the right side is the vector Laplacian, not Laplacian applied on ...