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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 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 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 the magnetic field B, takes the form:
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by =, where is a position in real space, is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and is time. (Here the physics definition for the wave vector is ...
This concept can be visualized by imagining a clock with a hand that turns at constant speed, making a full turn every seconds, and is pointing straight up at time . The phase φ ( t ) {\displaystyle \varphi (t)} is then the angle from the 12:00 position to the current position of the hand, at time t {\displaystyle t} , measured clockwise .
The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
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 ...
Since the amplitude to travel from x to y after a time t + t ' can be considered in two steps, the propagator obeys the composition identity, (;) (; ′) = (; + ′) , which can be interpreted as follows: the amplitude to travel from x to z in time t + t ' is the sum of the amplitude to travel from x to y in time t, multiplied by the amplitude ...