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The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions ...
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.
is the speed of light (i.e. phase velocity) in a medium with permeability μ, and permittivity ε, and ∇ 2 is the Laplace operator. In a vacuum, v ph = c 0 = 299 792 458 m/s, a fundamental physical constant. [1] The electromagnetic wave equation derives from Maxwell's equations.
Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299 792 458 m/s [2]). Known as electromagnetic radiation , these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays .
Numerical simulation of the Fisher–KPP equation. In colors: the solution u(t,x); in dots : slope corresponding to the theoretical velocity of the traveling wave.. In mathematics, KPP–Fisher equation (named after Andrey Kolmogorov, Ivan Petrovsky, Nikolai Piskunov [1] and Ronald Fisher [2]) also known as the KPP equation, Fisher equation or Fisher–KPP equation is the partial differential ...
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions (using the squared scalar wave velocity).
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 ...
N-wave type solutions of the Burgers equation, starting from the initial condition (,) = / (+) /. Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation [ 1 ] occurring in various areas of applied mathematics , such as fluid mechanics , [ 2 ] nonlinear acoustics , [ 3 ...