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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 equation is named after Hermann von Helmholtz, who studied it in 1860. [2]
The Kirchhoff–Helmholtz integral combines the Helmholtz equation with the Kirchhoff integral theorem [1] to produce a method applicable to acoustics, [2] seismology [3] and other disciplines involving wave propagation.
To simplify the notation, let = ˙ and define a collection of n 2 functions Φ j i by =. Theorem. (Douglas 1941) There exists a Lagrangian L : [0, T] × TM → R such that the equations (E) are its Euler–Lagrange equations if and only if there exists a non-singular symmetric matrix g with entries g ij depending on both u and v satisfying the following three Helmholtz conditions:
The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.
The Sommerfeld radiation condition is used to solve uniquely the Helmholtz equation. For example, consider the problem of radiation due to a point source in three dimensions, so the function in the Helmholtz equation is () = (), where is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the ...
Thus, one may obtain the function u(x) through knowledge of the Green's function in equation 1 and the source term on the right-hand side in equation 2. This process relies upon the linearity of the operator L .
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form), including fluid mechanics, acoustics, electromagnetics (where the technique is known as method of moments or abbreviated as MoM), [1] fracture mechanics, [2] and contact mechanics.
Since the Helmholtz equation is a prototypical equation for modeling the spatial variation of classical waves, Mathieu functions can be used to describe a variety of wave phenomena. For instance, in computational electromagnetics they can be used to analyze the scattering of electromagnetic waves off elliptic cylinders, and wave propagation in ...
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