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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.
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]
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 x 0 {\displaystyle x_{0}} in three dimensions, so the function f {\displaystyle f} in the Helmholtz equation is f ( x ) = δ ( x − x 0 ) , {\displaystyle f(x)=\delta (x-x_{0}),} where δ ...
The differential equations given above for the oblate radial and angular wave functions can be obtained from the corresponding equations for the prolate spheroidal wave functions by the substitution of for and for . The notation for the oblate spheroidal functions reflects this relationship.
These expansions still solve the original Helmholtz equations for E and B because for a divergence-free field F, ∇ 2 (r ⋅ F) = r ⋅ (∇ 2 F). The resulting expressions for a generic electromagnetic field are:
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.
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