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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 inhomogeneous Helmholtz equation is the equation + = (),, where ƒ : R n → C is a function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation , and would be identical if the plus sign (in front of the k term) were switched to a minus sign.
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 ...
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature . The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process ...
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.
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:
This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T 2 with knowledge of just the standard Gibbs free energy change of formation and the standard enthalpy change of formation for the individual components. Also, using the reaction isotherm equation, [8] that is
Originally, the spheroidal wave functions were introduced by C. Niven, [21] which lead to a Helmholtz equation in spheroidal coordinates. Monographs tying together many aspects of the theory of spheroidal wave functions were written by Strutt, [ 22 ] Stratton et al., [ 23 ] Meixner and Schafke, [ 24 ] and Flammer.