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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.
Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) [1] is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that ...
In 1882, Gustav Kirchhoff analyzed Fresnel's theory in a rigorous mathematical formulation, as an approximate form of an integral theorem. [3]: 375 Very few rigorous solutions to diffraction problems are known however, and most problems in optics are adequately treated using the Huygens-Fresnel principle. [3]: 370
Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem, [3] uses Green's second identity to derive the solution of the homogeneous scalar wave equation at an arbitrary spatial position P in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface as the boundary of some volume including P.
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.
For any nonzero loss, the fields decay exponentially with distance and the surface integral vanishes, regardless of whether the medium is homogeneous. Since the left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero losses, it must also vanish in the limit as the losses go to zero.
The model proposed by Kirchhoff [16] and Rayleigh [17] was based on the free-streamline theory of Helmholtz [18] and consists of a steady wake behind the body. Assumptions applied to the wake region include: flow velocities equal to the body velocity, and a constant pressure.
This is the Fresnel diffraction integral; it means that, if the Fresnel approximation is valid, the propagating field is a spherical wave, originating at the aperture and moving along z. The integral modulates the amplitude and phase of the spherical wave. Analytical solution of this expression is still only possible in rare cases.