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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 ...
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.
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.
Kirchhoff and Helmholtz showed, that the radiation of sound from a limited source region can be described by enclosing this source region by a control surface - the so-called Kirchhoff surface. Then the sound field inside or outside the surface, where no sources are allowed and the wave operator on the left hand side applies, can be produced as ...
Its mathematical basis is the Kirchhoff–Helmholtz integral. It states that the sound pressure is completely determined within a volume free of sources, if sound pressure and velocity are determined in all points on its surface.
Luigi Mangione, accused in the killing of UnitedHealthcare CEO Brian Thompson, will plead not guilty, according to his lawyer, Thomas Dickey.
See also Spectral concentration problem.) Pioneering work in this area was performed by Slepian and Pollak, [4] Landau and Pollak, [5] [6] and Slepian. [7] [8] Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions". [9]