Green functions Helmholtz equation

The Green s function of the modified Helmholtz equation is defined in analogy to Eq. (B.2),... 

Here V denotes the outward normal gradient on a. It can easily be verified from the defining differential equation that the Helmholtz Green function is given by the expansion... 

This problem can be avoided by expressing the MST equations in terms of the square matrix S C = CVS, Hermitian in consequence of the surface-matching theorem. This matrix has full rank because it is contracted over the larger index lg. From the definitions of the C and S matrices, the matrix product S C is a specific integral involving the Helmholtz Green function [281],... 

For energy k2 in Rydberg units, the Green function of the Helmholtz equation, with given boundary conditions, satisfies... 

When coordinate x = X/x + r lies in cell r/x and x = Xv + r lies in a different cell r , a two-center expansion of the Green function is needed. Since N /fx — X/x) is a regular solution of the Helmholtz equation in a sphere of radius X/( — Xv about the origin of a displaced cell v,... 

Let us now find the Green s function of the corresponding Helmholtz equation (13.56). We can solve this problem by applying the Fourier transform to expression... 

Note that the definition of the Green s function can be extended in the case of a variable velocity distribution c = c (r). The corresponding Green s function G (r r o ) is the fundamental solution of the Helmholtz equation for the given velocity c(r) ... 

We can also introduce the Green s function g z z lj) for the 1-D Helmholtz equation (13.57). This function is a solution of the following equation... 

Differential equation (15.48) can be transformed into an integral equation for the scattered field, using the 1-D Green s function g (z z oj) for the 1-D Helmholtz equation, which is dependent on the position of the source z, the observation point z, and the frequency ur. This function satisfies the 1-D Helmholtz equation... 

Thus, in the spectral approach, wavefield extrapolation from one horizontal level to another is implemented by multiplying its spectrum by the complex conjugate of the vertical derivative of the spatial spectrum of the Green s function for the Helmholtz equation. 

The Green s function corresponding to the inhomogeneous Helmholtz equation is the spherical Bessel function yo K Ir - s ) of the second kind, so that the integral form of Eq. (7.57) reads... 

T. Rother, M. Kahnert, A. Doicu, J. Wauer, Surface Green s function of the Helmholtz equation in spherical coordinates, PIER 38, 47 (2002)... 

Once again, Heras [13] gave another formulation of Helmholtz theorem now based on Green s retarded function G = 8(f — ts — R/c)/R for a value B5(r5. ts) function of space and retarded time ts = t — R/c. Heras obtained the following equation ... 

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