Explicit representation of Green function for 3Dimensional exterior Helmholtz equation

TL;DR

This paper develops explicit algebraic formulas for solutions and Green functions of the 3D exterior Helmholtz equation, enabling precise boundary and operator constructions.

Contribution

It introduces explicit algebraic representations for solutions, Green functions, and operators related to the exterior Helmholtz problem in three dimensions.

Findings

Constructed orthogonal solutions sequence with explicit coefficients

Derived explicit normal derivative of the Dirichlet Green function

Proved uniform boundedness of normalized coefficients

Abstract

We have constructed a sequence of solutions of the Helmholtz equation forming an orthogonal sequence on a given surface. Coefficients of these functions depend on an explicit algebraic formulae from the coefficient of the surface. Moreover, for exterior Helmholtz equation we have constructed an explicit normal derivative of the Dirichlet Green function. In the same way the Dirichlet-to-Neumann operator is constructed. We proved that normalized coefficients are uniformly bounded from zero.