The Helmholtz Equation
Part of the Applied Mathematical Sciences book series (AMS, volume 144)
The acoustic wave equation described the propagation of the sound in a medium like the air. It results, e.g., from the equation of the compressible gas dynamic, also called the compressible Navier-Stokes equations. In the case of small displacements of the gas, a linearization of these equations leads to an equation for the displacement and the small variation of the pressure in the gas. In the case of a homogeneous medium of mean density ρ 0, this is the well-known above system of equationsWhere c is the speed of sound in the medium, the speed of displacement in the medium and p the pressure, which is supposed to be isotropic. Eliminating the displacement , we obtain a scalar wave equation for the pressure p:
KeywordsDirichlet Problem Spherical Harmonic Laplace Equation Neumann Problem Helmholtz Equation
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© Springer Science+Business Media New York 2001