Abstract
This chapter discusses the Helmholtz and wave equations. One common type of eigenvalue problem is that governed by the Helmholtz equation: ∇2ø+λ2 ø = 0. For example, the vibration of a membrane, the oscillatory or seiche motion of an enclosed body of water, and the propagating modes of an electromagnetic waveguide are all governed by this equation. If the medium is nonhomogeneous and anisotropic, the Helmholtz equation can be stated in terms of properties referred to local principal axes. The Ritz finite element solution of the Helmholtz equation is closely similar to the solution for the Laplace equation. The electromagnetic field modes for a hollow, uniform waveguide filled with homogeneous and isotropic dielectric are determined by solving the two-dimensional scalar Helmholtz equation. As an alternative to a direct finite element formulation, the wave equation can be decomposed by the method of separation of variables into the Helmholtz equation and an ordinary differential equation.
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