High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries

• Main Author: Clifford O. Bloom
• Format: Article
• Language: English
• Published: Hindawi Limited 1996-01-01
• Series: Mathematical Problems in Engineering
• Subjects: High frequency radiation; scattering; global approximate solution; uniform asymptotic approximation; caustics; geometrical optics; inhomogeneous medium; anisotropic medium; reduced wave equation.
• Online Access: http://dx.doi.org/10.1155/S1024123X96000385

The asymptotic behavior as λ→∞ of the function U(x,λ) that satisfies the reduced wave equation Lλ[U]=∇⋅(E(x)∇U)+λ2N2(x)U=0 on an infinite 3-dimensional region, a Dirichlet condition on ∂V , and an outgoing radiation condition is investigated. A function UN(x,λ) is constructed that is a global approximate solution as λ→∞ of the problem satisfied by U(x,λ) . An estimate for WN(x,λ)=U(x,λ)−UN(x,λ) on V is obtained, which implies that UN(x,λ) is a uniform asymptotic approximation of U(x,λ) as λ→∞, with an error that tends to zero as rapidly as λ−N(N=1,2,3,...). This is done by applying a priori estimates of the function WN(x,λ) in terms of its boundary values, and the L2 norm of rLλ[WN(x,λ)] on V. It is assumed that E(x), N(x), ∂V and the boundary data are smooth, that E(x)−I and N(x)−1 tend to zero algebraically fast as r→∞, and finally that E(x) and N(x) are slowly varying; ∂V may be finite or infinite.