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

The asymptotic behavior as λ → ∞ of the function U ( x , λ ) that satisfies the reduced wave equation L λ [ U ] = ∇ ⋅ ( E ( x ) ∇ U ) + λ 2 N 2 ( x ) U = 0 on an infinite 3-dimensional region, a Dirichlet condition on ∂ V , and an outgoing radiation condition is investigated. A function U N ( x , λ ) is constructed that is a global approximate solution as λ → ∞ of the problem satisfied by U ( x , λ ) . An estimate for W N ( x , λ ) = U ( x , λ ) − U N ( x , λ ) on V is obtained, which implies that U N ( 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 W N ( x , λ ) in terms of its boundary values, and the L 2 norm of r L λ [ W N ( 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. The solution U ( x , λ ) can be interpreted as a scalar potential of a high frequency acoustic or electromagnetic field radiating from the boundary of an impenetrable object of general shape. The energy of the field propagates through an inhomogeneous, anisotropic medium; the rays along which it propagates may form caustics. The approximate solution (potential) derived in this paper is defined on and in a neighborhood of any such caustic, and can be used to connect local “geometrical optics” type approximate solutions that hold on caustic free subsets of V . The result of this paper generalizes previous work of Bloom and Kazarinoff [C. O. BLOOM and N. D. KAZARINOFF, Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions , SPRINGER VERLAG, NEW YORK, NY, 1976].