HIGH FREQUENCY ASYMPTOTIC SOLUTIONS OF THE REDUCED WAVE EQUATION ON INFINITE REGIONS WITH NON-CONVEX BOUNDARIES

CLIFFORD O. BLOOM

《Mathematical Problems in Engineering》 1996卷 (1996/12) Pp. 333-365

https://doi.org/10.1155/S1024123X96000385

摘要

The asymptotic behavior as λ → ∞ of the function U (x, λ) that satisfies the reduced wave equation L_λ[U] = ▽．(E(x) ▽U)+λ^2N^2(X)U = 0 on an infinite 3-dimensional region, a Dirichlet condition on əv, 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_N(x,λ) on V is obtained, which 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 ot 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 rL_λ[W_N(x,λ)] on V. V. It is assumed that E(x), N(x) əv and the boundary data are smooth, that E(x)- - I and N(x) - I tend to zero algebraically fast as r → ∞, and finally that E(x) and N(x) are slowly varying; əv 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].

關鍵字

High frequency radiation ； scattering ； global approximate solution ； uniform asymptotic approximation ； caustics ； geometrical optics ； inhomogeneous medium ； anisotropic medium ； reduced wave equation