Abstract: Our results are always parameterless, that is, explicit functions of the problem variables found without using rays, ray equations, or ray parameters. When c = sinhz, we find a series solution of the reduced wave equation ▿2u + ω2c−2u = 0 characterizing radiation from a point source at (x0, y0, z0) in three dimensions. Ignoring the shadow region (x − x0)2 + (y − y0)2 > π2, we use formal methods to find the first three terms of a series for the radiation solution, beginning with the term exp[iτ(ω2 − 1/4)1/2]{sinρ/[ρc(z)c(z0)]}1/2 cschτ, τ defined by cosh τ = (coshz coshz0 − cosρ)/(sinhz sinhz0) and ρ = [(x − x0)2 + (y − y0)2]1/2. In all of two-dimensional (x, z) space except the shadow region |x − x0| > π, formal methods and reexamination of Cohn's Riemann functions show that Qλ(coshτ) is an excellent approximation to the radiation solution. Here Qλ is a Legendre function, λ = 1/2 + i(ω2 − 1/4)1/2, and ρ = x − x0. Special and limiting cases of the above give analogous results for c equal to coshz, cosz, ez, z, and 1. We obtain Pekeris' exact solutions when c(z) = z, and the classical solutions when c(z) = 1.
Publication Year: 1973
Publication Date: 1973-12-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 3
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