Computation of a green's function for the three-dimensional linearized transient gravity waves problem

Abstract

We have studied the problem of calculating the function of four variables G(r,z,ζ;μ) = − 1 4π 1 (r 2 + (z − ζ) 2) 1 2 + 1 (r 2 + (z − ζ) 2) 1 2 +2 ∫ 0 ∞ (s − μ) cosh((z +1)s) cosh((ζ + 1)s) μ cosh s + s sinh s e −sJ 0(rs)ds , where r ⩾, 0, −1 ⩽ z, ζ ⩽ 0, μ ⩾ 0, cosh, sinh are the elementary hyperbolic functions, and J 0 is the regular Bessel function of zero order. Using only three methods of computation requiring Gauss quadratures with less than 20 nodes or summation of a truncated series with less than 20 terms, an accuracy of five decimal digits and often six decimal digits is obtained and covers the domain r ⩾, 0,−1 ⩽ z, ζ ⩽ 0, μ ⩽ 0, r 2 + ( z − ζ) 2 ≠ 0. G is the Green's function of the Laplacian in the strip −1 ⩽ z ⩽ 0 with the boundary conditions ∂ G/∂z = 0 for z = −1 and ∂ G/∂z + μG = 0 for z = 0 and can be used in the numerical solution of the three-dimensional linearized transient gravity waves problem by coupling the techniques of finite elements and integral representation.