GREENS FUNCTIONS FOR UNBOUNDED CONSTANT GRADIENT MEDIA

Results of Pekeris and Goodman are used in a Fourier synthesis that gives the response to a signal φ(t) from a point source in an unbounded medium with constant sound speed gradient. That response is [formulla omitted], where * denotes convolution, H (t) is the unit step function, τ is the travel time between source and receiver, r is the distance from the source to the receiver, and γ is one half the magnitude of the gradient. The quantity to the right of the * in the above expression is the Green's function for a point source. The Green's function for a plane source is 2πvH (tâτ)J0[γ(t2âτ2)12], where v is the geometric mean of the sound speeds at the source and receiver. The Green's function for a line source is approximately 2γH (tâτ) (sinh2γtâsinh2γτ)â12. As γ â 0, all intermediate and final results approach known results for an unbounded medium with uniform sound speed. © 1969, Acoustical Society of America. All rights reserved.