Etude numérique de la fonction de Green scalaire d'une cavité à l'aide d'une nouvelle équation intégrale

We study numerically, with the aid of an IBM-370 computer, the Green's functions of a cavity afforded by the solutions of a new integral equation (B. T. Darling and J. A. Imbeau. Can. J. Phys. 56, 387 (1978)). A number of prolate spheroidal cavities whose eccentricities cover the complete range zero to one are employed, and the solutions are subject to the Dirichlet and von Neumann conditions at the surface. We use the Gauss–Legendre integration formula to replace the integral equation by a set of linear algebraic equations. The Green's function is evaluated by substituting the solution of this set in the formula of Helmholtz, using the same integration formula. Criteria for the optimization of this procedure also are developed and employed. The Green's function can be determined to high precision except in the immediate vicinity of the surface of the cavity where it suffers a well-known discontinuity. We also explore the use of the Helmholtz formula itself in the exterior region as an integral equation to obtain the Green's function of the cavity. We find that although the precision of the solution is much less than that afforded by the precedingly mentioned integral equation the precision is still within the range of practical application. All calculations used double precision arithmetic (16 significant digits on the IBM-370).