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).

Une nouvelle équation intégrale pour l'étude de la radiation scalaire dans une cavité

We derive an integral equation of the first kind connecting the surface values and the normal derivative for a regular solution inside a closed cavity of the Helmholtz equation. This integral equation has two advantages over the usual limit form of integral equations where the field point must lie on the boundary and the kernel is singular, namely, the field point may be anywhere inside or outside the cavity, and the kernel is regular. Analytic solution of our integral equation is obtained for the special cases of monopole and of dipole sources at the center of a sphere (Dirichlet's condition). The next paper will apply this integral equation to prolate spheroidal cavities.