Scalar multiple scattering effects due to a random distribution of spheres are considered in detail. Transformation from a volume to a surface integral allows one to take account of the ``hole corrections'' involved in the equation of multiple scattering, and yields a secular equation for the propagation constant K of the composite medium. In the low-frequency limit a result is given which appears to be exact over the entire range 0 ≤ δ ≤ 1, where δ is the fractional volume occupied by scatterers. Also in this limit, the boundary conditions appropriate to the boundary of the composite medium are established from examination of the total transmitted and reflected fields.

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The cutoff wave numbers knm and the field of two-conductor, perfectly conducting wave guides are determined analytically. Three types of wave guides are considered: Eccentric circular conductors of radii R•, R2 and distance d between their axes, elliptic inner with circular outer conductor, and circular inner with elliptic outer conductor. The electromagnetic field is expressed in the first case in terms of circular cylindrical wave functions referred to both axes in combination with related addition theorems, while in the last two cases, both circular and elliptical cylindrical wave functions are used, which are further connected with one another by well-known expansion formulas. When the solutions are specialized to small eccentricities, kd in the first case and h = ka/2 in the last two cases (where a is the interfocal distance of the elliptic conductor), exact, closed-form expressions are obtained for the coefficients gnm in the resulting relations knm(d ) = knm(0)[1 + gnm(knmd) 2 + .-.] and knm(h ) = knm(0)[1 + grim h2 + '''] for the cutoff wave numbers of the corresponding wave guides. Similar expressions are obtained for the field. Numerical results for all types of modes, comparisons, and certain generalizations are also included.

The improper integrals, appearing in the course of evaluating the vector potential A and the electric field E inside a current-carrying region, are carefully examined. It is found that the integrals exist and have a well-defined meaning only when the current density function satisfies a Holder condition. A definite and precise way of evaluating them is derived and A, E are shown to satisfy the usual inhomogeneous equations.

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