Spherical Harmonics

Abstract

The singular solutions introduced in the last chapter are particular cases of a class of functions known as spherical harmonics. In this chapter, we shall develop these functions and some related harmonic potential functions in a more formal way. In particular, we shall identify:-

(i) Finite polynomial potentials expressible in the alternate forms P_n(x, y, z), P_n(r, z) cos(mθ) or R_nP_n(cos β) cos(mθ), where Pn represents a polynomial of degree n.

(ii) Potentials that are singular only at the origin.

(iii) Potentials including the factor ln(R+z) that are singular on the negative z-axis (z < 0, r = 0).

(iv) Potentials including the factor ln{(R+z)/(R?z)} that are singular everywhere on the z-axis.

(v) Potentials including the factor ln(r) and/or negative powers of r that are singular everywhere on the z-axis.

All of these potentials can be obtained in axisymmetric and non-axisymmetric forms. When used in solutions A,B and E of Tables 21.1, 21.2, the bounded potentials (i) provide a complete set of functions for the sphere, cylinder or cone with prescribed surface tractions or displacements on the curved surfaces. These problems are three-dimensional counterparts of those considered in Chapters 5, 8 and 11. Problems for the hollow cylinder and cone can be solved by supplementing the bounded potentials with potentials (v) and (iv) respectively. Axisymmetric problems for the hollow sphere or the infinite body with a spherical hole can be solved using potentials (i,ii), but these potentials do not provide a complete solution to the non-axisymmetric problem1. Functions (ii) and (iii) are useful for problems of the half space, including crack and contact problems.