Relaxation methods applied to engineering problems. XI. Problems governed by the ‘quasi-plane-potential equation'

Abstract

The ‘quasi-plane-potential equation’ ∂/∂x (X∂ψ/∂x) + ∂/∂y (x∂ψ/∂y) +Z = 0 is discussed in relation to problems where Z and x are known functions of x and y . It governs ( inter alia ) the small transverse displacement of a membrane in which the tension T ∝X, and its finite-difference approximation governs the small transverse displacements of nodal points of a net in which, similarly, the string tension T varies from node to node. (Equilibrium in the directions of x and y can be maintained, both in the membrane and in the net, by forces acting in those directions and accordingly having no effect on the transverse equilibrium.) The relaxational treatment, based on this mechanical (net) analogue, reduces when X is constant to the treatment developed, in earlier papers of this series, for problems governed by the ‘plane-harmonic’ (Poisson) equation, ∂ 2 ψ/∂x 2 + ∂ 2 ψ/∂ y 2 + Z = 0 Thus an opportunity is afforded for a review of various improvements which have been incorporated in the technique first propounded in Part III. In particular, this paper for the first time uses systematically the device of ‘graded nets’, i.e. nets of which the mesh-side a is smaller in some parts than in others.