III. Problems involving two independent variables

Abstract

1. Problems already treated in this series (Bradfield and Southwell 1937; Black and Southwell 1938) have been concerned both with systems of restricted freedom (stress determination in framed structures; current partition in electrical networks; adjustment of errors of observations) and with continuous systems governed by equations in one variable (transversely loaded beams). For the latter tow lines of attack have been successful: either the governing equations can be replaced by an approximate equation in finite differences which is soluble, or by an application of Relaxation Methods it can be satisfield as it stands, not at all but a a finite number of "sections", i. e. for a finite number of values of the independent variable. Either procedure leads to tabulated values of the function investigated, for values of the independent variable separated by small intervals. Solutions thus presented are of more immediate value than the mathematical expressions given by orthodox analysis, which may require much labour to be expended in numerical computation before their significance can be appreciated. Tested in certain cases which can be solved exactly, both methods seem capable of giving more than sufficient accuracy for practical purposes. Methods which permit a comparable treatment of problems in two dimensions will have still greater value, because here the power of orthodox methods is more restricted and in general their solutions are more difficult to interpret numerically. We must expect to encounter greater difficulties, seeing that terminal conditions are now replaced by conditions relating to every point of some geometrical boundary; and it is significant that much attention has been devoted to the development of experimental methods, bases on analogies whereby equations made in another, because mathematically the same equation describes the phenomena of both. The "membrane analogue" or Prandtl (1903), utilized in the "soap-film method" of Taylor and Griffith (1917) for solving de Saint-Venant's problems of torsion and flexure (Southwell 1936, Appendix to Chapter xi), and the electrical method used by Relf (1924) and others to determine hydrodynamic streamlines, may be cited as examples. Mechanical equation-solvers of the kinds developed in recent years by Bush, Hartree, Mallock and others do not seem likely to provide much assistance in this field.