w(x,y) = Acos(kx) + Bcos(Vk2 - a2x)cos(ay),

The general solutions of partial differential equations contain arbitrary functions equivalent to an infinite set of constants. For boundary problems in which the boundary does not coincide with curves or surfaces of a coordinate system in which the partial differen­ tial equation is separable into ordinary differential equations, these constants may be used to satisfy the given physical boundary condition on arbitrary given boundary curves and in­ terfaces. Thus it is possible to calculate wave propagation and eigenfrequencies of straight and bent glass fibers and wave-guides of various cross sections like Cassinian curves etc. The basic idea The new method has been invented to solve non-separable boundary problems. Problems of this kind arise when the boundary on which certain boundary conditions have to be satisfied cannot be described by coordinate lines or surfaces of a coordinate system in which the re­ spective partial differential equation is separable. For instance Maxwell's equations cannot be separated into ordinary differential equations for toroidal coordinates. The main idea of the new method consists now in the following: The general solution of a partial differential equation of order n contains n arbitrary functions: If it is possible to construct such an arbitrary function as a sum of degenerate eigenfunctions, that means functions belonging to one and the same eigenvalue, then the partial amplitudes of the eigenfunctions representing the arbitrary function may be used to satisfy the boundary conditions given by the physical problem on an arbitrarily given curve or interface. A simple example As an example for the new method we will solve the eigenvalue problem of a circular mem­ brane of radius R in Cartesian coordinates, that is in straight line coordinates. Here the boundary curve, a circle, does not coincide with coordinate lines. We choose this example to show that problems can be solved in which the boundary curve is not a coordinate curve. The eigenvalue k of this problem is very well known from the zero of the Bessel function Jo. With the ansatz w(x,y)exp(icot), k = w/c the wave equation for membrane oscillations is transformed into Helmholtz equation Wxx + wyy + k2w = O