Non-separable solutions of the Helmholtz wave equation

A set of solutions not obtainable by the method of separation of variables is presented for the vector Helmholtz wave equation in circular cylindrical coordinates limited to non-angular dependence. These are constructed of Bessel and trigonometric functions. For example, if A is the vector, the r r -component of the simplest member of the set is \[ A r = C 1 [ m r J 0 ( p r ) cos ⁡ ( m z ) + p z J 1 ( p r ) sin ⁡ ( m z ) ] e − i w t , {A_r} = {C_1}\left [ {mr{J_0}\left ( {pr} \right )\cos \left ( {mz} \right ) + pz{J_1}\left ( {pr} \right )\sin \left ( {mz} \right )} \right ]{e^{ - iwt}}, \] where C 1 {C_1} is an arbitrary constant, m m and p p are propagation constants, and ω \omega is angular frequency. Brief reference is made to three-dimensional solutions in rectangular coordinates.