Two wave interaction in a cylinder

The interaction of two acoustical waves in a cylinder is studied in the quadratic approximation. In cylinder coordinates, the usual perturbation techniques in the separation of variables method leads inevitably to a series of overdetermined systems of linear algebraic equations for unknown coefficients (in contrast to Cartesian coordinates). However, if we formally introduce a new function W(r) (dependent on a radius r) satisfying the first system of this series, all these overdetermined systems become compatible (remaining overdetermined) for a special case of the non-linear acoustical wave equation. Using the new function W(r) and quadratic polynomials of the Bessel functions of radius, we express explicitly the coefficients of the resulting harmonics and obtain solutions describing two wave interactions. The solutions are found with the same accuracy as the non-linear acoustical equation is derived. As a consequence, a general boundary problem can be solved explicitly in these terms.