Separation of variables for the nonlinear wave equation in cylindrical coordinates

Some classical types of nonlinear wave motion in cylindrical coordinates are studied within the quadratic approximation. When cylindrical coordinates are used, the usual perturbation techniques inevitably lead to overdetermined systems of linear algebraic equations for the unknown coefficients (in contrast to the case for Cartesian coordinates). However, we show that these overdetermined systems are compatible for the special case of the nonlinear acoustic wave equation and express the coefficients of the first two harmonics explicitly as polynomials in Bessel functions of the radius and in trigonometric functions of the angle. This gives a series of solutions to the nonlinear acoustic wave equation which are found with the same accuracy as the equation is derived with.