Some classical types of periodic wave motion in the circular cylinder coordinates are studied within quadratic approximation using a Fourier expansion along the moving axial coordinate. When the nonlinear quadratic terms in a wave equation are arbitrary, the usual perturbation techniques used in the cylinder coordinates leads to the overdetermined systems of linear algebraic equations for unknown coefficients. However, we show that these systems are compatible for the special case of the nonlinear acoustic wave equation and express explicitly the coefficients of the first three harmonics as polynomials of the Bessel functions of radius and of the trigonometric functions of angle. It gives a series of solutions to the nonlinear acoustic wave equation, which are periodic in time and found with the same accuracy as the equation is derived. In the two-dimensional case, this technique gives a solution to the nonlinear shallow water equation. These solutions can be used for describing the nonlinear waves propagating inside or outside of a cylinder. They can be considered as nonlinear corrections to the classical linear solutions, which were studied and used in a series of classical books (see, for example [1] and [2]). In some sense we obtain generalization of method of separation of variables for the nonlinear equation.
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