Variant of the Nonlinear Wave Equations Describing Cylindrical Axisymmetrical Waves

New nonlinear wave equations are derived using the Murnaghan five-constant elastic potential. A feature of these equations is the following two assumptions: the elastic deformation process is physically nonlinear (geometric nonlinearity is neglected) and the deformation is geometrically axisymmetric and described by cylindrical coordinates. Therefore, the system of wave equations contains only two coupled equations. Such a statement allows us to use these new equations to analyze surface waves propagating along a circular cylindrical cavity in an elastic medium. Another feature of the nonlinear equations is that every equation includes the classical linear part. The nonlinear terms of the equations are quadratically nonlinear and contain twenty-three types of nonlinearities in the first equation and twenty-two types in the second equation.