It is known that wave propagation in nonlinear continues media, such as acoustic waves in solids, water waves, and solitary waves in arteries, can be reduced to a third order ordinary differential equations. They can be cast in a general third order ODE as  
However, having an ODE as a reduced model for a phenomenon expressible by a partial differential equation lacks a proof to grantee for having a periodic solution. A third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the above general third-order ODE. However, the equation is too general. In this paper we examine the following more specific equation  
and prove a new theorem to establish the sufficient condition for its periodicity. To obtain the periodicity conditions, the Schauder’s fixed-point theorem is implemented. A numerical method is also developed for rapid convergence.
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