Time-Periodic Linearized Solutions of the Compressible Euler Equations and a Problem of Small Divisors

It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [27], we derive exact linearized solutions that exhibit the simplest possible periodic wave structure that can balance compression and rarefaction along characteristics in the nonlinear Euler problem. These linearized waves exhibit interesting phase and group velocities analogous to linear dispersive waves. Moreover, when the spacial period is incommensurate with the time period, the sound speed is incommensurate with the period, and a new periodic wave pattern is observed in which the sound waves move in a quasi-periodic trajectory though a periodic configuration of states. This establishes a new way in which nonlinear solutions that exist arbitrarily c...

• Type: Journal
• Year: 2011
• Where: SIAMMA
• Authors: Blake Temple, Robin Young
• Keywords: Compressible Euler Equations, Linearized Solutions, SIAMMA 2011, Wave