We formally derive the simplest possible periodic wave structure consistent with time-periodic sound wave propagation in the $3 × 3$ nonlinear compressible Euler equations. The construction is based on identifying the simplest periodic pattern with the property that compression is counter-balanced by rarefaction along every characteristic. Our derivation leads to an explicit description of shock-free waves that propagate through an oscillating entropy field without breaking or dissipating, indicating a new mechanism for dissipation free transmission of sound waves in a nonlinear problem. The waves propagate at a new speed, (different from a shock or sound speed), and sound waves move through periods at speeds that can be commensurate or incommensurate with the period. The period determines the speed of the wave crests, (a sort of observable group velocity), but the sound waves move at a faster speed, the usual speed of sound, and this is like a phase velocity. It has been unknown since the time of Euler whether or not time-periodic solutions of the compressible Euler equations, which propagate like sound waves, are physically possible, due mainly to the ubiquitous formation of shock waves. A complete mathematical proof that waves with the structure derived here actually solve the Euler equations exactly, would resolve this long standing open problem.
"A Paradigm for Time-periodic Sound Wave Propagation in the Compressible Euler Equations." Methods Appl. Anal. 16 (3) 341 - 364, September 2009.