# Preprint 2008-033

# A Paradigm for Time-Periodic Sound Wave Propagation in the Compressible Euler Equations

## Blake Temple and Robin Young

**Abstract:**
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.