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.