  We present a numerical and theoretical investigation of nonlinear spectral
energy cascade of decaying finite-amplitude planar acoustic waves in a
single-component ideal gas at standard temperature and pressure (STP). We
analyze various one-dimensional canonical flow configurations: a propagating
traveling wave (TW), a standing wave (SW), and randomly initialized Acoustic
Wave Turbulence (AWT). We use shock-resolved mesh-adaptive direct numerical
simulations (DNS) of the fully compressible one-dimensional Navier-Stokes
equations to simulate the spectral energy cascade in nonlinear acoustic waves.
We also derive a new set of nonlinear acoustics equations truncated to second
order and the corresponding perturbation energy corollary yielding the
expression for a new perturbation energy norm $E^{(2)}$. Its spatial average,
<$E^{(2)}$> satisfies the definition of a Lyapunov function, correctly
capturing the inviscid (or lossless) broadening of spectral energy in the
initial stages of evolution -- analogous to the evolution of kinetic energy
during the hydrodynamic break down of three-dimensional coherent vorticity --
resulting in the formation of smaller scales. Upon saturation of the spectral
energy cascade i.e. fully broadened energy spectrum, the onset of viscous
losses causes a monotonic decay of <$E^{(2)}$> in time.
