  Gravitational waves from a phase transition associated with the generation of
the masses of elementary particles are within the reach of future space-based
detectors such as LISA. A key determinant of the resulting power spectrum, not
previously studied, is the lifetime of the acoustic turbulence which follows.
We study decaying acoustic turbulence using numerical simulations of a
relativistic fluid in two dimensions. Working in the limit of non-relativistic
bulk velocities, with an ultra-relativistic equation of state, we find that the
energy spectrum evolves towards a self-similar broken power law, with a
high-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ by
the inverse width of the shock waves. Our model for the decay of acoustic
turbulence can be extended to three dimensions using the universality of the
high-$k$ power law and the evolution laws for the kinetic energy and the
integral length scale. It is used to build an estimate for the gravitational
wave power spectrum resulting from a collection of shock waves, as might be
found in the aftermath of a strong first order phase transition in the early
universe. The power spectrum has a peak wavenumber set by the initial length
scale of the acoustic waves, and a new secondary scale at a lower wavenumber
set by the integral scale after a Hubble time. Between these scales a
distinctive new power law appears. Our results allow more accurate predictions
of the gravitational wave power spectrum for a wide range of early universe
phase transition scenarios.
