  We consider the stochastic background of gravity waves produced by
first-order cosmological phase transitions from two types of sources: colliding
bubbles and hydrodynamic turbulence. First we discuss the fluid mechanics of
relativistic spherical combustion. We then numerically collide many bubbles
expanding at a velocity $v$ and calculate the resulting spectrum of
gravitational radiation in the linearized gravity approximation. Our results
are expressed as simple functions of the mean bubble separation, the bubble
expansion velocity, the latent heat, and the efficiency of converting latent
heat to kinetic energy of the bubble walls. We also estimate the gravity waves
produced by a Kolmogoroff spectrum of turbulence and find that the
characteristic amplitude of gravity waves produced is comparable to that from
bubble collisions. Finally, we apply these results to the electroweak
transition. Using the one-loop effective potential for the minimal electroweak
model, the characteristic amplitude of gravity waves produced is $h\simeq
1.5\times 10^{-27}$ at a characteristic frequency of $4.1\times 10^{-3} \,\rm
Hz$ corresponding to $\Omega \sim10^{-22}$ in gravity waves, far too small for
detection. Gravity waves from more strongly first-order phase transitions,
including the electroweak transition in non-minimal models, have better
prospects for detection, though probably not by LIGO.
