We present details of numerical simulations of the gravitational radiation produced by a first order thermal phase transition in the early universe .
We confirm that the dominant source of gravitational waves is sound waves generated by the expanding bubbles of the low-temperature phase .
We demonstrate that the sound waves have a power spectrum with a power-law form between the scales set by the average bubble separation ( which sets the length scale of the fluid flow L _ { \text { f } } ) and the bubble wall width .
The sound waves generate gravitational waves whose power spectrum also has a power-law form , at a rate proportional to L _ { \text { f } } and the square of the fluid kinetic energy density .
We identify a dimensionless parameter \tilde { \Omega } _ { \text { GW } } characterising the efficiency of this “ acoustic ” gravitational wave production whose value is 8 \pi \tilde { \Omega } _ { \text { GW } } \simeq 0.8 \pm 0.1 across all our simulations .
We compare the acoustic gravitational waves with the standard prediction from the envelope approximation .
Not only is the power spectrum steeper ( apart from an initial transient ) but the gravitational wave energy density is generically larger by the ratio of the Hubble time to the phase transition duration , which can be 2 orders of magnitude or more in a typical first order electroweak phase transition .