  In a vacuum first-order phase transition, gravitational waves are generated
from collision of bubbles of the true vacuum. The spectrum from such collisions
takes the form of a broken power law. We consider a toy model for such a phase
transition, where the dynamics of the scalar field depends on a single
parameter $\overline{\lambda}$, which controls how thin the bubble wall is at
nucleation and how close to degenerate the vacua are relative to the barrier.
We extend on our previous work by performing a series of simulations with a
range of $\overline{\lambda}$. The peak of the gravitational-wave power
spectrum varies by up to a factor of $1.3$, which is probably an unobservable
effect. We find that the ultraviolet (UV) power law in the gravitational-wave
spectrum becomes steeper as $\overline{\lambda} \rightarrow 0$, varying between
$k^{-1.4}$ and $k^{-2.2}$ for the $\overline{\lambda}$ considered. This
provides some evidence that the form of the underlying effective potential of a
vacuum first-order phase transition could be determined from the
gravitational-wave spectrum it produces.
