  We study gravitational wave (GW) production in strongly supercooled
cosmological phase transitions, taking particular care of models featuring a
complex scalar field with a U$(1)$ symmetric potential. We perform lattice
simulations of two-bubble collisions to properly model the scalar field
gradients, and compute the GW spectrum sourced by them using the thin-wall
approximation in many-bubble simulations. We find that in the U$(1)$ symmetric
case the low-frequency spectrum is $\propto\omega$ whereas for a real scalar
field it is $\propto\omega^3$. In both cases the spectrum decays as
$\omega^{-2}$ at high frequencies.
