  Models with singlet fields coupling to the Higgs can enable a strongly first
order electroweak phase transition, of interest for baryogenesis and gravity
waves. We improve on previous attempts to self-consistently solve for the
bubble wall properties -- wall speed $v_w$ and shape -- in a highly predictive
class of models with $Z_2$-symmetric singlet potentials. A new algorithm is
implemented to determine $v_w$ and the wall profiles throughout the singlet
parameter space in the case of subsonic walls, focusing on models with strong
enough phase transitions to satisfy the sphaleron washout constraint for
electroweak baryogenesis. We find speeds as low as $v_w \cong 0.1$ in our scan
over parameter space, and the singlet must be relatively light to have a
subsonic wall, $m_s \lesssim 135$ GeV.
