  The gravitational-wave energy-density spectra from cosmological first-order
phase transitions crucially depend on the terminal wall velocity of asymptotic
bubble expansion when the driving force from the effective potential difference
is gradually balanced by the backreaction force from the thermal plasma. Much
attention has previously focused on the backreaction force acting on the bubble
wall alone but overlooked the backreaction forces on the sound shell and
shock-wave front, if any, which have been both numerically and analytically
accomplished in our previous studies but only for a bag equation of state. In
this paper, we will generalize the backreaction force on bubble expansion
beyond the simple bag model.
