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The stochastic gravitational-wave backgrounds (SGWBs) from the cosmological
first-order phase transitions (FOPTs) serve as a promising probe for the new
physics beyond the standard model of particle physics. When most of the bubble
walls collide with each other long after they had reached the terminal wall
velocity, the dominated contribution to the SGWBs comes from the sound waves
characterized by the efficiency factor of inserting the released vacuum energy
into the bulk fluid motions. However, the previous works of estimating this
efficiency factor have only considered the simplified case of the constant
sound velocities in both symmetric and broken phases, either for the bag model
with equal sound velocities or $\nu$-model with different sound velocities in
the symmetric and broken phases, which is unrealistic from a viewpoint of
particle physics. In this paper, we propose to solve the fluid EoM with an
iteration method when taking into account the sound-velocity variation across
the bubble wall for a general and realistic equation of state (EoS) beyond the
simple bag model and $\nu$-model. We have found a suppression effect for the
efficiency factor of bulk fluid motions, though such a suppression effect could
be negligible for the strong FOPT, in which case the previous estimation from a
bag EoS on the efficiency factor of bulk fluid motions still works as a good
approximation.

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