  For a cosmological first-order phase transition in the early Universe, the
associated stochastic gravitational wave background is usually dominated by
sound waves from plasma fluid motions, which have been analytically modeled as
a random superposition of freely propagating sound shells but with the force by
the scalar field that produces the self-similar profile removed. In this
Letter, we propose a new analytic sound shell model by focusing on the forced
propagating contribution from the initial collision stage of sound shells when
their self-similar profiles are still maintained by the moving bubble walls. We
reproduce the causal $k^3$ scaling in the infrared consistent with numerical
simulations, and also recover the broad dome in the power spectrum first
observed in numerical simulations. The total sound waves should contain both
contributions from forced collisions and free propagation of sound shells at
early and late stages of the phase transition, respectively.
