  Context. The frequencies, lifetimes, and eigenfunctions of solar acoustic
waves are affected by turbulent convection, which is random in space and in
time. Since the correlation time of solar granulation and the periods of
acoustic waves ($\sim$5 min) are similar, the medium in which the waves
propagate cannot a priori be assumed to be time independent. Aims. We compare
various effective-medium solutions with numerical solutions in order to
identify the approximations that can be used in helioseismology. For the sake
of simplicity, the medium is one dimensional. Methods. We consider the Keller
approximation, the second-order Born approximation, and spatial homogenization
to obtain theoretical values for the effective wave speed and attenuation
(averaged over the realizations of the medium). Numerically, we computed the
first and second statistical moments of the wave field over many thousands of
realizations of the medium (finite-amplitude sound-speed perturbations are
limited to a 30 Mm band and have a zero mean). Results. The effective wave
speed is reduced for both the theories and the simulations. The attenuation of
the coherent wave field and the wave speed are best described by the Keller
theory. The numerical simulations reveal the presence of coda waves, trailing
the coherent wave packet. These late arrival waves are due to multiple
scattering and are easily seen in the second moment of the wave field.
Conclusions. We find that the effective wave speed can be calculated,
numerically and theoretically, using a single snapshot of the random medium
(frozen medium); however, the attenuation is underestimated in the frozen
medium compared to the time-dependent medium. Multiple scattering cannot be
ignored when modeling acoustic wave propagation through solar granulation.
