  The bubbles involved in sonochemistry and other applications of cavitation
oscillate inertially. A correct estimation of the wave attenuation in such
bubbly media requires a realistic estimation of the power dissipated by the
oscillation of each bubble, by thermal diffusion in the gas and viscous
friction in the liquid. Both quantities and calculated numerically for a single
inertial bubble driven at 20 kHz, and are found to be several orders of
magnitude larger than the linear prediction. Viscous dissipation is found to be
the predominant cause of energy loss for bubbles small enough. Then, the
classical nonlinear Caflish equations describing the propagation of acoustic
waves in a bubbly liquid are recast and simplified conveniently. The main
harmonic part of the sound field is found to fulfill a nonlinear Helmholtz
equation, where the imaginary part of the squared wave number is directly
correlated with the energy lost by a single bubble. For low acoustic driving,
linear theory is recovered, but for larger drivings, namely above the Blake
threshold, the attenuation coefficient is found to be more than 3 orders of
magnitude larger then the linear prediction. A huge attenuation of the wave is
thus expected in regions where inertial bubbles are present, which is confirmed
by numerical simulations of the nonlinear Helmholtz equation in a 1D standing
wave configuration. The expected strong attenuation is not only observed but
furthermore, the examination of the phase between the pressure field and its
gradient clearly demonstrates that a traveling wave appears in the medium.
