  Feedback from active galactic nuclei (AGN) is believed to be the most
promising solution to the cooling flow problem in cool-core clusters, though
how exactly the jet energy is transformed into heat is a subject of debate.
Dissipation of sound waves is considered as one of the possible heating
mechanisms; however, its relative contribution to heating remains unclear. To
estimate the energy budget for heating, we perform 3D hydrodynamic simulations
of AGN jet injections in a Perseus-like cluster and quantify the amount of
energy stored in the forms of weak shocks and waves. We find that, for a single
jet injection with typical parameters in cool-core clusters, $\sim9\%$ of the
total jet energy is stored in compressional waves (including both shocks and
waves). However, due to the destructive effects among randomly phased waves as
well as the dissipation of shock energies, in our simulations including
self-regulated AGN feedback, no more than $3\%$ of the total injected energy
goes into compressional waves. We further separate the energy contribution from
shocks and waves and find that, for a single outburst, the shocks can only
contribute to $\sim20-30\%$ of the total compressional energy in the inner
radii, and they dissipate very quickly as they travel outward. However, because
of the repeated generation of shocks by multiple AGN outbursts, in the
self-regulated case shocks completely dominate over sound waves in the inner
region and can still provide $\sim40-50\%$ of the total compressional energy
even at outer radii. Our results suggest that the production of sound waves is
not as efficient as what was found in previous single-outburst simulations, and
thus sound wave dissipation may be a subdominant source of heating in cool-core
clusters.
