  Typical flows in stellar interiors are much slower than the speed of sound.
To follow the slow evolution of subsonic motions, various sound-proof equations
are in wide use, particularly in stellar astrophysical fluid dynamics. These
low-Mach number equations include the anelastic equations. Generally, these
equations are valid in nearly adiabatically stratified regions like stellar
convection zones, but may not be valid in the sub-adiabatic, stably stratified
stellar radiative interiors. Understanding the coupling between the convection
zone and the radiative interior is a problem of crucial interest and may have
strong implications for solar and stellar dynamo theories as the interface
between the two, called the tachocline in the Sun, plays a crucial role in many
solar dynamo theories. Here we study the properties of gravity waves in
stably-stratified atmospheres. In particular, we explore how gravity waves are
handled in various sound-proof equations. We find that some anelastic
treatments fail to conserve energy in stably-stratified atmospheres, instead
conserving pseudo-energies that depend on the stratification, and we
demonstrate this numerically. One anelastic equation set does conserve energy
in all atmospheres and we provide recommendations for converting low-Mach
number anelastic codes to this set of equations.
