  Here we suggest an alternative understanding of the surface gravity wave
propagation mechanism based on the baroclinic torque, which operates to
translate the interfacial vorticity anomalies at the air-water interface. We
demonstrate how the non-Boussinesq term of the baroclinic torque acts against
the Boussinesq one to hinder wave propagation. By standard vorticity inversion
and mirror imaging, we then show how the existence of the bottom boundary
affects the two types of torque. Since the opposing non-Boussinesq torque
results solely from the mirror image, it vanishes in the deep water limit and
its magnitude is half of the Boussinesq torque in the shallow water limit. This
reveals that Boussinesq approximation is valid in the deep water limit, even
though the density contrast between air and water is large. The mechanistic
roles, played by the Boussinesq and non-Boussinesq parts of the baroclinic
torque, remain obscured in the standard derivation where the time-dependent
Bernoulli equation is implemented instead of the interfacial vorticity
equation. Finally, we note on passing that the Virial theorem for surface
gravity waves can be obtained solely from considerations of the dynamics at the
air-water interface.
