  It is a deceptively simple question to ask how acoustic disturbances
propagate in a non--homogeneous flowing fluid. If the fluid is barotropic and
inviscid, and the flow is irrotational (though it may have an arbitrary time
dependence), then the equation of motion for the velocity potential describing
a sound wave can be put in the (3+1)--dimensional form: d'Alembertian psi = 0.
That is partial_mu(sqrt{-g} g^{mu nu} partial_nu psi)/sqrt{-g} = 0. The
acoustic metric --- g_{mu nu}(t,x) --- governing the propagation of sound
depends algebraically on the density, flow velocity, and local speed of sound.
Even though the underlying fluid dynamics is Newtonian, non--relativistic, and
takes place in flat space + time, the fluctuations (sound waves) are governed
by a Lorentzian spacetime geometry.
