2.6 Regaining geometric acoustics

Up to now, we have been developing general machinery to force acoustics into Lorentzian form. This can be justified either with a view to using fluid mechanics to teach us more about general relativity, or to using the techniques of Lorentzian geometry to teach us more about fluid mechanics.

For example, given the machinery developed so far, taking the short wavelength/high frequency limit to obtain geometrical acoustics is now easy. Sound rays (phonons) follow the null geodesics of the acoustic metric. Compare this to general relativity, where in the geometrical optics approximation, light rays (photons) follow null geodesics of the physical spacetime metric. Since null geodesics are insensitive to any overall conformal factor in the metric [445, 275, 670], one might as well simplify life by considering a modified conformally-related metric

⌊ ⌋ − (c2 − v20) ... − vj0 | ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅| hμν ≡ || . || . (111 ) ⌈ − vi0 .. δij ⌉
This immediately implies that, in the geometric acoustics limit, sound propagation is insensitive to the density of the fluid. In this limit, acoustic propagation depends only on the local speed of sound and the velocity of the fluid. It is only for specifically wave-related properties that the density of the medium becomes important.

We can rephrase this in a language more familiar to the acoustics community by invoking the eikonal approximation. Express the linearised velocity potential, ϕ1, in terms of an amplitude, a, and phase, φ, by ϕ1 ∼ aeiφ. Then, neglecting variations in the amplitude a, the wave equation reduces to the eikonal equation

μν h ∂μφ∂νφ = 0. (112 )
This eikonal equation is blatantly insensitive to any overall multiplicative prefactor (conformal factor). As a sanity check on the formalism, it is useful to re-derive some standard results. For example, let the null geodesic be parameterised by μ x (t) ≡ (t;x(t)). Then the null condition implies
dxμ-dxν- hμν dt dt = 0 dxi dxi dxi ⇐ ⇒ − (c2 − v20) − 2vi0----+ --------= 0 ∥ ∥ dt dt dt ∥∥dx- ∥∥ ⇐ ⇒ ∥ dt − v0∥ = c. (113 )
Here the norm is taken in the flat physical metric. This has the obvious interpretation that the ray travels at the speed of sound, c, relative to the moving medium.

Furthermore, if the geometry is stationary (steady flow) one can do slightly better. Let x μ(s ) ≡ (t(s);x(s)) be some null path from x 1 to x 2, parameterised in terms of physical arc length (i.e., ||dx∕ds || ≡ 1). Then the tangent vector to the path is

( ) dxμ dt dxi ----= --;---- . (114 ) ds ds ds
The condition for the path to be null (though not yet necessarily a null geodesic) is
dx-μdx-ν gμν ds ds = 0. (115 )
Using the explicit algebraic form for the metric, this can be expanded to show
( )2 ( i) ( ) − (c2 − v2) dt- − 2vi dx-- dt- + 1 = 0. (116 ) 0 ds 0 ds ds
Solving this quadratic
∘ ------------------ ( ) i( dxi) 2 2 ( idxi)2 dt- −-v0--ds--+----c-−--v0 +--v0ds--- ds = c2 − v2 . (117 ) 0
Therefore, the total time taken to traverse the path is
∫ x2 ∫ {∘ ----------------------- } T [γ ] = (dt∕ds )ds = ---1--- (c2 − v2)ds2 + (vidxi)2 − vidxi . (118 ) x1 γ c2 − v20 0 0 0
If we now recall that extremising the total time taken is Fermat’s principle for sound rays, we see that we have checked the formalism for stationary geometries (steady flow) by reproducing the discussion of Landau and Lifshitz [372, p. 262].13
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