Theorem. If a fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent) then the equation of motion for the velocity potential describing an acoustic disturbance is identical to the d’Alembertian equation of motion for a minimally-coupled massless scalar field propagating in a (3+1)-dimensional Lorentzian geometryUnder these conditions, the propagation of sound is governed by an acoustic metric – . This acoustic metric describes a (3+1)-dimensional Lorentzian (pseudo–Riemannian) geometry. The metric depends algebraically on the density, velocity of flow, and local speed of sound in the fluid. Specifically (Here is the identity matrix.) In general, when the fluid is non-homogeneous and flowing, the acoustic Riemann tensor associated with this Lorentzian metric will be nonzero.
Comment. It is quite remarkable that even though the underlying fluid dynamics is Newtonian, nonrelativistic, and takes place in flat space-plus-time, the fluctuations (sound waves) are governed by a curved (3+1)-dimensional Lorentzian (pseudo-Riemannian) spacetime geometry. For practitioners of general relativity this observation describes a very simple and concrete physical model for certain classes of Lorentzian spacetimes, including (as we shall later see) black holes. On the other hand, this discussion is also potentially of interest to practitioners of continuum mechanics and fluid dynamics in that it provides a simple concrete introduction to Lorentzian differential geometric techniques.
Now linearise these equations of motion around some assumed background . Setdefined to be these linearised fluctuations in the dynamical quantities. Note that this is the standard definition of (linear) sound and more generally of acoustical disturbances. In principle, of course, a fluid mechanic might really be interested in solving the complete equations of motion for the fluid variables . In practice, it is both traditional and extremely useful to separate the exact motion, described by the exact variables, , into some average bulk motion, , plus low amplitude acoustic disturbances, . See, for example, [370, 372, 443, 577].
Since this is a subtle issue that we have seen cause considerable confusion in the past, let us be even more explicit by asking the rhetorical question: “How can we tell the difference between a wind gust and a sound wave?” The answer is that the difference is to some extent a matter of convention – sufficiently low-frequency long-wavelength disturbances (wind gusts) are conventionally lumped in with the average bulk motion. Higher-frequency, shorter-wavelength disturbances are conventionally described as acoustic disturbances. If you wish to be hyper-technical, we can introduce a high-pass filter function to define the bulk motion by suitably averaging the exact fluid motion. There are no deep physical principles at stake here – merely an issue of convention. The place where we are making a specific physical assumption that restricts the validity of our analysis is in the requirement that the amplitude of the high-frequency short-wavelength disturbances be small. This is the assumption underlying the linearization programme, and this is why sufficiently high-amplitude sound waves must be treated by direct solution of the full equations of fluid dynamics.
Linearizing the continuity equation results in the pair of equations22) determines , and Equation (23) then determines . Thus this wave equation completely determines the propagation of acoustic disturbances. The background fields , and , which appear as time-dependent and position-dependent coefficients in this wave equation, are constrained to solve the equations of fluid motion for a barotropic, inviscid, and irrotational flow. Apart from these constraints, they are otherwise permitted to have arbitrary temporal and spatial dependencies.
Now, written in this form, the physical import of this wave equation is somewhat less than pellucid. To simplify things algebraically, observe that the local speed of sound is defined by24) is easily rewritten as 24) and is a much more promising stepping-stone for further manipulations. The remaining steps are a straightforward application of the techniques of curved space (3+1)-dimensional Lorentzian geometry.
Now in any Lorentzian (i.e., pseudo–Riemannian) manifold the curved space scalar d’Alembertian is given in terms of the metric by[213, 446, 597, 445, 275, 670].) The inverse metric, , is pointwise the matrix inverse of , while . Thus one can rewrite the physically derived wave Equation (24) in terms of the d’Alembertian provided one identifies 26), expanding the determinant in minors yields inverse (contravariant) acoustic metric [445, pp. 505–508].) The (covariant) acoustic metric is then read off by inspection
We have presented the theorem and proof, which closely follows the discussion in , in considerable detail because it is a standard template that can be readily generalised in many ways. This discussion can then be used as a starting point to initiate the analysis of numerous and diverse physical models.
Living Rev. Relativity 14, (2011), 3
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