2.3 Physical acoustics

It is well known that for a static homogeneous inviscid fluid the propagation of sound waves is governed by the simple wave equation [370Jump To The Next Citation Point, 372Jump To The Next Citation Point, 443Jump To The Next Citation Point, 577Jump To The Next Citation Point]
∂2tϕ = c2∇2 ϕ. (5 )
Generalizing this result to a fluid that is non-homogeneous, or to a fluid that is in motion, possibly even in non-steady motion, is more subtle than it at first would appear. To derive a wave equation in this more general situation we shall start by adopting a few simplifying assumptions to allow us to derive the following theorem.

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 geometry

--1-- (√ --- μν ) Δ ϕ ≡ √ −-g∂ μ − gg ∂νϕ = 0. (6 )
Under these conditions, the propagation of sound is governed by an acoustic metric gμν(t,x). 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
⌊ 2 2 .. T ⌋ | − (c − v ) . − v | ρ-|⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅| gμν(t,x ) ≡ c |⌈ .. |⌉ . (7 ) − v . I
(Here I is the 3 × 3 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.

Proof. The fundamental equations of fluid dynamics [370Jump To The Next Citation Point, 372Jump To The Next Citation Point, 443Jump To The Next Citation Point, 577Jump To The Next Citation Point] are the equation of continuity

∂tρ + ∇ ⋅ (ρv ) = 0, (8 )
and Euler’s equation (equivalent to F = ma applied to small lumps of fluid)
dv ρ ---≡ ρ[∂tv + (v ⋅ ∇ )v] = f. (9 ) dt
Start the analysis by assuming the fluid to be inviscid (zero viscosity), with the only forces present being those due to pressure.4 Then, for the force density, we have
f = − ∇p. (10 )
Via standard manipulations the Euler equation can be rewritten as
( ) 1- 1-2 ∂tv = v × (∇ × v ) − ρ∇p − ∇ 2v . (11 )
Now take the flow to be vorticity free, that is, locally irrotational. Introduce the velocity potential ϕ such that v = − ∇ ϕ, at least locally. If one further takes the fluid to be barotropic (this means that ρ is a function of p only), it becomes possible to define
∫ p dp′ 1 h(p) = ----; so that ∇h = -∇p. (12 ) 0 ρ(p′) ρ
Thus, the specific enthalpy, h(p), is a function of p only. Euler’s equation now reduces to
1- 2 − ∂tϕ + h + 2(∇ ϕ ) = 0. (13 )
This is a version of Bernoulli’s equation.

Now linearise these equations of motion around some assumed background (ρ0,p0,ϕ0). Set

ρ = ρ0 + 𝜖ρ1 + O (𝜖2), (14 ) 2 p = p0 + 𝜖p1 + O (𝜖 ), (15 ) ϕ = ϕ0 + 𝜖ϕ1 + O (𝜖2). (16 )
Sound is defined 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 (ρ,p,ϕ ). In practice, it is both traditional and extremely useful to separate the exact motion, described by the exact variables, (ρ,p,ϕ), into some average bulk motion, (ρ0,p0,ϕ0 ), plus low amplitude acoustic disturbances, (𝜖ρ1, 𝜖p1,𝜖ϕ1 ). See, for example, [370Jump To The Next Citation Point, 372Jump To The Next Citation Point, 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 equations

∂ ρ + ∇ ⋅ (ρ v ) = 0, (17 ) t 0 0 0 ∂tρ1 + ∇ ⋅ (ρ1v0 + ρ0v1) = 0. (18 )
Now, the barotropic condition implies
( 2) p1- 2 h(p) = h p0 + 𝜖p1 + O (𝜖) = h0 + 𝜖ρ + O (𝜖 ). (19 ) 0
Use this result in linearizing the Euler equation. We obtain the pair
− ∂tϕ0 + h0 + 1(∇ ϕ0)2 = 0. (20 ) 2 − ∂ϕ + p1− v ⋅ ∇ ϕ = 0. (21 ) t 1 ρ0 0 1
This last equation may be rearranged to yield
p1 = ρ0 (∂tϕ1 + v0 ⋅ ∇ ϕ1). (22 )
Use the barotropic assumption to relate
ρ = ∂ρp = ∂ρρ (∂ ϕ + v ⋅ ∇ ϕ ). (23 ) 1 ∂p 1 ∂p 0 t 1 0 1
Now substitute this consequence of the linearised Euler equation into the linearised equation of continuity. We finally obtain, up to an overall sign, the wave equation:
( ) ( ) − ∂t ∂ρρ0(∂tϕ1 + v0 ⋅ ∇ ϕ1 ) + ∇ ⋅ ρ0∇ ϕ1 − ∂ρ-ρ0v0(∂tϕ1 + v0 ⋅ ∇ ϕ1) = 0. (24 ) ∂p ∂p
This wave equation describes the propagation of the linearised scalar potential ϕ1. Once ϕ1 is determined, Equation (22View Equation) determines p1, and Equation (23View Equation) then determines ρ1. Thus this wave equation completely determines the propagation of acoustic disturbances. The background fields p0, ρ0 and v0 = − ∇ ϕ0, 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 by

c−2 ≡ ∂ρ. (25 ) ∂p
Now construct the symmetric 4 × 4 matrix
⌊ .. j ⌋ | − 1 . − v 0 | μν ρ0-|⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅| f (t,x ) ≡ c2 |⌈ i .. 2 ij i j |⌉ . (26 ) − v0 . (c δ − v0v0)
(Greek indices run from 0 – 3, while Roman indices run from 1 – 3.) Then, introducing (3+1)-dimensional space-time coordinates, which we write as μ i x ≡ (t;x ) , the above wave Equation (24View Equation) is easily rewritten as
∂ (fμν∂ ϕ ) = 0. (27 ) μ ν 1
This remarkably compact formulation is completely equivalent to Equation (24View Equation) 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 gμν(t,x) by

1 (√ --- μν ) Δ ϕ ≡ √-−-g∂μ − gg ∂νϕ . (28 )
(See, for example, [213, 446, 597, 445Jump To The Next Citation Point, 275Jump To The Next Citation Point, 670Jump To The Next Citation Point].) The inverse metric, gμν(t,x ), is pointwise the matrix inverse of gμν(t,x), while g ≡ det(gμν). Thus one can rewrite the physically derived wave Equation (24View Equation) in terms of the d’Alembertian provided one identifies
√--- − gg μν = fμν. (29 )
This implies, on the one hand,
√ --- det(fμν) = ( − g )4g− 1 = g. (30 )
On the other hand, from the explicit expression (26View Equation), expanding the determinant in minors yields
μν ( ρ0)4 [ 2 2 2] [ 2] [ 2] ρ40- det(f ) = c2 ⋅ (− 1) ⋅ (c − v0) − (− v0) ⋅ c ⋅ c = − c2. (31 )
Thus,
ρ40- √ --- ρ20- g = − c2; − g = c . (32 )
Therefore, we can pick off the coefficients of the inverse (contravariant) acoustic metric
⌊ . j ⌋ − 1 .. − v0 μν -1--|| ⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅|| g (t,x) ≡ ρ c |⌈ i .. 2 ij i j|⌉ . (33 ) 0 − v0 . (c δ − v0v0)
We could now determine the metric itself simply by inverting this 4 × 4 matrix (and if the reader is not a general relativist, proceeding in this direct manner is definitely the preferred option). On the other hand, for general relativists it is even easier to recognise that one has in front of one a specific example of the Arnowitt–Deser–Misner split of a (3+1)-dimensional Lorentzian spacetime metric into space + time, more commonly used in discussing initial value data in general relativity. (See, for example, [445Jump To The Next Citation Point, pp. 505–508].) The (covariant) acoustic metric is then read off by inspection
⌊ ⌋ − (c2 − v20) ... − vj0 ρ ||⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅|| gμν ≡ -0-| . | . (34 ) c ⌈ − vi0 .. δij ⌉
Equivalently, the acoustic interval (acoustic line-element) can be expressed as
2 μ ν ρ0-[ 2 2 i i j j ] ds ≡ gμνdx dx = c − c dt + (dx − v0dt)δij(dx − v 0dt) . (35 )
This completes the proof of the theorem. □

We have presented the theorem and proof, which closely follows the discussion in [624Jump To The Next Citation Point], 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.


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