ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence October 3, 2016, Axel Brandenburg

Handout 12: Acoustic cavity

In Handout 11 we derived the following eigenvalue problem for eigenvalue ω^{2},

(ω^{2} − c_{s}^{2}k^{2}) Φ − gΦ′ + c_{s}^{2}Φ′′ = 0  (1)

with boundary condition Φ = 0 at the bottom and ∇ · ξ = 0, or (k^{2} − d^{2}/dz^{2})Φ = 0, at the top. We assume a polytropic stratification with

c_{s}^{2} = c_{s0}^{2} − (γ − 1) gz.  (2)

We can bring Equation (1) into Sturm-Liouville form by substituting

Φ(z) = f(z) ϕ(z),  (3)

so

Φ′ = f′ϕ + fϕ′  (4)

and

Φ′′ = f′′ϕ + 2f′ϕ′ + fϕ′′.  (5)

Thus, we have

(ω^{2} − c_{s}^{2}k^{2}) fϕ − g(f′ϕ + fϕ′) + c_{s}^{2}(f′′ϕ + 2f′ϕ′ + fϕ′′) = 0  (6)

We now choose f such that the ϕ′ term vanishes, i.e.,

−gf + 2c_{s}^{2}f′ = 0,  (7)

so that

ln f = \frac{1}{2} ∫ \frac{g}{c_{s}^{2}} dz.  (8)

Using Equation (2), we have dc_{s}^{2}/dz = −(γ − 1)g, so

ln f = \frac{1}{γ − 1} ∫ \frac{dc_{s}^{2}}{2c_{s}^{2}} = \frac{1}{γ − 1} ∫ \frac{dc_{s}}{c_{s}} = \frac{ln c_{s}}{γ − 1}  (9)

or simply f = c_{s}^{1/(γ−1)} = c_{s}^{n}, where n = 1/(γ − 1) is the polytropic index.

Having eliminated the ϕ′ term, our differential equation takes the form

(ω^{2} − c_{s}^{2}k^{2}) fϕ − gf′ϕ + c_{s}^{2}(f′′ϕ + fϕ′′) = 0,  (10)

or

(ω^{2}/c_{s}^{2} − k^{2} + f′′/f − gf′/f) ϕ + ϕ′′ = 0,  (11)

where f′/f = nc′_{s}/c_{s} and f′′/f = n(n − 1)c_{s}^{′2}/c_{s}^{2} + nc′′_{s}/c_{s}. Let us now introduce the abbreviation K(z)^{2} = ω^{2}/c_{s}^{2} − k^{2} + f′′/f − gf′/f, so we have

K^{2}ϕ + ϕ′′ = 0.  (12)

We now proceed by solving Equation (12) using WKB. If K was constant, the solution would be of the form ϕ = e^{iK}. Thus, we make the ansatz

ϕ(z) = A(z) e^{iφ(z)},  (13)

where A(z) is a slowly varying function. Let us first calculate the first and second derivatives, so

ϕ′ = (A′ + iAφ′) e^{iφ},  (14)
ϕ′′ = (A′′ + iA′φ′ + iAφ′′) e^{iφ} + (A′ + iAφ′) iφ′ e^{iφ},  (15)

i.e.,

ϕ′′ = [(A′′ − Aφ^{′2}) + i(2A′φ′ + Aφ′′)] e^{iφ}  (16)

Inserting this into Equation (12), canceling the e^{iφ} term everywhere, and demanding that real and imaginary parts vanish separately, we have

(K^{2} − φ^{′2})A + A′′ = 0, 2A′φ′ + Aφ′′ = 0.  (17)

We now make use of the fact that A is slowly varying, so we drop A′′ in favor of the (K^{2} − φ^{′2})A term, which then implies that

φ′ = ±K  (18)

and therefore

φ(z) = ± ∫ ^{z} K(z′) dz′  (19)

Integrating the second part of Equation (17) yields

d ln A/dz = −\frac{1}{2}d ln φ′/dz = −\frac{1}{2}d ln K/dz,  (20)

so

A = A_{0}K^{−1/2}.  (21)

and therefore Equation (22) yields

ϕ(z) = \frac{A_{0}}{\sqrt{K}} e^{±i ∫ K(z′) dz′}  (22)

Figure 1: Eigenfunctions from WKB (red) and the eigenvalue solver.

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