16C (Fluid Dynamics)

Irrotationality gives

u_{i} = \nabla ϕ_{i}

and

\nabla^{2} ϕ_{i} = 0

. At

z = ζ

, the exact kinematic conditions are

ζ_{t} + ϕ_{i x} ζ_{x} = ϕ_{i z},

p_{1} = p_{2}

. Bernoulli equation in each fluid is

p_{i} = - ρ_{i} (ϕ_{i t} + \frac{1}{2} ∣\nabla ϕ_{i} ∣^{2} + g z) + C_{i} (t) .

Linearize at

z = 0

and write

ζ = Z e^{i (k x - ω t)}

. Decay at infinity gives

ϕ_{1} = A_{1} e^{- k z} e^{i (k x - ω t)}, ϕ_{2} = A_{2} e^{k z} e^{i (k x - ω t)} .

The kinematic conditions give

- iω Z = - k A_{1} = k A_{2}

ρ_{1} (- iω A_{1} + g Z) = h o_{2} (- iω A_{2} + g Z) .

Elimination yields

ω^{2} = g k \frac{ρ _{2} - ρ _{1}}{ρ _{1} + ρ _{2}}, F (r) = \frac{1 - r}{1 + r} .

Stable waves require the lower fluid to be denser,

ρ_{2} > ρ_{1}

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