= Solution
Write $V_0=\Delta U/2$, $z_\pm=\pm h/2$, $B=g\Delta\rho/(2\rho_0)$ and $d=B/(2k)$. The velocity is linear through all three layers, so $U(z_\pm)=\pm V_0$ and $U'$ has no jump. Each interface has density drop $\Delta\rho/2$. The <jump conditions for stratified inviscid shear flow> become
$$
[\widehat w]=0,\qquad [\widehat w']=-\frac{B}{(U(z_\pm)-c)^2}\widehat w(z_\pm).
$$
Between interfaces, the <Taylor–Goldstein equation> is $\widehat w''-k^2\widehat w=0$. A decaying solution can therefore be written
$$
\widehat w(z)=b_+e^{-k|z-z_+|}+b_-e^{-k|z-z_-|}.
$$
Its derivative jumps are $-2kb_\pm$, while its values at the interfaces are $b_\pm+e^{-kh}b_\mp$. Hence the coefficients satisfy
$$
\begin{pmatrix}(V_0-c)^2-d&-de^{-kh}\\-de^{-kh}&(-V_0-c)^2-d\end{pmatrix}\begin{pmatrix}b_+\\b_-\end{pmatrix}=0.
$$
The determinant must vanish. With $\widetilde c=c/V_0$, $\alpha=kh/2$ and $J=g\Delta\rho h/(\rho_0\Delta U^2)$, one has $d/V_0^2=J/(2\alpha)$. Expanding gives the <dispersion relation for two density interfaces in uniform shear>:
$$
\boxed{\widetilde c^4-\left(2+\frac J\alpha\right)\widetilde c^2+\frac{(2\alpha-J)^2-J^2e^{-4\alpha}}{4\alpha^2}=0.}
$$
Back to article page
