= Solution
To recover the printed dispersion relation, interpret the surface maintained flat as an imposed flat upper boundary for the perturbation, so its normal velocity is zero. This is the <rigid-lid approximation>. Work with perturbation <velocity potentials> $\phi_-=A e^{ky}e^{ikx+\sigma t}$ below the interface and $\phi_+=B\cosh[k(y-h)]e^{ikx+\sigma t}$ in the upper layer. They solve the <Laplace equation>, decay as $y\to-\infty$, and obey $\partial_y\phi_+(h)=0$.
The linear <kinematic boundary conditions> at the interface give $(\sigma+ikU)\eta_0=kA$ and $\sigma\eta_0=-kB\sinh(kh)$. Equality of pressures, using the linear <Bernoulli equation>, gives $(\sigma+ikU)A=\sigma B\cosh(kh)$. The equal fluid densities cancel the interface gravitational restoring term. Eliminating $A,B,\eta_0$ yields
$$
\boxed{(\sigma+ikU)^2+\sigma^2\coth(kh)=0.}
$$
With $q=\tanh(kh)$, the roots are
$$
\boxed{\sigma=\frac{Uk}{1+q}(-iq\pm\sqrt q),\qquad\operatorname{Re}\sigma=\pm\frac{Uk\sqrt{\tanh(kh)}}{1+\tanh(kh)}.}
$$
For every $k>0$ one root has positive real part. Thus the <vortex sheet> has a <Kelvin-Helmholtz instability> at every wavelength in this model. If $kh\gg1$, $q\to1$ and the growing rate is $Uk/2$. If $kh\ll1$, $q\sim kh$ and it is $Uk\sqrt{kh}$.
There is a physical qualification to the wording. A dynamically deformable <free surface> at finite gravity is not exactly a flat lid. Its linear conditions combine to $\sigma^2\phi_+(h)+g\partial_y\phi_+(h)=0$. Writing the upper potential as $B\cosh(ky)+C\sinh(ky)$ and using the same interface conditions instead gives
$$
(\sigma+ikU)^2[\sigma^2+gk\tanh(kh)]+\sigma^2[\sigma^2\tanh(kh)+gk]=0.
$$
The stated quadratic follows in the strong-gravity flat-surface limit, or with the flat condition imposed exactly. \b[It is not the exact dispersion relation of a freely moving upper surface at arbitrary finite $g$.] This makes explicit the boundary interpretation needed for the requested formula.
Back to article page
