= Solution
Take $k>0$; reflection in $x$ supplies the result for negative <wavenumber>. Since each region has uniform base velocity, the perturbation is an incompressible, irrotational flow with a <velocity potential> satisfying the <Laplace equation>. Its decaying outer forms are
$$
\phi_+=A_+e^{-k(y-h)}e^{ikx+\sigma t},
\qquad
\phi_-=A_-e^{k(y+h)}e^{ikx+\sigma t}.
$$
For a <varicose mode of a planar jet>, reflection symmetry makes the inner potential proportional to $\cosh(ky)$; for a <sinuous mode of a planar jet>, it is proportional to $\sinh(ky)$.
At the upper interface, the linearized <kinematic boundary condition> is
$$
\sigma\eta=\partial_y\phi_+(h),
\qquad
(\sigma+ikU)\eta=\partial_y\phi_{\rm in}(h).
$$
The first equation gives $\phi_+(h)=-\sigma\eta/k$. The second gives
$$
\phi_{\rm in}(h)=\frac{\sigma+ikU}{k}\eta
\begin{cases}
\coth(kh),&\text{varicose},\\
\tanh(kh),&\text{sinuous}.
\end{cases}
$$
The linearized <Unsteady Bernoulli equation> gives $p'=-\rho(\partial_t+U\partial_x)\phi$. Applying <pressure continuity> therefore produces the <equal-density top-hat planar-jet dispersion relation>
$$
\boxed{\sigma^2+F(\sigma+ikU)^2=0,}
\qquad
F=
\begin{cases}
\coth(kh),&\text{varicose},\\
\tanh(kh),&\text{sinuous}.
\end{cases}
$$
Solving the <quadratic equation> gives
$$
\sigma=\frac{kU}{1+F}\left(\pm\sqrt F-iF\right).
$$
Equivalently,
$$
\boxed{
\begin{aligned}
\sigma_{\rm var}
&=\pm\frac{kU}{2}\sqrt{1-e^{-4kh}}
-\frac{ikU}{2}(1+e^{-2kh}),\\
\sigma_{\rm sin}
&=\pm\frac{kU}{2}\sqrt{1-e^{-4kh}}
-\frac{ikU}{2}(1-e^{-2kh}).
\end{aligned}}
$$
The plus sign is the unstable <Kelvin-Helmholtz instability> branch. In particular, the two modes have exactly the same <growth rate>,
$$
\boxed{\sigma_R(k)=\frac{kU}{2}\sqrt{1-e^{-4kh}}.}
$$
Its limiting forms are
$$
\boxed{
\sigma_R\sim U\sqrt h\,k^{3/2}\quad(kh\to0),
\qquad
\sigma_R\sim\frac{Uk}{2}\quad(kh\to\infty).}
$$
Thus neither type grows faster in this equal-density idealization, although their phase speeds differ. The unbounded large-$k$ growth is the familiar short-wave pathology of an inviscid <vortex sheet>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-2-planar-jet-growth-rate.png]
{title=Growth rate of varicose and sinuous planar-jet disturbances}
{description=The two modes have the same dimensionless growth curve, with the long-wave and short-wave asymptotes shown.}
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