= Solution
The factor $\eta_{xx}/(1+\eta_x^2)^{3/2}$ is the signed <curvature> of the graph, so the pressure jump is surface tension times curvature, the planar <Young–Laplace equation>.
Take total velocity potentials $\Phi_-=\phi_-$ and $\Phi_+=Ux+\phi_+$. Each disturbance potential satisfies the <Laplace equation>. The interface moves with the fluid: for $F=y-\eta(x,t)$, the material equation $F_t+v\cdot\nabla F=0$ gives $\eta_t+u\eta_x=v_y$. Keeping only first-order terms and evaluating at $y=0$ yields
$$
\boxed{\phi_{-,y}=\eta_t,\qquad\phi_{+,y}=\eta_t+U\eta_x.}
$$
The unsteady <Bernoulli equation> is $\Phi_t+|\nabla\Phi|^2/2+p/\rho=C(t)$, with any common gravitational contribution canceling across this equal-density interface. Subtracting the basic flow and linearizing gives $p'_-= -\rho\phi_{-,t}$ and $p'_+=-\rho(\phi_{+,t}+U\phi_{+,x})$. The linearized pressure jump is $T\eta_{xx}$. Hence
$$
\boxed{\phi_{-,t}-(\phi_{+,t}+U\phi_{+,x})=(T/\rho)\eta_{xx}.}
$$
Use complex modes with real parts understood:
$$
\eta=Ae^{i(kx-\omega t)},\qquad
\phi_-=B_-e^{ky}e^{i(kx-\omega t)},\quad
\phi_+=B_+e^{-ky}e^{i(kx-\omega t)}.
$$
The decaying vertical factors solve the Laplace equation in their respective half-planes. The kinematic conditions give $B_-=-i\omega A/k$ and $B_+=i(\omega-kU)A/k$. Substitution in the dynamic condition gives
$$
\boxed{\omega^2+(\omega-kU)^2=\frac T\rho k^3,\qquad
\omega=\frac{kU}{2}\pm\sqrt{\frac{Tk^3}{2\rho}-\frac{k^2U^2}{4}}.}
$$
When the discriminant is nonnegative, the <phase speeds> are $U/2\pm\sqrt{Tk/(2\rho)-U^2/4}$. They are two dispersive capillary-wave branches advected by the mean velocity $U/2$.
When $\boxed{U^2>2Tk/\rho}$, the frequencies are complex conjugates and one mode grows exponentially, with growth rate $k\sqrt{U^2/4-Tk/(2\rho)}$. This is <Kelvin-Helmholtz instability>. Surface tension stabilizes sufficiently short waves, while for any nonzero shear, sufficiently long waves are unstable in this gravity-free equal-density model. A real cosine with real frequency describes the stable branches; the unstable case is represented by the real part of the corresponding growing complex mode.
Back to article page
