A homogeneous Poisson process of rate $\lambda >0$ is an integer-valued counting process with $N_0=0$ almost surely, nondecreasing right-continuous paths with unit jumps, and independent stationary increments satisfyingIncrements over disjoint time intervals are independent. In particular,The finite Poisson counts ensure that there is no explosion in finite time. The increment formula also gives stochastic continuity, since the probability of any arrival in an interval of length $h$ is $1-e^{-\lambda h}\to 0$.

A material interface obeys $D(y-\eta (x,t))/Dt=0$. In each fluid this gives ${\varphi }_{jy}={\eta }_t+(U_j+{\varphi }_{jx}){\eta }_x$ at $y=\eta$, where $U_1=U$ and $U_2=V$. Discarding products of perturbations and evaluating at the undisturbed interface givesFor an inviscid irrotational fluid, unsteady Bernoulli is ${\Phi }_t+\frac{1}{2}|\nabla \Phi {|}^2+p/\rho +gy=C(t)$, with total potential $\Phi =U_{j}x+{\varphi }_j$. Subtract the undisturbed relation and linearize at the moving interface. The pressure perturbation there is $-{\rho }_j({\varphi }_{jt}+U_j{\varphi }_{jx}+g\eta )$. With no surface tension the two pressures agree, givingUse complex amplitudes and take real parts at the end. For $k>0$, solutions of Laplace's equation decaying away from the interface have the formPut ${\Omega }_1=\omega -Uk$ and ${\Omega }_2=\omega -Vk$. The kinematic conditions imply $B_1=-i{\Omega }_{1}A/k$ and $B_2=i{\Omega }_{2}A/k$. Substituting into the pressure condition givesA nonzero amplitude therefore requiresTo determine stability, let $R={\rho }_1+{\rho }_2$ and $\overline{U}=({\rho }_{1}U+{\rho }_{2}V)/R$. Completing the square yieldsFor $U\ne V$, the right-hand side is negative exactly whenThen the two frequencies have opposite nonzero imaginary parts. The root with positive imaginary part gives exponential growth through $e^{-i\omega t}$. In terms of wavelength $\lambda =2\pi /k$, the unstable range isThis is the short-wavelength Kelvin-Helmholtz instability for a density-stable interface without surface tension. Equality is the neutral threshold; if $U=V$, the gravitational term is positive and this shear instability is absent.