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This paper is concerned with the 2-dim two-phase interface Euler equationlinearized at a pair of monotone shear flows in both fluids. We extend theHoward's Semicircle Theorem and study the eigenvalue distribution of thelinearized Euler system. Under certain conditions, there are exactly twoeigenvalues for each fixed wave number $k\in \mathbb{R}$ in the whole complexplane. We provide sufficient conditions for spectral instability arising fromsome boundary values of the shear flow velocity. A typical mode is theocean-air system in which the density ratio of the fluids is sufficientlysmall. We give a complete picture of eigenvalue distribution for a certainclass of shear flows in the ocean-air system.

Instability and spectrum of the linearized two-phase fluids interface problem at shear flows