Abstract
We consider the linear stability of inviscid parallel shear flow with a free surface. Gravity is included in the analysis, but surface tension is not. We focus on several specific profiles including Poiseuille flow and various profiles with inflection points. Instabilities are found in all the profiles we studied. At neutral limiting modes, the wave speed of the disturbance is equal to one of three choices: the fluid speed at the bottom, an extremum of the fluid speed, or an inflectional value. The results help to clarify conflicting claims in prior literature.
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