Gravity waves on shear flows

The eigenvalue problem for gravity waves on a shear flow of depth h and noninflected velocity profile U(y) (typically parabolic) is revisited, following Burns (1953) and Yih (1972). Complementary variational formulations that provide upper and lower bounds to the Froude number F as a function of the wave speed c and wavenumber k are constructed. These formulations are used to improve Burns's longwave approximation and to determine Yih's critical wavenumber k., for which the wave is stationary (c = 0) and to which k must be inferior for the existence of an upstream running wave.