On the Development of Caustics in Shear Flows Over Rigid Walls

In this paper the scattered field produced by a source inside a shear layer of arbitrary profile, flowing above an infinite, rigid wall, is examined. The shear layer is topped by a uniform flow. A representation of the solution is obtained in terms of a pair of functions that satisfy a homogeneous second-order ordinary differential equation, with variable coefficients. The asymptotic form of these functions is obtained in the high-frequency limit. The representation yields, in this limit, a pair of parametric equations whose solutions describe the formation of infinite families of caustics inside the shear layer and downstream of the source. This is in agreement with previous results found by employing ray-theory techniques. Applications are given to the linear profile. The advantages of the present method and its application to other physical problems are discussed.