Some Results for Surface Gravity Waves on Shear Flows

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Abstract
This work attempts to fill some gaps in the subject of steady surface gravity waves on two-dimensional flows in which the velocity varies with depth, as is the case for waves propagating on a flowing stream. Following most previous work the theory is basically inviscid, for the shear is assumed to be produced by external effects: the theory examines the non-viscous interaction between wave disturbances and the shear flow. In particular, some results are obtained for the dispersion relationship for small waves on a flow of arbitrary velocity distribution, and this is generalized to include the decay from finite disturbances into supercritical flows. An exact operator equation is developed for all surface gravity waves for the particular case of flow with constant vorticity; this is solved to give first-order equations for solitary and cnoidal waves in terms of channel flow invariants. Exact numerical solutions are obtained for small waves on some typical shear flows, and it is shown how the theory can predict the growth of periodic waves upon a stream by the development of a fully-turbulent velocity profile in flow which was originally irrotational and supercritical. Results from all sections of this work show that shear is an important quantity in determining the propagation behaviour of waves and disturbances. Small changes in the primary flow may alter the nature of the surface waves considerably. They may in fact transform the waves from one type to another, corresponding to changes in the flow between super- and sub-critical states directly caused by changes in the velocity profile.

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