Some Results for Surface Gravity Waves on Shear Flows

This work attempts to fill some gaps in the subject of steadysurface gravity waves on two-dimensional flows in which thevelocity varies with depth, as is the case for waves propagatingon a flowing stream. Following most previous work the theoryis basically inviscid, for the shear is assumed to be producedby external effects: the theory examines the non-viscous interactionbetween wave disturbances and the shear flow. In particular,some results are obtained for the dispersion relationship forsmall waves on a flow of arbitrary velocity distribution, andthis is generalized to include the decay from finite disturbancesinto supercritical flows. An exact operator equation is developedfor all surface gravity waves for the particular case of flowwith constant vorticity; this is solved to give first-orderequations for solitary and cnoidal waves in terms of channelflow invariants. Exact numerical solutions are obtained for small waves on sometypical shear flows, and it is shown how the theory can predictthe growth of periodic waves upon a stream by the developmentof a fully-turbulent velocity profile in flow which was originallyirrotational and supercritical. Results from all sections of this work show that shear is animportant quantity in determining the propagation behaviourof waves and disturbances. Small changes in the primary flowmay alter the nature of the surface waves considerably. Theymay in fact transform the waves from one type to another, correspondingto changes in the flow between super- and sub-critical statesdirectly caused by changes in the velocity profile.