Surface waves in flowing water
- 25 January 1972
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of Fluid Mechanics
- Vol. 51 (2), 209-220
- https://doi.org/10.1017/s002211207200117x
Abstract
Surface waves in flowing water and their stability are studied. With U(y) denoting the mean velocity and d the depth of water, the following results are obtained: (i) in the plane of the complex wave velocity, c = cr + ici, all eigenvalues c with a positive ci lie within a semicircle which has as its diameter the range of the velocity U(y) of the primary flow, y being the vertical co-ordinate. (ii) If U″(y) does not change sign and U is monotonic in the field of flow, singular neutral modes (for which c = U somewhere in the field of flow) are impossible and the flow is stable. (iii) If U is analytic and U″ vanishes at the point or points where U is equal to the same constant Uc and where U′ is not zero then at least one neutral mode exists with c = Uc, provided U(d) ≠ Uc. (iv) If U is monotonic and U″/(U—c) is finite and non-zero at the critical point (c real), where U″ vanishes, then the neutral mode mentioned in (iii) above is contiguous with unstable modes, (v) If U″ < 0 and U′ [ges ] 0 there are waves with c [les ] U(0), with a finite maximum wavenumber kc corresponding to c = U(0) and with c decreasing monotonically to a finite c0 for k = 0. (vi) If U″ < 0 and U′ [ges ] 0 waves of all wavenumbers can travel with c > U(d). The eigenvalue c for any k is bounded.This publication has 5 references indexed in Scilit:
- Gravity waves over a non-uniform flowJournal of Fluid Mechanics, 1969
- Stability of Liquid Flow down an Inclined PlanePhysics of Fluids, 1963
- Wave formation in laminar flow down an inclined planeJournal of Fluid Mechanics, 1957
- Gravity waves in flowing waterProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1955
- Long waves in running waterMathematical Proceedings of the Cambridge Philosophical Society, 1953