Surface waves in flowing water

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 = c r + i c i , all eigenvalues c with a positive c i 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 U c and where U′ is not zero then at least one neutral mode exists with c = U c , provided U(d) ≠ U c . (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 k c corresponding to c = U(0) and with c decreasing monotonically to a finite c 0 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.