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- Title
THE BIFURCATION OF LONG WAVES IN THE PARAMETER SPACE OF PRESSURE HEAD AND FLOWFORCE.
- Authors
DOOLE, S. H.
- Abstract
The bifurcation of steady waves from (irrotational) inviscid streamflows is considered. The flux Q is scaled to unity to leave two quantities R (pressure head) and S (flowforce) parametrizinig the wavetrain. In a well-known paper, Benjamin and Lighthill (Proc. R. Soc. A 224 (1954) 448–460) presented calculations within a novel version of cnoidal wave theory which suggested that the coordinate pairs of R and S lie inside the cusped locus traced by the sub- and supercritical streamflows, and conjectured that this was the case for all irrotational water waves. Recently, the author described explicitly how wave branches, representing (Stokes) periodic waves bifurcating from the streamflow branch, point locally inside the streamflow cusp in the (R, S) diagram. In addition, accurate numerics showed how these constant-period branches extend globally towards the wave of greatest height. The main results were that the large-amplitude Stokes' branches lie surprisingly close to the subcritical stream branch, and that the transition from the ‘long-wave’ region towards the ‘deep-water’ limit is characterized by an extreme geometry of the wave branches (and how they sit inside each other). In this paper, the streamfunction formulation is used to consider the large-period wave case. A third-order ‘hypercnoidal’ expansion gives a local description of the incipient wave branch behaviour and demonstrates explicitly how these branches also point inside the streamflow cusp. In addition, this long-wave (R, S) theory predicts the largely tangential nature and non-crossing of the global large-period wave branches observed in the numerics described previously. Finally, it is shown how this tangential nature of wave branches close to the cusp persists in the presence of constant vorticity.
- Subjects
BIFURCATION theory; STREAMFLOW; STOKES equations; HAMILTONIAN systems; FLUID dynamics
- Publication
Quarterly Journal of Mechanics & Applied Mathematics, 1997, Vol 50, Issue 1, p17
- ISSN
0033-5614
- Publication type
Article
- DOI
10.1093/qjmam/50.1.17