The pressures head and flowforce parameter space for waves with constant vorticity

Abstract

The bifurcation of steady waves from inviscid streamflows in the presence of constant vorticity is studied. The flux Q is scaled to unity which leaves two natural quantities R (pressure head) and S (flowforce) parametrizing the wavetrain. In a seminal paper, Benjamin and Lighthill presented calculations within an (irrotational) 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, using regular expressions for the streamfunction and profile, how branches representing irrotational (Stokes) periodic waves, bifurcating from the streamflow branch, point locally inside the streamflow cusp in (R, S) parameter space. In addition, accurate numerics showed how these constant-period branches extend globally towards the wave of greatest height, remaining surprisingly close to the subcritical stream branch, even for large amplitudes. We investigate the effects of constant vorticity upon this physically important parameter-space representation for the first time