On Cnoidal Waves and Bores

For a train of gravity waves of finite amplitude in a frame of reference in which they are stationary, it is suggested that, in addition to the volume flow per unit span Q and the total head R, one may usefully study a third constant S, the rate of flow of horizontal momentum (corrected for pressure force, and divided by the density). The values of Q, R and S probably determine the wave-train uniquely; this is proved explicitly for long waves in a new presentation of the 'cnoidal wave' theory (section 3). The combinations of Q, R and S which are possible in the general case are illustrated by a diagram (figure 2) in which the co-ordinates are r = R/R_c and s = S/S_c; here suffix c refers to 'critical' flow at the volume rate Q. There are two barriers on this diagram beyond which no stationary waves, or other steady flows, are possible; at the left-hand barrier (corresponding to uniform subcritical flows) the amplitude tends to zero; at the right-hand barrier (corresponding to uniform supercritical flows) the wave-length tends to infinity (case of the solitary wave). The diagram needs to be completed by a third barrier corresponding to 'waves of greatest height'. This starts at the point marked Z and moves to infinity within the unshaded region without ever intersecting the left-hand barrier. The diagram may be used to determine what losses of momentum and energy a given stream can undergo, without departing from the condition of steady flow. Thus, one can determine the maximum wave resistance on a cylindrical obstacle athwart a given subcritical stream, or on a two-dimensional 'step' in the bed, in the absence of energy dissipation. Again, one can study the transition to a wave-train behind a bore. Such a wave-train (present in all but very strong bores) is shown to be capable of absorbing almost the whole energy which according to classical theory is liberated at the bore, although some minute residual dissipation of energy still appears to be necessary.