ENERGY-DISTRIBUTION FOR WAVES IN TRANSCRITICAL FLOWS OVER A BUMP

Undisturbed water in a two-dimensional long channel obtains mechanical energy from a moving bump on the bottom of the channel. When the bump moves to the left at a speed near the critical shallow water wave vel ocity (gH)(1/2), the free surface of the water consists of a soliton z one upstream, and a uniform depression zone and a wake zone downstream . Lee, Yates and Wu [J. Fluid Mech. 199, 569-593 (1989)] computed the drag on the bump and the total energy of the water waves. In this pape r, we answer the question how the total energy is distributed among th e zones of the upstream solitons, the downstream depression and the do wnstream wakes. From the energy distribution formulas derived in Secti on 3, we conclude that: (i) The energy of the downstream wake is a dec reasing function of the Froude number F and contains almost all the en ergy when F is small but still in the transcritical range; (ii) the so liton energy is an increasing function of F and contains most energy o f the system when F is large but still in the transcritical range; (ii i) the depression energy does not vary significantly with F; (iv) the soliton energy is smaller (greater) than the depression energy when th e Froude number is small (large respectively); and (v) the wake energy is greater (smaller) than the depression energy when the Froude numbe r is small (large respectively). Hence our results analytically show t hat the drag on a vessel moving at a transcritical speed is mainly due to the waves ahead of the vessel when its cruising speed is large and the waves behind the vessel when its speed is low. These conclusions agree with the pertinent concepts of moving vessel designs.