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 velocity (gH) ‘12, the free surface of the water consists of a soliton zone 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 paper, we answer the question how the total energy is distributed among the zones of the upstream solitons, the downstream depression and the downstream wakes. From the energy distribution formulas derived in Section 3, we conclude that: (i) The energy of the downstream wake is a decreasing function of the Fmude number F and contains almost all the energy when F is small but still in the transcritical range; (ii) the soliton energy is an increasing function of F and contains most energy of the system when F is large but still in the transcritical range; (iii) the depression energy does not vary significantly with F; (iv) the soliton energy is smaller (greater) than the depression energy when, the Froude number is small (large respectively); and (v) the wake energy is greater (smaller) than tbe depression energy when the Froude number is small (large respectively). Hence our results analytically show that the drag on a vessel moving at a transcritical speed is mainly due to tbe 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.