On the excitation of long nonlinear water waves by a moving pressure distribution

Abstract

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.

Steady free-surface flow over spatially periodic topography

“…The dimensionless form of the steady fKdV equation is given by (Akylas 1984;Binder et al 2005;Cole 1985;Dias & Vanden-Broeck 2002;Grimshaw & Smyth 1986;Shen 1993) η xx + 9 2 η 2 − 6µη = −3σ(x), (1.1) where η(x) is the free-surface elevation relative to a uniform stream, σ(x) is a prescribed forcing representing the channel bottom topography, and µ is a real parameter related to the Froude number. Some background detail on the form of (1.1) is given in Other Supplementary Material (OSM) Appendix A.…”

Effects of channel cross-sectional geometry on long wave generation and propagation

“…In the last two decades, various studies were carried out by Wu and Wu,11,12 Akylas, 13 Ertekin, Webster and Wehausen, 14 Mei, 15 Lee, Yates and Wu, 16 Ertekin, Qian and Wehausen, 17 Teng and Wu 18,19 and others to investigate the nonlinear phenomenon of periodic production of upstreamradiating solitary waves ͑also called run-away solitons͒ by disturbances steadily moving at critical speeds in rectangular channels. Results showed that, for a rectangular channel, whether the disturbances ͑such as submerged moving objects͒ are two-or three-dimensional, the run-away solitons generated by them are invariably two-dimensional with a uniform crest across the channel.…”

Ship-induced solitary Riemann waves of depression in Venice Lagoon

“…It is at times present at fairly low Froude numbers (down to 0.13 [13]) but is much more pronounced at moderate and high depth Froude numbers. It becomes often evident as a region of depression of nearly uniform depth [14][15][16], causes the draw-down effect (squat [17][18][19][20][21][22]) usually restricted to the navigation channel and may form structures similar to undular bore [23][24][25].…”