TY - JOUR

T1 - Supercritical surface gravity waves generated by a positive forcing

AU - Choi, J. W.

AU - Sun, S. M.

AU - Whang, S. I.

N1 - Funding Information:
The research reported here was supported by MIC, Korea, under the ITRC support program. SMS was partially supported by the National Science Foundation. JWC appreciates the support of Korea University, Korea. The authors are grateful for the comments and suggestions given by the reviewers.

PY - 2008/11

Y1 - 2008/11

N2 - Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ε{lunate} with ε{lunate} > 0 a small parameter, then a time-dependent forced Korteweg-de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ≥ 0 (or F ≥ 1, called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ0 of λ. For λ ≥ λ0 there are steady solutions, while for 0 ≤ λ < λ0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ0, the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 < λ < λ0 a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.

AB - Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ε{lunate} with ε{lunate} > 0 a small parameter, then a time-dependent forced Korteweg-de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ≥ 0 (or F ≥ 1, called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ0 of λ. For λ ≥ λ0 there are steady solutions, while for 0 ≤ λ < λ0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ0, the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 < λ < λ0 a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.

KW - Forced gravity waves

KW - Supercritical surface waves

UR - http://www.scopus.com/inward/record.url?scp=50849117783&partnerID=8YFLogxK

U2 - 10.1016/j.euromechflu.2008.01.006

DO - 10.1016/j.euromechflu.2008.01.006

M3 - Article

AN - SCOPUS:50849117783

VL - 27

SP - 750

EP - 770

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

SN - 0997-7546

IS - 6

ER -