The exact solution of a non-linear boundary-value problem of the theory of waves on the surface of a liquid of finite depth

✍ Scribed by Ye. A. Karabut

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 567 KB
• Volume: 73
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/j.jappmathmech.2009.11.006

✦ Synopsis

A non-linear boundary-value problem of the theory of waves on the surface of a heavy ideal incompressible liquid, which arises as a result of the expansion of the required functions in amplitude, taking quadratic terms into account, is investigated. A solution is constructed, on the one hand, suitable for describing long waves, and on the other, matched to the Stokes expansion (i.e., with the expansion in amplitude of the first order of infinitesimals). A function is sought which conformally maps a strip into the plane of the complex potential in the flow region. An exact solution is obtained for this problem, defined by fairly simply formulae. This solution, in the limit of long and short waves, gives linear sinusoidal waves and cnoidal waves respectively.