A Numerical Solution To A Nonlinear Wave Problem Using Boundary Integral Equation Method With Analysis Of Limit Cases

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)


Applied Marine Physics/Ocean Engineering


A numerical solution for periodic nonlinear irrotational surface gravity waves on a horizontal sea floor with analysis of limit cases is developed by an application of the Boundary Integral Equation Method (BIEM). The investigation includes the limit waves with or without currents. For the limit waves, the wave crest ceases to be rounded and become angled, with an included angle of 120 degrees.The nonlinear wave problem is to solve a mixed Dirichlet-Newman boundary value problem for the Laplace equation with nonlinear boundary condition on an unknown free surface boundary. Mathematically, the solution is based upon the Reimann mapping theorem, the Schwarz-Christoffel transformation, and Fatou's therem on radial limits.The existing solutions to this problem are based on Fourier expansion. Such series expansions cannot rigorously represent a limit wave since the angled wave crest with tangent discontinuity would theoretically require an infinite number of terms. The present investigation directly solves the problem in a physical domain and, as a result, the solution obtained is exact in a numerical sense.A systematic discussion of the Boundary Integral Equation Method is presented. For numerical applications, a special treatment is employed at every physical corner where discontinuity of normal derivative exists (A corner is defined as a point on a boundary domain where geometrical direction changes abruptly).The results are graphically presented. The set-down due to the existence of waves is discussed. For a limit wave without a current, the BIEM solutions are given in terms of wave period and water depth (or relative water depth). The dimensionless wave height is plotted as the variation of the relative water depth. The maximum wave steepness in deep water is found to be 0.1402. When a current simultaneously exists with a limit wave, wave properties are determined by finding an equivalent limit wave without a current. The results are presented as functions of the relative water depth and dimensionless current speed.When compared with other limit wave solutions, the BIEM solution requires a less complicated computer process, yet yields an accurate solution for limit waves with or without currents. (Abstract shortened with permission of author.)


Engineering, Marine and Ocean

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