Unidirectional waves over a slowly varying bottom, part I: derivation of a KdVtype of equation
Groesen, E. van and Pudjaprasetya, S.R. (1993) Unidirectional waves over a slowly varying bottom, part I: derivation of a KdVtype of equation. Wave Motion, 18 (4). pp. 345370. ISSN 01652125

PDF
1MB 
Abstract:  The exact equations for surface waves over an uneven bottom can be formulated as a Hamiltonian system, with the total energy of the fluid as Hamiltonian. If the bottom variations are over a length scale L that is longer than the characteristic wavelength ¿, approximating the kinetic energy for the case of "rather long and rather low" waves gives Boussinesq type of equations. If in the case of an even bottom one restricts further to unidirectional waves, the Kortewegde Vries (KdV) is obtained. For slowly varying bottom this unidirectionalization will be studied in detail in this part I, in a very direct way which is simpler than other derivations found in the literature. The surface elevation is shown to be described by a forced KdVtype of equation. The modification of the obtained KdVequation shares the property of the standard KdVequation that it has a Hamiltonian structure, but now the structure map depends explicitly on the spatial variable through the bottom topography. The forcing is derived explicitly, and the order of the forcing, compared to the first order contributions of dispersion and nonlinearity in KdV, is shown to depend on the ratio between ¿ and L; for very mild bottom variations, the forcing is negligible. For localized topography the effect of this forcing is investigated. In part II the distortion of solitary waves will be studied. 
Item Type:  Article 
Copyright:  © 1993 Elsevier Science B.V. 
Faculty:  Electrical Engineering, Mathematics and Computer Science (EEMCS) 
Research Group:  
Link to this item:  http://purl.utwente.nl/publications/30259 
Official URL:  http://dx.doi.org/10.1016/01652125(93)90065N 
Export this item as:  BibTeX EndNote HTML Citation Reference Manager 
Repository Staff Only: item control page
Metis ID: 140899