An Introduction to Nonlinear Waves (original) (raw)

A coupled-mode approach to nonlinear waves in finite depth. Viscous bottom boundary-layer flow

A weakly dissipative free-surface flow model is presented, based on the potential flow approach previously developed by the authors . The potential flow model is derived with the aid of Luke's (1967) variational principle, in conjunction with a complete vertical expansion, leading to a non-linear coupled-mode system of horizontal equations. The latter coupled-mode system models the evolution of nonlinear water waves over a general bathymetry in intermediate and shallow water depth conditions. The consistent coupledmode system has been applied to numerical investigation of families of steady travelling wave solutions in constant depth showing good agreement with known solutions both in the Stokes and the cnoidal wave regimes. In the present work, the above coupled-mode model is linked with laminar bottom boundary layer equations, permitting the investigation of viscous effects on wave propagation up to leading-order.

Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation

Wave Motion, 1993

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 e, 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 uni-directional waves, the Korteweg-de Vries (KdV) is obtained. For slowly varying bottom this uni-directionalization will be studied in detail in this pan 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 KdV-type of equation. The modification of the obtained KdV-equation shares the property of the standard KdV-equation 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.

Fully nonlinear interfacial waves in a bounded two-fluid system

Proquest Dissertations and Theses Thesis New Jersey Institute of Technology 2003 Publication Number Aai3177204 Isbn 9780542164156 Source Dissertation Abstracts International Volume 66 05 Section B Page 2612 158 P, 2003

We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of twoand three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize shortwave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions BIOGRAPHICAL SKETCH

Surface waves on shear currents: solution of the boundary-value problem

Journal of Fluid Mechanics, 1993

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terns of an infinite series in powers of a certain parameter E , which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that E be less than unity.