Small Amplitude Steady Water Waves With Vorticity

The problem of describing two-dimensional traveling water waves is considered. The water region is of finite depth and the interface between the region and the air is given by the graph of a function. We assume the flow to be incompressible and neglect the effects of surface tension. However we assume the flow to be rotational so that the vorticity distribution is a given function depending on the values of the stream function of the flow. The presence of vorticity increases the complexity of the problem and also leads to a wider class of solutions. First we study unidirectional waves with vorticity and verify the Benjamin-Lighthill conjecture for flows whose Bernoulli constant is close to the critical one. For this purpose it is shown that every wave, whose slope is bounded by a fixed constant, is either a Stokes or a solitary wave. It is proved that the whole set of these waves is uniquely parametrised (up to translation) by the flow force which varies between its values for the supercritical and subcritical shear flows of constant depth. We also study large-amplitude unidirectional waves for which we prove bounds for the free-surface profile and for Bernoulli’s constant. Second, we consider small-amplitude waves over flows with counter currents. Such flows admit layers, where the fluid flows in different directions. In this case we prove that the initial nonlinear free-boundary problem can be reduced to a finite-dimensional Hamiltonian system with a stable equilibrium point corresponding to a uniform stream. As an application of this result, we prove the existence of non-symmetric wave profiles. Furthermore, using a different method, we prove the existence of periodic waves with an arbitrary number of crests per period.

Consider a two-dimensional body of water with constant density which lies below a vacuum. The ocean bed is assumed to be impenetrable, while the boundary which separates the fluid and the vacuum is assumed to be a free boundary. Under the assumption that the vorticity is only bounded and measurable, we prove that for any upstream velocity field, there exists a continuous curve of large-amplitude solitary wave solutions. This is achieved via a local and global bifurcation construction of weak solutions to the elliptic equations which constitute the steady water wave problem. We also show that such solutions possess a number of qualitative features; most significantly that each solitary wave is a symmetric, monotone wave of elevation.

Comprehensive, self-contained, and clearly written, this book describes the macroscopic equilibrium and stability of high temperature plasmas.

This overview of some of the main results and recent developments in nonlinear water waves presents fundamental aspects of the field and discusses several important topics of current research interest. It contains selected information about water-wave motion for which advanced mathematical study can be pursued, enabling readers to derive conclusions that explain observed phenomena to the greatest extent possible. The author discusses the underlying physical factors of such waves and explores the physical relevance of the mathematical results that are presented. The material is an expanded version of the author's lectures delivered at the NSF-CBMS Regional Research Conference in the Mathematical Sciences organized by the Mathematics Department of the University of Texas-Pan American in 2010.

This book was published in 2004. The Interaction of Ocean Waves and Wind describes in detail the two-way interaction between wind and ocean waves and shows how ocean waves affect weather forecasting on timescales of 5 to 90 days. Winds generate ocean waves, but at the same time airflow is modified due to the loss of energy and momentum to the waves; thus, momentum loss from the atmosphere to the ocean depends on the state of the waves. This volume discusses ocean wave evolution according to the energy balance equation. An extensive overview of nonlinear transfer is given, and as a by-product the role of four-wave interactions in the generation of extreme events, such as freak waves, is discussed. Effects on ocean circulation are described. Coupled ocean-wave, atmosphere modelling gives improved weather and wave forecasts. This volume will interest ocean wave modellers, physicists and applied mathematicians, and engineers interested in shipping and coastal protection.

When a fluid is in periodic wave motion, a fluid particle is carried by a velocity field varying from place to place. At different instants the location of the particle differs and so does the velocity field in its immediate neighborhood. As a result the time-averaged velocity of a particle may be different from the local velocity field. In particular, a fluid particle may have a net mean drift even if the local velocity field has zero mean; this is indeed the case in irrotational gravity waves. In a viscous fluid, the wave-induced Reynolds stress imparts a steady momentum to the fluid; a steady shear is set up to balance it and hence a further mean velocity field results. The sum of these two steady currents provides the total drift by which a fluid particle migrates, and is termed the mass transport velocity. It is of importance to the study of sediment motion in coastal waters. The present report describes a coordinated inquiry into both theoretical and experimental aspects of mass transport by waves. In accordance with the division of effort, it is separated into two parts. However, nearly all ideas expressed and actions taken in both parts have been influenced by extensive mutual discussions. Part I (Theory) begins with a review of the basic assumptions underlying existing theories. General formulas of mass transport velocity components throughout the Stokes boundary layer near a solid body are then derived; details of two examples are calculated. The three-dimensional mass transport distribution throughout the cross section of a wave tank is worked out for progressive waves of very small amplitudes. The effects of finite width is studied with the assumption that vorticity is diffused by molecular viscosity throughout the entire cross section. For a wave obliquely incident and reflected from a vertical sea wall, the structure in the second boundary layer between the Stokes layer and the inviscid core is investigated. This is appropriate for amplitudes much greater than the Stokes layer thickness. Part II (Experiments), were intended in part to check and to evaluate the theoretical deductions in Part I. In particular, extensive measurements were made for the longitudinal mass transport velocity in a progressive wave in a long tank with a smooth bottom. For standing waves and partially standing waves, possible features of erosion and deposition were observed by spreading (1) a small amount of sand on a smooth bottom and (2) a thick layer of sand on the bottom. The relevance of mass transport very near the bottom to the bed load transport is discussed in the light of the real beach environment.

First published in 1957, this is a classic monograph in the area of applied mathematics. It offers a connected account of the mathematical theory of wave motion in a liquid with a free surface and subjected to gravitational and other forces, together with applications to a wide variety of concrete physical problems. A never-surpassed text, it remains of permanent value to a wide range of scientists and engineers concerned with problems in fluid mechanics. The four-part treatment begins with a presentation of the derivation of the basic hydrodynamic theory for non-viscous incompressible fluids and a description of the two principal approximate theories that form the basis for the rest of the book. The second section centers on the approximate theory that results from small-amplitude wave motions. A consideration of problems involving waves in shallow water follows, and the text concludes with a selection of problems solved in terms of the exact theory. Despite the diversity of its topics, this text offers a unified, readable, and largely self-contained treatment.

This volume brings together four lecture courses on modern aspects of water waves. The intention, through the lectures, is to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. Water waves of large amplitude can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. Taking advantage of insights from physical observation, experimental evidence and numerical simulations, classical and modern mathematical approaches can be used to gain insight into their dynamics. The book presents several avenues and offers a wide range of material of current interest. The lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of water wave phenomena. In addition, some of the material can be used by those who are already familiar with one branch of the study of water waves, to learn more about other areas.