Showing papers on "Longitudinal wave published in 2001"

Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches

Abstract: Shunted piezoelectric patches are periodically placed along rods to control the longitudinal wave propagation in these rods. The resulting periodic structure is capable of filtering the propagation of waves over specified frequency bands called stop bands. The location and width of the stop bands can be tuned, using the shunting capabilities of the piezoelectric materials, in response to external excitations and to compensate for any structural uncertainty. A mathematical model is developed to predict the response of a rod with periodic shunted piezoelectric patches and to identify its stop band characteristics. The model accounts for the aperiodicity, introduced by proper tuning of the shunted electrical impedance distribution along the rod. Disorder in the periodicity typically extends the stop bands into adjacent propagation zones and, more importantly, produces the localization of the vibration energy near the excitation source. The conditions for achieving localized vibration are established and the localization factors are evaluated for different levels of disorder on the shunting parameters. The numerical predictions demonstrate the effectiveness and potentials of the proposed treatment that requires no control energy and combines the damping characteristics of shunted piezoelectric films, the attenuation potentials of periodic structures, and the localization capabilities of aperiodic structures. The theoretical investigations presented in this paper provide the guidelines for designing tunable periodic structures with high control flexibility where propagating waves can be attenuated and localized.

On the deep water wave motion

Abstract: The problem of the propagation of surface waves over deep water is considered. We present a rigorous approach towards the only known (non-trivial) explicit solution to the governing equations for water waves - Gerstner's wave. Some properties of this solution, and how these relate to some basic conclusions about water waves that may be observed experimentally, are discussed.

Waves in locally periodic media

Abstract: We review the theory of wave propagation in one dimension through a medium consisting of N identical “cells.” Surprisingly, exact closed-form results can be obtained for arbitrary N. Examples include the vibration of weighted strings, the acoustics of corrugated tubes, the optics of photonic crystals, and, of course, electron wave functions in the quantum theory of solids. As N increases, the band structure characteristic of waves in infinite periodic media emerges.

The Korteweg–de Vries–Zakharov–Kuznetsov equation for electron-acoustic waves

Abstract: Motivated by a recent paper [Phys. Plasmas 7, 2987 (2000)] highlighting the potential importance of the electron-acoustic wave in interpreting the solitary waves observed by high time resolution measurements of the electric field in the auroral region, the effect of a magnetic field on weakly nonlinear electron-acoustic waves is investigated. A Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation is derived for a plasma comprised of cool and hot electrons and a species of fluid ions. Two models are employed for the ions: magnetized and unmagnetized. When the ions are magnetized the frequency constraints imposed upon the electron-acoustic wave packet prove to be too limiting to be of general use. The second model, which treats the ions as a stationary neutralizing background, overcomes the restrictions imposed by the former and is more fitting for the frequency domain of the electron-acoustic wave. Plane and ellipsoidal soliton solutions are admitted by the KdV-ZK equation, the latter perhaps able to explain some of the two dimensional features of the solitary waves observed in the Earth’s high altitude auroral region. Both models for the ions predict only negative potential solitons. It is discussed how the plasma model might be adapted to produce positive potential solitons.

Edge waves along a sloping beach

Abstract: We construct a family of explicit rotational solutions to the nonlinear governing equations for water waves, describing edge waves propagating over a planesloping beach. A detailed analysis of the edge wave dynamics and of the run-up pattern is made possible by the use of the Lagrangian approach to the water motion.

Guided waves energy velocity in absorbing and non-absorbing plates

Abstract: This paper presents a study of the velocity of the propagation of energy in guided waves in plates. The motivation of the work comes from the practical observation that the conventional approach to predicting the velocities of pulses or wave packets, that is, the simple group velocity calculation, breaks down when the guided waves are attenuative. The conventional approach is therefore not valid for guided waves in absorbing materials or for leaky waves. The paper presents a theoretical derivation of an expression to predict the energy velocity of guided waves in an isotropic plate, based on the integration of the Poynting energy vectors. When applied to modes with no attenuation, it is shown analytically from this expression that the energy velocity is always identical to the group velocity. On the other hand, when applied to attenuative modes, numerical integration of the expression to yield the true energy velocity shows that this can differ quite significantly from the group velocity. Experimental validation of the expression is achieved by measuring the velocity of wave packets in an absorbing plate, under such conditions when the energy velocity differs substantially from the group velocity. Excellent agreement is found between the predictions and the measurements. The paper also shows the Poynting vectors in the various model studies, and some interesting phenomena relating to their directions.

A laboratory study of the focusing of transient and directionally spread surface water waves

Abstract: This paper describes a new laboratory study in which a large number of waves, of varying frequency and propagating in different directions, were focused at one point in space and time to produce a large transient wave group. A focusing event of this type is believed to be representative of the evolution of an extreme ocean wave in deep water. Measurements of the water–surface elevation and the underlying water–particle kinematics are compared with both a linear solution and a second–order solution based on the sum of the interactions first identified by Longuet–Higgins & Stewart. Comparisons between these data confirm that the directionality of the wavefield has a profound effect upon the nonlinearity of a large wave event. If the sum of the wave amplitudes generated at the wave paddles is held constant, an increase in the directional spread of the wavefield leads to lower maximum crest elevations. Conversely, if the generated wave amplitudes are increased until the onset of wave breaking, at or near the focal position, an increase in the directional spread allows larger limiting waves to evolve. An explanation of these results lies in the redistribution of the wave energy within the frequency domain. In the most nonlinear wave cases, neither the water–surface elevation nor the water–particle kinematics can be explained in terms of the free waves generated at the wave paddles and their associated bound waves. Indeed, the laboratory data suggest that there is a rapid widening of the free–wave regime in the vicinity of a large wave event. For a constant input–amplitude sum, these important spectral changes are shown to be strongly dependent upon the directionality of the wavefield. These findings explain the very large water–surface elevations recorded in previous unidirectional wave studies and the apparent contrast between unidirectional results and recent field data in which large directionally spread waves were shown to be much less nonlinear. The present study clearly demonstrates the need to incorporate the directionality of a wavefield if extreme ocean waves are to be accurately modelled and their physical characteristics explained.

Dynamics of Internal Gravity Waves in the Ocean

Abstract: Foreword Preface Acknowledgments Introduction Part I: Principles of Thermohydrodynamic Description of Internal Gravity Waves in the Ocean 1 Brief Information about Oceanic Thermohydrodynamics 2 Equations of the Theory of Internal Waves Part II: Linear Theory of Internal Waves 3 Linear Theory of Propagation of Internal Waves in the Undisturbed Horizontally Homogeneous Ocean 4 Shear Flow Influence on Internal Waves Propagation 5 Propagation of Internal Waves in Horizontally Inhomogeneous Ocean 6 Basic Sources of Internal Waves Generation in the Ocean Part III: Nonlinear Theory of Internal Waves 7 Hamiltonian Formalism for the Description of Oceanic Internal Waves 8 Long Weakly Nonlinear Internal Waves 9 Propagation of Weakly Nonlinear Internal Wave Packets in the Ocean 10 Resonant Interactions of Wave Triads and Kinetic Equation for Oceanic Internal Waves' Spectrum Part IV: Some Information on Internal Wave Observations in the Ocean 11 On Statistical Description of Natural Internal Waves Data 12 Basic Experimental Facts of Internal Waves' Behaviour in the Ocean References Author Index Topic Index

Strain Solitons in Solids and How to Construct Them

Abstract: Preface Introduction List of Symbols NONLINEAR WAVES IN ELASTIC SOLIDS Basic Definitions Physical and Geometrical Nonlinearity Compressibility, Dispersion, and Disipation in Wave Guides MATHEMATICAL DESCRIPTION OF GENERAL DEFORMATION WAVE PROBLEM Action Functional and the Lagrange Formalism Coupled Equations of Long Wave Propagation One-Dimensional Quasi Hyperbolic Equation Main Assumptions and 2-D Coupled Equations Waves in a Wave Guide Embedded in External Medium DIRECT METHODS AND FORMAL SOLUTIONS Nonlinear Hyperbolic and Evolution Equations Conservation Laws Some Notices in Critical Points Analysis for an O.D.E. New Approach to a Solution for an Autonomous Dissipative Nonlinear Equation A General Theorem of Reduction Dissipative Equations with Polynomial Nonlinearity Elliptic Function Solutions to Higher Order Problems Example for a Nonlinear Reaction-Diffusion Problem NONLINEAR STRAIN WAVES IN ELASTIC WAVE GUIDES Features of Longitudinal Waves in a Rod Experiments in Nonlinear Waves in Solids Solitons in Inhomogeneous Rods Experiments in Soliton Propagation in the Non-Uniform Rod NONLINEAR WAVES IN COMPLEX WAVE GUIDES Longitudinal Nonlinear Waves in Elastic Plate Longitudinal Waves in Rods Embedded in Surrounding Medium Nonlinear Waves in Layers on the Elastic Half Space NUMERICAL SIMULATION OF SOLITARY WAVES IN SOLIDS Numerical Simulation of Non-Stationary Deformation Waves Solitary Waves in a Homogenous Rod Solitary Waves in a Nonuniform Rod Solitary Waves in Complex Rods CONCLUSIVE REMARKS AND TENTATIVE APPLICATIONS APPENDIX INDEX

Surface waves in fibre-reinforced anisotropic elastic media

Abstract: The aim of this paper is to investigate surface waves in anisotropic fibre-reinforced solid elastic media. First, the theory of general surface waves has been derived and applied to study the particular cases of surface waves — Rayleigh, Love and Stoneley types. The wave velocity equations are found to be in agreement with the corresponding classical result when the anisotropic elastic parameters tends to zero. It is important to note that the Rayleigh type of wave velocity in the fibre-reinforced elastic medium increases to a considerable amount in comparison with the Rayleigh wave velocity in isotropic materials.

Elastic waves in anisotropic laminates

Abstract: FUNDAMENTS OF WAVES IN ELASTIC SOLIDS Introduction Formulation of Longitudinal Wave in a Bar Free Wave Motion in Infinite Bars Free Wave Motion in a Finite Bar Forces Wave Motion in an Infinite Bar Forced Wave Motion in a Finite Bar Transient Waves in an Infinite Bar Remarks WAVES IN PLATES OF FUNCTIONALLY GRADED MATERIAL Introduction Element of Linear Property Variation Boundary and Continuity Conditions Transient Response Evaluation of Confluent Hypergeometric Function Examples Remarks FREE WAVE MOTION IN ANISOTROPIC LAMINATES Introduction Basic Equations Derivation of Dispersion Equation Strain Energy Distribution Examples Remarks FORCED WAVE MOTION IN COMPOSITE LAMINATES Introduction Basic Equations Boundary and Interface Conditions Displacement in the Wavenumber Domain A Technique for the Inverse Fourier Integration Response in Time Domain Poles and Complex Paths Examples Remarks CHARACTERISTICS OF WAVES IN COMPOSITE LAMINATES Introduction Dispersion Equation Group Velocities Phase Velocity Surface Phase Slowness Surface Phase Wave Surface Group Velocity Surface Group Slowness Surface Group Wave Surface Examples Remarks FREE WAVE MOTION IN ANISOTROPIC LAMINATED BARS: FINITE STRIP ELEMENT METHOD Introduction System Equation Examples Remarks FREE WAVE MOTION IN COMPOSITE LAMINATED BARS: SEMI-EXACT METHOD Introduction System Equation Examples of Harmonic Waves in Bars Edge Waves in Semi-Infinite Laminates Remarks TRANSIENT WAVES IN COMPOSITE LAMINATES Introduction HNM Formulation Equation in Wavenumber Domain Displacement in Wavenumber Domain Response in Space-Time Domain Response to Line Time-Step Load Response to Point Time-Step Load Techniques for Inverse Fourier Integral Response to Transient Load of Arbitrary Time Function Remarks WAVES IN FUNCTIONALLY GRADED PLATES Introduction Dynamic System Equation Dispersion Relation Group Velocity Response Analysis Two-Dimensional Problem Computational Procedure Dispersion Curves Transient Response to Line Time-Step Loads Remarks WAVES IN ANISOTROPIC FUNCTIONALLY GRADED PIEZOELECTRIC PLATES Introduction Basic Equations Approximated Governing Equations Equations in Transform Domain Characteristics of Waves in FBPM Plates Transient Response Analysis Interdigital Electrodes Excitation Displacement and Electrostatics Potential Response Computation Procedure Dispersion Curves Excitation of Time-Step Shear Force in y Direction Excitation of a Line Electrode Excitation of Interdigital Electrodes Remarks STRIP ELEMENT METHOD FOR STRESS WAVES IN ANISOTROPIC SOLIDS Introduction System Equation SEM for Static Problems (Flamant's Problem) SEM for Dynamic Problems Remarks WAVE SCATTERING BY CRACKS IN COMPOSITE LAMINATES Introduction Governing Differential Equations Particular Solution Application of the SEM to Cracked Laminates Solution in the Time Domain Examples of Scattered Wave Fields Characterization of Horizontal Cracks Characterization of Vertical Surface-Breaking Cracks Characterization of Middle Interior Vertical Cracks Characterization of Arbitrary Interior Vertical Cracks Remarks WAVES SCATTERING BY FLAWS IN COMPOSITE LAMINATES Introduction Applications of the SEM to Plates Containing Flaws Examples for Wave Scattering in Laminates SH Waves in Sandwich Plates Strip Element Equation for SH Waves Particular Solution Complementary Solution General Solution SH Waves Scattered by Flaws Remarks BENDING WAVES IN ANISOTROPIC LAMINATED PLATES Introduction Governing Equation Strip element Equation Assembly of Element Equations Static Problems for Orthotropic Laminated Plates Wave Motion in Anisotropic Laminated Plates CHARACTERISTICS OF WAVES IN COMPOSITE CYLINDERS Introduction Basic Equations Dispersion Relations Examples Remarks WAVE SCATTERING BY CRACKS IN COMPOSITE CYLINDERS Introduction Basic Equations Axisymmetric Strip Element Examples Remarks INVERSE IDENTIFICATION OF IMPACT LOADS USING ELASTIC WAVES Introduction Two-dimensional Line Load Two-dimensional Extended Load Three-dimensional Concentrated Load Examples Remarks INVERSE DETERMINATION OF MATERIAL CONSTANTS OF COMPOSITE LAMINATES Introduction Inverse Operation Uniform-Micro Genetic Algorithms Examples Remarks

Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. Part 1. Dynamics of steep three-dimensional waves

Abstract: We develop an efficient high-order boundary-element method with the mixed-Eulerian-Lagrangian approach for the simulation of fully nonlinear three-dimensional wave-wave and wave-body interactions. For illustration, we apply this method to the study of two three-dimensional steep wave problems. (The application to wave-body interactions is addressed in an accompanying paper: Liu, Xue & Yue 2001.) In the first problem, we investigate the dynamics of three-dimensional overturning breaking waves. We obtain detailed kinematics and full quantification of three-dimensional effects upon wave plunging. Systematic simulations show that, compared to two-dimensional waves, three-dimensional waves generally break at higher surface elevations and greater maximum longitudinal accelerations, but with smaller tip velocities and less arched front faces. For the second problem, we study the generation mechanism of steep crescent waves. We show that the development of such waves is a result of three-dimensional (class II) Stokes wave instability. Starting with two-dimensional Stokes waves with small three-dimensional perturbations, we obtain direct simulations of the evolution of both L 2 and L 3 crescent waves. Our results compare quantitatively well with experimental measurements for all the distinct features and geometric properties of such waves.

The kinetic shell model of coronal heating and acceleration by ion cyclotron waves: 2. Inward and outward propagating waves

Abstract: We extend the kinetic shell model of the cyclotron resonant interaction between coronal hole protons and outward propagating ion cyclotron waves presented in the first paper of this series [Isenberg et al., 2001]. That work showed that the resonant dissipation of outward propagating waves produced proton distributions that were unstable to the generation of inward propagating waves. Here we include the kinetic shell interaction with the inward waves, assuming that both the wave generation and wave dissipation proceed much faster than all other processes in the collisionless coronal hole. In this case, the entire proton distribution will be resonant with one set of waves or the other and is thus composed of constant-energy shells in both the sunward and antisunward regions of velocity space. The evolution of the distribution as the plasma flows away from the Sun is then described by following the motion of these shells under the action of the nonresonant forces in the coronal hole. We find that the distribution consists of a core population which circulates through velocity space and a halo which continually expands to higher energy. The halo population is shown to be essential to obtain acceleration of the bulk proton plasma through its response to the mirror force. We present an illustrative calculation of this system which assumes the waves to be dispersionless. We find for this case that the halo particles soon reach extremely high energies, leading to a continuous, rather than declining, acceleration of the plasma. We suggest that these properties are due to the dispersionless assumption and that an improved model incorporating wave dispersion may give more reasonable quantitative results.

Distortion of the Wave Profiles in an Elastoplastic Body upon Spalling

Abstract: The distortion of wave profiles in measuring the spall strength of elastoplastic materials is analyzed. An expression for the velocity of an elastic compression wave that overtakes a plastic rarefaction wave is obtained. It is shown that, depending on the ratio between the stress gradients in the plastic rarefaction wave and the overtaking compression wave, the front velocity of the compressive wave varies in the limits between the velocities of the longitudinal perturbations and the perturbations of volume expansion or compression.

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

Abstract: The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biot's theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.

Two dimensional hydrothermal waves in an extended cylindrical vessel

Abstract: We report experimental results for hydrothermal waves instability in thermocapillary-driven flow in an extended cylindrical geometry. The waves are shown to appear via a supercritical I o instability. At larger fluid depth, the conventional predictions of Smith and Davis - planar waves referred to as HW1 - are observed. For smaller depth, i.e. for larger aspect ratios, a new kind of spatial behavior is reported which may be interpreted as a new instability. This new instability is localized in the center of the cell and its spatial structure close to onset resembles targets, i.e. purely radial waves. We refer to these hydrothermal waves as HW2.

Alternate Evans Functions and Viscous Shock Waves

Abstract: The Evans function is known as a helpful tool for locating the spectrum of some variational differential operators. This is of special interest regarding the stability analysis of traveling waves, ...

On the character of acoustic waves at the interface between hard and soft solids and liquids.

Abstract: Laser ultrasonics is used to optically excite and detect acoustic waves at the interface between a liquid and a solid or coated solid. Several case studies show that this technique is feasible to investigate experimentally the theoretically predicted fundamental properties of different aspects of interface waves at liquid–solid interfaces and to characterize the elastic properties of soft solids. The theoretical prediction that the leaky Rayleigh (LR)-type root of the characteristic determinant becomes forbidden when the shear velocity of the solid lies below the bulk velocity of the liquid was experimentally confirmed. The depth profiling and nondestructive testing potential of Scholte waves was experimentally illustrated and explained by the properties of the wave displacement profile.

Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches

Abstract: Shunted piezoelectric patches are periodically placed along rods to control the longitudinal wave propagation in these rods. The resulting periodic structure is capable of filtering the propagation of waves over specified frequency bands called stop bands. The location and width of the stop bands can be tuned, using the shunting capabilities of the piezoelectric materials, in response to external excitations and to compensate for any structural uncertainty. A mathematical model is developed to predict the response of a rod with periodic shunted piezoelectric patches and to identify its stop band characteristics. The model accounts for the aperiodicity, introduced by proper tuning of the shunted electrical impedance distribution along the rod. Disorder in the periodicity typically extends the stop-bands into adjacent propagation zones and more importantly, produces the localization of the vibration energy near the excitation source. The conditions for achieving localized vibration are established and the localization factors are evaluated for different levels of disorder on the shunting parameters. The numerical predictions demonstrated the effectiveness and potentials of the proposed treatment that requires no control energy and combines the damping characteristics of shunted piezoelectric films, the attenuation potentials of periodic structures, and the localization capabilities of aperiodic structures. The theoretical investigations presented in this work provide the guidelines for designing tunable periodic structures with high control flexibility where propagating waves can be attenuated and localized.

Linear Elastic Waves

Abstract: Wave propagation and scattering are among the most fundamental processes that we use to comprehend the world around us. While these processes are often very complex, one way to begin to understand them is to study wave propagation in the linear approximation. This is a book describing such propagation using, as a context, the equations of elasticity. Two unifying themes are used. The first is that an understanding of plane wave interactions is fundamental to understanding more complex wave interactions. The second is that waves are best understood in an asymptotic approximation where they are free of the complications of their excitation and are governed primarily by their propagation environments. The topics covered include reflection, refraction, the propagation of interfacial waves, integral representations, radiation and diffraction, and propagation in closed and open waveguides. Linear Elastic Waves is an advanced level textbook directed at applied mathematicians, seismologists, and engineers.

Signatures of mode conversion and kinetic Alfvén waves at the magnetopause

Abstract: It has been suggested that resonant mode conversion of compressional MHD waves into kinetic Alfven waves at the magnetopause can explain the abrupt transition in wave polarization from compressional to transverse commonly observed during magnetopause crossings [Johnson and Cheng, 1997b]. We analyze magnetic field data for magnetopause crossings as a function of magnetic shear angle (defined as the angle between the magnetic fields in the magnetosheath and magnetosphere) and compare with the theory of resonant mode conversion. The data suggest that amplification in the transverse magnetic field component at the magnetopause is not significant up to a threshold magnetic shear angle. Above the threshold angle significant amplification results but with weak dependence on magnetic shear angle. Waves with higher frequency are less amplified and have a higher threshold angle. These observations are qualitatively consistent with theoretical results obtained from the kinetic-fluid wave equations.

Modal Analysis of Hollow Cylindrical Guided Waves and Applications

Abstract: Dispersion behavior of guided waves in hollow cylinders (cylindrical waves) was evaluated theoretically and experimentally. Observed dispersion behavior suggests an assignment, different from the traditional one, of longitudinal (L-), flexural (F-) and torsional (T-) modes which are consistent with Lamb waves and shear-horizontal (SH) mode waves. The L- and F-modes of the cylindrical waves have characteristics which are asymptotic to Lamb waves and to waves in a solid cylinder. Experimentally, wide-band cylindrical waves in aluminum pipes were generated using a laser-ultrasonic method. Wavelet transform of the cylindrical wave signals was utilized for time-frequency analysis in order to compare them with the theoretical dispersion curves. For the L(0, 1), F(1, 1), F(2, 1), L(0, 2), F(1, 2) and F(2, 2) modes of the cylindrical waves, which were efficiently excited, theoretical and experimental dispersion curves agree with each other.

Supersonic Dislocation Kinetics from an Augmented Peierls Model.

Abstract: The controversial issue of whether dislocations can travel faster than shear or longitudinal waves is investigated. The Peierls model, modified to account for drag and gradient effects, furnishes a kinetic relation between the applied shear stress and speed of uniformly moving dislocations. This relation predicts intersonic and supersonic speeds at high enough stress, but also regimes of unstable motion, in agreement with recent atomistic simulations.

Ionospheric reflection of small‐scale Alfvén waves

Abstract: The transfer of energy from the magnetosphere to the ionosphere occurs through complex mechanisms that often involve shear Alfven waves, implying that the interaction of these waves with the ionosphere is fundamental to our understanding of the coupling process. Indeed, a variety of problems that involve magnetosphere-ionosphere coupling assume that the ionosphere acts as a perfectly reflecting boundary, although we know that this is not the case in general. In this work, we calculate the reflection coefficient of shear Alfven waves incident on the ionosphere under three different ionospheric profiles. Our results indicate that 1) under all profiles considered, reflection of shear Alfven waves with perpendicular lengths of ∼ 1 km or less is negligible and 2) under a profile that might be typical of a nightside ionosphere, reflection of shear Alfven waves is significant only for waves with frequencies below ∼ 0.4 Hz and λ x above ∼ 2 km; reflection of waves with λ x less than ∼ 2 km is poor.

Experiments on focusing unidirectional water waves

Abstract: Results of four groups of experiments involving transient, mechanically generated water waves in a narrow wave tank are described. The purpose of these experiments was to investigate the limitations of the validity of linear theory predictions of the spatiotemporal structure of the surface elevation in focal regions. For unidirectional surface gravity waves, focusing occurs as a result of long waves overtaking short waves. Surprisingly, in our measurements, nonlinear effects are stronger in deep water than in intermediate depth water and are stronger in nonfocusing wave trains than in focusing wave trains. These trends can be explained by the observation that the dominant source of nonlinear interaction in our measurements was the Benjamin-Feir instability, which acts only over a limited duration in focusing wave trains, only in wave trains whose bandwidth is narrow, and only in deep water. Under conditions in which the Benjamin-Feir instability does not act (as is expected to be the case in the ocean), predictions that take into account amplitude-dependent dispersion but otherwise neglect nonlinear effects are in good agreement with measurements for wave trains with (ka)max slightly in excess of 0.30.