Showing papers on "Longitudinal wave published in 2007"
Book•
22 Oct 2007
TL;DR: In this article, a variational formulation of the wave equation is proposed for continuous systems with axially translating strings, and the initial value problem is solved using Laplace transform and forced vibration analysis.
Abstract: Preface . 1 Vibrations of strings and bars. 1.1 Dynamics of strings and bars: the Newtonian formulation. 1.2 Dynamics of strings and bars: the variational formulation. 1.3 Free vibration problem: Bernoulli's solution. 1.4 Modal analysis. 1.5 The initial value problem: solution using Laplace transform. 1.6 Forced vibration analysis. 1.7 Approximate methods for continuous systems. 1.8 Continuous systems with damping. 1.9 Non-homogeneous boundary conditions. 1.10 Dynamics of axially translating strings. Exercises. References. 2 One-dimensional wave equation: d'Alembert's solution. 2.1 D'Alembert's solution of the wave equation. 2.2 Harmonic waves and wave impedance. 2.3 Energetics of wave motion. 2.4 Scattering of waves. 2.5 Applications of the wave solution. Exercises. References. 3 Vibrations of beams. 3.1 Equation of motion. 3.2 Free vibration problem. 3.3 Forced vibration analysis. 3.4 Non-homogeneous boundary conditions. 3.5 Dispersion relation and flexural waves in a uniform beam. 3.6 The Timoshenko beam. 3.7 Damped vibration of beams. 3.8 Special problems in vibrations of beams. Exercises. References. 4 Vibrations of membranes. 4.1 Dynamics of a membrane. 4.2 Modal analysis. 4.3 Forced vibration analysis. 4.4 Applications: kettledrum and condenser microphone. 4.5 Waves in membranes. Exercises. References. 5 Vibrations of plates. 5.1 Dynamics of plates. 5.2 Vibrations of rectangular plates. 5.2.1 Free vibrations. 5.3 Vibrations of circular plates. 5.4 Waves in plates. 5.5 Plates with varying thickness. Exercises. References. 6 Boundary value and eigenvalue problems in vibrations. 6.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations. 6.2 Forced vibrations. 6.3 Some discretization methods for free and forced vibrations. References. 7 Waves in fluids. 7.1 Acoustic waves in fluids. 7.2 Surface waves in incompressible liquids. Exercises. References. 8 Waves in elastic continua. 8.1 Equations of motion. 8.2 Plane elastic waves in unbounded continua. 8.3 Energetics of elastic waves. 8.4 Reflection of elastic waves. 8.5 Rayleigh surface waves. 8.6 Reflection and refraction of planar acoustic waves. Exercises. References. A The variational formulation of dynamics. References. B Harmonic waves and dispersion relation. B.1 Fourier representation and harmonic waves. B.2 Phase velocity and group velocity. References. C Variational formulation for dynamics of plates. References. Index.
203 citations
TL;DR: In this paper, the second order harmonic of a Lamb wave propagating in a metallic plate is measured using a hybrid wedge generation and laser interferometric detection system, and the results show that the proposed procedure is able to characterize the inherent material nonlinearity of two different aluminum plates whose absolute nonlinearness parameters are known from longitudinal wave measurements.
Abstract: The objective of this research is to develop an accurate and reliable procedure to measure the second order harmonic of a Lamb wave propagating in a metallic plate. There are two associated complications in measuring these nonlinear Lamb waves, namely, their inherent dispersive and multimode natures. To overcome these, this research combines a time-frequency representation with a hybrid wedge generation and laser interferometric detection system. The effectiveness of the proposed procedure is demonstrated by characterizing the inherent material nonlinearity of two different aluminum plates whose absolute nonlinearity parameters are known from longitudinal wave measurements.
198 citations
TL;DR: In this article, an effective medium theory is developed which goes beyond the quasistatic limit to accurately predict the unusual properties of certain elastic metamaterials in two dimensions.
Abstract: An effective medium theory is developed which goes beyond the quasistatic limit to accurately predict the unusual properties of certain elastic metamaterials in two dimensions. The theory's validity is numerically verified through band-structure calculations for three different elastic metamaterials. The theory shows that the effective bulk modulus ${\ensuremath{\kappa}}_{e}$, shear modulus ${\ensuremath{\mu}}_{e}$, and mass density ${\ensuremath{\rho}}_{e}$ can be made negative near resonances by choosing appropriate resonant scatterers, leading to eight possible types of wave propagation. The theory not only provides a convenient tool to search for new metamaterials with desired properties, it also gives a unified physical picture of these properties. Two examples are presented: one possesses large band gaps at low frequencies; the other exhibits two regions of negative refraction, i.e., one for both longitudinal and transverse waves and the other for longitudinal waves only.
179 citations
Book•
20 Apr 2007
TL;DR: This book discusses the interaction of Elastic Longitudinal Waves, Visco-Elastic Waves, and Elastic-Visco-Plastic Waves with One-Dimensional Stress Waves in Bars, and Numerical Methods for Stress Wave Propagation.
Abstract: Chapter 1. Introduction Chapter 2. Elementary Theory of One-Dimensional Stress Waves in Bars Chapter 3. Interaction of Elastic Longitudinal Waves Chapter 4. Interaction of Elastic-Plastic Longitudinal Waves in Bars Chapter 5. Rigid Unloading Approximation Chapter 6. One-Dimensional Visco-Elastic Waves and Elastic-Visco-Plastic Waves Chapter 7. One-Dimensional Strain Plane Waves Chapter 8. Spherical Waves and Cylindrical Waves Chapter 9. Elastic Plastic Waves Propagating in Flexible Strings Chapter 10. Elastic Plastic Waves Propagating in Beams under Transverse Impact (Bending Wave Theory) Chapter 11. General Theory for Linear Elastic Waves Chapter 12. Numerical Methods for Stress Wave Propagation
175 citations
TL;DR: In this article, the experimental observation of a direct correlation between the acoustic nonlinearity measured with Lamb waves and the level of plasticity in a metal specimen was reported, which implies that even though Lamb waves are multimodal and dispersive, they will interact with a material's plasticity, similar to longitudinal and Rayleigh waves; there is a fundamental relationship between material plasticity and acoustic non-linearity that is independent of wave type.
Abstract: This letter reports on the experimental observation of a direct correlation between the acoustic nonlinearity measured with Lamb waves and the level of plasticity in a metal specimen. This correlation implies that even though Lamb waves are multimodal and dispersive, they will interact with a material’s plasticity in a manner similar to longitudinal and Rayleigh waves; there is a fundamental relationship between material plasticity and acoustic nonlinearity that is independent of wave type. As a result, Lamb waves can be used to quantitatively assess plasticity driven material damage using established higher harmonic generation techniques.
163 citations
TL;DR: In vitro experiments and an elastic wave model were used to analyze how stress is induced in kidney stones by lithotripsy and to test the roles of individual mechanisms-spallation, squeezing, and cavitation.
Abstract: In vitro experiments and an elastic wave model were used to analyze how stress is induced in kidney stones by lithotripsy and to test the roles of individual mechanisms—spallation, squeezing, and cavitation. Cylindrical U30 cement stones were treated in an HM-3-style lithotripter. Baffles were used to block specific waves responsible for spallation or squeezing. Stones with and without surface cracks added to simulate cavitation damage were tested in glycerol (a cavitation suppressive medium). Each case was simulated using the elasticity equations for an isotropic medium. The calculated location of maximum stress compared well with the experimental observations of where stones fractured in two pieces. Higher calculated maximum tensile stress correlated with fewer shock waves required for fracture. The highest calculated tensile stresses resulted from shear waves initiated at the proximal corners and strengthened along the side surfaces of the stone by the liquid-borne lithotripter shock wave. Peak tensile stress was in the distal end of the stone where fracture occurred. Reflection of the longitudinal wave from the distal face of the stone—spallation—produced lower stresses. Surface cracks accelerated fragmentation when created near the location where the maximum stress was predicted.
137 citations
TL;DR: In this article, a combination of Rayleigh and longitudinal waves is applied to evaluate crack parameters before impregnation as well as to determine the final repair effectiveness, and the proposed tomography at different frequencies, demonstrating that higher frequencies lead to more accurate characterization.
Abstract: Surface opening cracks are common defects in large civil structures like bridges. They allow penetration of water or other agents that result in loss of durability earlier than expected. Their repair can be conducted by the injection of epoxy material that seals the crack sides keeping out any aggressive substances in addition to the recovery of strength. In order to evaluate crack parameters before impregnation as well as to determine the final repair effectiveness, a combination of Rayleigh and longitudinal waves is applied. Rayleigh waves demonstrate the filling condition of the material into the shallow layer near the surface while tomography using longitudinal waves through the thickness yields information about the area inside the structure. Wave propagation dispersion features are exploited by the proposed tomography at different frequencies, demonstrating that higher frequencies lead to more accurate characterization.
127 citations
TL;DR: In this article, the authors studied the influence of the train cross-sectional shape and length on the compression wave produced by the vehicle entering the confined area, and found that the crosssectional shape was not influential as long as the blockage ratio, namely the ratio between the train and tunnel crosssectional areas, is constant.
119 citations
TL;DR: In this article, the feasibility of an absolute ultrasonic measurement of the tightening force in bolts was investigated using both longitudinal and transverse ultrasonic waves (bi-wave method) to remove the time-of-flight measurement of waves in the unstressed state.
Abstract: The feasibility of an absolute ultrasonic measurement of the tightening force in bolts was investigated. For this purpose, we used both longitudinal and transverse ultrasonic waves (bi-wave method) to remove the time-of-flight measurement of waves in the unstressed state. To avoid the systematic calibration tests (generally carried out before tightening control), an experimental method has been applied. This method allows determination of the effective length under stress and the difference between longitudinal and transverse acoustoelastic coefficients of a tightened bolt. The results show that the proposed method can provide absolute measurement of the tightening force without either loosening the bolt or performing systematically calibration tests with an uncertainty estimated on the order of ±10% at 95% reliability.
106 citations
TL;DR: In this paper, the modified spatial autocorrelation technique (MSPAC) was extended to three-component analysis (3c-MSPCA) for the determination of Love wave dispersion curves.
Abstract: SUMMARY
Analysis of seismic ambient vibrations is becoming a widespread approach to estimate subsurface shear wave velocity profiles. However, the common restriction to vertical component wavefield data does not allow investigations of Love wave dispersion and the partitioning between Rayleigh and Love waves. In this study we extend the modified spatial autocorrelation technique (MSPAC) to three-component analysis (3c-MSPAC). By determination of Love wave dispersion curves, this technique provides additional information for the determination of shear wave velocity–depth profiles. Furthermore, the relative fraction of Rayleigh waves in the total portion of surface waves on the horizontal components is estimated. Tests of the 3c-MSPAC method are presented using synthetic ambient vibration waveform data. Different types of surface waves are simulated as well as different modes. In addition, different spatial distributions of sources are used. We obtain Rayleigh and Love wave dispersion curves for broad frequency bands in agreement with the models used for waveform simulation. The same applies for the relative fraction of Rayleigh waves. Dispersion curves are observed at lower frequencies for Love waves than for Rayleigh waves. While 3c-MSPAC has clear advantages for determination of Love waves velocities, 3c-MSPAC and conventional vertical frequency–wavenumber analysis complement each other in estimating the Rayleigh wave dispersion characteristics. Inversions of the dispersion curves for the shear wave velocity–depth profile show that the use of Love wave velocities confirms the results derived from Rayleigh wave velocities. In the presence of higher mode surface waves, Love waves even can improve results. Application of 3c-MSPAC to ambient vibration data recorded during field measurements (Pulheim, Germany) show dominance of Love waves in the wavefield. Existing shear wave profiles for this site are consistent with models obtained from inversion of Rayleigh and Love wave dispersion curves.
103 citations
TL;DR: In this paper, the formation of large-focused or near-focused waves in both unidirectional and directional sea-states was investigated by applying both the fully nonlinear wave model of Bateman et al. and the Zakharov's evolution equation.
Abstract: This paper concerns the formation of large-focused or near-focused waves in both unidirectional and directional sea-states. When the crests of wave components of varying frequency superimpose at one point in space and time, a large, transient, focused wave can occur. These events are believed to be representative of the largest waves arising in a random sea and, as such, are of importance to the design of marine structures. The details of how such waves form also offer an explanation for the formation of the so-called freak or rogue waves in deep water. The physical mechanisms that govern the evolution of focused waves have been investigated by applying both the fully nonlinear wave model of Bateman et al . (Bateman et al . 2001 J. Comput. Phys . 174 , 277–305) and the Zakharov's evolution equation (Zakharov 1968 J. Appl. Mech. Tech. Phys . 9 , 190–194). Aspects of these two wave models are complementary, and their combined use allows the full nonlinearity to be considered and, at the same time, provides insights into the dominant physical processes.
In unidirectional seas, it has been shown that the local evolution of the wave spectrum leads to larger maximum crest elevations. In contrast, in directional seas, the maximum crest elevation is well predicted by a second-order theory based on the underlying spectrum, but the shape of the largest wave is not. The differences between the evolution of large waves in unidirectional and directional sea-states have been investigated by analysing the results of Bateman et al . (2001) using a number of spectral analysis techniques. It has been shown that during the formation of a focused wave event, there are significant and rapid changes to the underlying wave spectrum. These changes alter both the amplitude of the wave components and their dispersive properties. Importantly, in unidirectional sea-states, the bandwidth of the spectrum typically increases; whereas, in directional sea-states it decreases.
The changes to the wave spectra have been investigated using Zakharov's equation (1968). This has shown that the third-order resonant effects dominate changes to both the amplitude of the wave components and the dispersive properties of the wave group. While this is the case in both unidirectional and directional sea-states, the consequences are very different. By examining these consequences, directional sea-states in which large wave events that are higher and steeper than second-order theory would predict have been identified. This has implications for the types of sea-states in which rogue waves are most likely to occur.
TL;DR: Results from the first statistical study of waveform capture data during 67 interplanetary (IP) shocks with Mach numbers ranging from approximately 1-6.m are described, consistent with theory suggesting IAWs provide the primary dissipation for low Mach number shocks.
Abstract: We describe results from the first statistical study of waveform capture data during 67 interplanetary (IP) shocks with Mach numbers ranging from approximately 1-6. Most of the waveform captures and nearly 100% of the large amplitude waves were in the ramp region. Although solitary waves, Langmuir waves, and ion acoustic waves (IAWs) are all observed in the ramp region of the IP shocks, large amplitude IAWs dominate. The wave amplitude is correlated with the fast mode Mach number and with the shock strength. The observed waves produced anomalous resistivities from approximately 1-856 Omega.m (approximately 10(7) times greater than classical estimates.) The results are consistent with theory suggesting IAWs provide the primary dissipation for low Mach number shocks.
TL;DR: In this article, the authors investigated the stability of small amplitude traveling waves, both in the defocusing and the focusing case, and showed that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave.
TL;DR: In this article, the propagation of axial and flexural waves in a multi-branched framed structure and that of seismic wave in a layered medium are reviewed in detail.
TL;DR: In this article, mean velocity distributions measured in several different wave flumes are presented, involving different types of mechanical wavemakers, channels of differing sizes, and two different end conditions.
Abstract: In this paper we present mean velocity distributions measured in several different wave flumes. The flows shown involve different types of mechanical wavemakers, channels of differing sizes, and two different end conditions. In all cases, when surface waves, nominally deep-water Stokes waves, are generated, counterflowing Eulerian flows appear that act to cancel locally, i.e. not in an integral sense, the mass transport associated with the Stokes drift. No existing theory of wave–current interactions explains this behaviour, although it is symptomatic of Gerstner waves, rotational waves that are exact solutions to the Euler equations. In shallow water (kH ≈ 1), this cancellation of the Stokes drift does not hold, suggesting that interactions between wave motions and the bottom boundary layer may also come into play.
TL;DR: In this paper, internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation, where the outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall.
Abstract: Internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation. The outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall. After initializing with a linear density profile, a mixed layer forms near the wall separated from the ambient stratification by a pycnocline. Two types of internal wave are observed. Waves with frequencies larger than the free-stream buoyancy frequency are seen in the pycnocline, and vertically propagating internal waves are observed in the outer layer with characteristic frequency and wavenumber spectra. Since a signature of the pycnocline waves is observed in the frequency spectrum of the mixed layer, these waves may affect the boundary-layer turbulence. The dominant outer-layer waves have a group velocity directed 35-60° from the vertical axis, which is consistent with previous laboratory studies. The energy flux associated with the radiated waves is small compared to the integrated dissipation in the boundary layer, but is of the same order as the integrated buoyancy flux. A linear model is proposed to estimate the decay in wave amplitude owing to viscous effects. Starting from the observed wave amplitudes at the bottom of the pycnocline, the model prediction for the spectral distribution of the outer layer wave amplitude compares favourably with the simulation results.
TL;DR: The governing equations for two-dimensional homogeneous, isotropic generalized thermoelastic half-space with voids are solved in context of Lord–Shulman theory and three compressional waves and a shear vertical wave are shown to exist.
TL;DR: In this article, a robust methodology for the analytical calculation of strains in flexible buried pipelines due to surface point-source blasts is introduced, where a method used to model wave propagation with radial attenuation and spherical front is presented.
TL;DR: In this article, the authors present an experimental study at Reynolds number ∼O(100) of internal gravity waves generated by a circular cylinder that oscillates horizontally (at a frequency Ω), thus mimicking barotropic tidal flow over bottom topography.
Abstract: Oscillatory tides flowing over rough topography on the ocean floor generate internal gravity waves, which are a major source of ocean mixing. Linear inviscid theory can describe waves generated by gentle topography with slopes that are less steep than the propagation angle of the internal waves; such topography is termed subcritical. However, a clear physical picture of internal waves generated by topography with slopes steeper than the angle of internal waves (supercritical topography) is lacking. In this paper we present an experimental study at Reynolds number ∼O(100) of internal gravity waves generated by a circular cylinder that oscillates horizontally (at a frequency Ω), thus mimicking barotropic tidal flow over bottom topography. Fundamental waves of frequency Ω emanate from locations on the cylinder where the topographic slope equals the slope of internal waves. For small oscillating amplitude A (weak forcing), our experimental results compare well with predictions of the viscous linear theory of ...
TL;DR: In this paper, a wave approach based on reflection, transmission and propagation of waves is presented for the analysis of structures containing curved beam elements, where the beams are assumed to undergo only in-plane motion, which is described by the sixth-order coupled differential equations based on Flugge's theory.
TL;DR: In this article, a longitudinal wave is radiated into a liquid when the liquid is placed on a surface acoustic wave (SAW) propagating surface, and the liquid's temperature increases, as the radiated longitudinal wave becomes attenuated inside the liquid.
Abstract: A longitudinal wave is radiated into a liquid when the liquid is placed on a surface acoustic wave (SAW) propagating surface. The radiated longitudinal wave induces liquid dynamics, such as vibrating and streaming. This phenomenon is called SAW streaming. The liquid's temperature increases, as the radiated longitudinal wave becomes attenuated inside the liquid. In this paper, the measurement and observation results of temperature and streaming in liquids are described. First, the sol–gel formation of agar–agar is observed. Second, a highly viscous liquid droplet is placed on the SAW propagation surface. However, the stable measurements are difficult. Therefore, a U-type cell is developed and the pattern of the streaming in this cell is observed. The results show that there is an association between the attenuation of a longitudinal wave and the temperature distribution.
TL;DR: In this paper, the authors investigated the reflection phenomenon of a set of coupled longitudinal waves from a free plane boundary of a thermo-elastic half-space with voids.
TL;DR: In this article, the diffraction of plane harmonic compressional waves (P wave) by a spherical nanoinclusion is studied theoretically using the surface/interface elasticity theory, and the results demonstrated that when the inclusion size shrinks to nanometers, surface and interface elasticity plays a significant role in the diffusion of elastic waves.
Abstract: Effects of surfaces/interfaces become prominent in micro- and nanosized materials and devices. In the present paper, the diffraction of plane harmonic compressional waves (P wave) by a spherical nanoinclusion is studied theoretically using the surface/interface elasticity theory. The results demonstrated that when the inclusion size shrinks to nanometers, surface/interface elasticity plays a significant role in the diffraction of elastic waves. For incident waves of different frequencies, the interface effects on the dynamic stress concentration around the spherical inclusion are examined in detail.
TL;DR: In this paper, the role of the Coriolis acceleration in the equations of motion that describe the behavior of waves and to evaluate its impact on angular momentum transport was examined, where the waves excited by core convection in a 3 M�, Pop I main sequence star were analyzed.
Abstract: Context. This is the third in a series of papers that deal with angular momentum transport by internal gravity waves. We concentrate on the waves excited by core convection in a 3 M� , Pop I main sequence star. Aims. Here, we want to examine the role of the Coriolis acceleration in the equations of motion that describe the behavior of waves and to evaluate its impact on angular momentum transport. Methods. We use the so-called traditional approximation of geophysics, which allows variable separation in radial and horizontal components. In the presence of rotation, the horizontal structure is described by Hough functions instead of spherical harmonics. Results. The Coriolis acceleration has two main effects on waves. It transforms pure gravity waves into gravito-inertial waves that have a larger amplitude closer to the equator, and it introduces new waves whose restoring force is mainly the conservation of vorticity. Conclusions. Taking the Coriolis acceleration into account changes the subtle balance between prograde and retrograde waves in nonrotating stars. It also introduces new types of waves that are either purely prograde or retrograde. We show in this paper where the local deposition of angular momentum by such waves is important.
TL;DR: In this paper, the authors consider the interaction of a normally incident time-harmonic longitudinal plane wave with a circular crack imbedded in a porous medium governed by Biot's equations of dynamic poroelasticity and find that the scattering that takes place is predominantly due to wave induced fluid flow between the pores and the crack.
TL;DR: In this paper, the authors studied the acoustic wave propagation in a semi-infinite, two-dimensional periodic elastic structure, i.e., a 2D phononic crystal, with a flat surface together with a line defect.
TL;DR: In this paper, the authors examined wave propagation in microtubules based on an orthotropic elastic shell model, with particular emphasis on the role of strongly anisotropic elastic properties of micro-tubules.
Abstract: For various cellular functions of microtubules, wave propagation along microtubules is one of the issues of major concern. In this article, general behavior of wave propagation in microtubules is examined based on an orthotropic elastic shell model, with particular emphasis on the role of strongly anisotropic elastic properties of microtubules. Strong anisotropy of microtubules is found to substantially lower both torsional and radial wave velocities, although it does not affect longitudinal wave velocity. In many cases, it is found that one of three wave velocities in orthotropic microtubules depends on wave vector nonmonotonically, and reaches a minimum velocity around a specific value of the wave vector. In particular, this interesting phenomenon would not exist if microtubules were isotropic. In addition, transverse bending waves of orthotropic microtubules always correspond to the lowest wave velocity, and can be determined by the (isotropic) elastic beam model provided the wavelength is long enough. Many of the results obtained in the present article have been absent from the literature on wave propagation in microtubules.
TL;DR: In this paper, the authors developed a technique to calculate the reflection, absorption, and transmission of electromagnetic waves by a bounded plasma region, which is referred to as Scattering Matrix Model (SMM).
Abstract: This paper aims at developing a technique to calculate the reflection, absorption, and transmission of electromagnetic waves by a bounded plasma region. The model chosen for this study is a magnetized, steady-state, two-dimensional, nonuniform plasma slab, which is presented by a number of parallel flat layers. It is assumed that the electron density is constant in each layer such that the overall electron density profile across the slab follows any prescribed distribution function. The proposed technique is referred to as Scattering Matrix Model (SMM). The fields in each layer are written in the form of summation of the appropriate eigen functions weighted by unknown scattering coefficients. These coefficients are determined via the application of the appropriate boundary conditions at each interface. The effect of varying the wave frequency and the plasma parameters on the reflected, transmitted, and absorbed powers are presented and discussed.
Book•
01 Jan 2007
TL;DR: In this article, Korteweg-de Vries equation solitons and undular bores were used to propagate the KdV soliton through a variable environment, and numerical methods for periodic and internal solitary waves were proposed.
Abstract: Chapter 1: Introduction Historical introduction Korteweg-de Vries equation Solitary waves: bifurcation from the linear spectrum Chapter 2: Korteweg-de Vries equation: solitons and undular bores Introduction Periodic travelling wave solutions and solitary waves Inverse scattering transform method and solitons Whitham modulation equations Undular bores Propagation of KdV soliton through a variable environment Chapter 3: Solitary waves in water: numerical methods and results Introduction Formulation Linear solutions Weakly nonlinear results Numerical methods for periodic waves Numerical methods for solitary waves Extensions Chapter 4: Internal solitary waves Introduction Approximate evolution equations Solitons and their properties Evolution of initial disturbances Solitons in variable media Conclusion Chapter 5: Solitary waves in rotating fluids Introduction Axisymmetric inertial solitary waves Internal gravity waves in a rotating channel Chapter 6: Planetary solitary waves Introduction Coherent structures and solitons Weakly nonlocal solitary waves: slow death by radiation The dynamical isolation of the crests of cnoidal waves: Gauss' cosine-lemniscate function, imbricate series and all that An exemplary model: equatorial solitary waves in the shallow-water equations Equatorial Kelvin solitary waves Vortices embedded in a shear zone and nonlinear critical latitude theory Summary Chapter 7: Envelope solitary waves Linear waves and group velocity Weakly nonlinear waves Envelope solitons Bifurcation to solitary waves Two-dimensional solitary waves