Wave Equation Model Research Articles

Channel waves in cross‐borehole data

Layered geological formations with large seismic velocity contrasts can effectively create channel waves in cross‐borehole seismic data. The existence of channel waves for such waveguides can be confirmed by ray tracing, wave equation modeling, and modal analysis. Channel wave arrivals are identified in cross‐borehole data recorded at Conoco’s Newkirk test facility. For these data, where velocity contrasts are about 2 to 1, tomography based on first arrival traveltimes, is limited due to problems with extreme ray bending and seismic shadow zones. However, it may be possible to extract geological information using channel wave information. The seismometer differencing method appears to be a promising approach for detecting waveguide boundaries by use of cross‐borehole data.

Weak shock interaction with a free-slip interface at low grazing angles

Theoretical/numerical investigations with the nonlinear progressive wave equation (NPE) model predict qualitative differences between behavior of a planar weak shock and a linear acoustic wave of similar time signature incident downward at small grazing angle onto a free-slip interface between two fluids. For the linear case, the model agrees well with an appropriate analytic benchmark solution. For the nonlinear cases considered, initial conditions evolve toward (a) supercritical reflection (steady wave with no Mach stem development) or (b) subcritical reflection (self-similar behavior with Mach stem development). When the horizontal phase speed c0 of the incident wave barely exceeds the linear sound speed cL of the lower medium, an intermediate state can be obtained in which the incident shock discontinuity terminates on the interface; wave energy below the interface runs ahead and reforms a deep penetrating shock ahead of and below the interface disturbance. When c0 is below cL, the shock discontinuity terminates at the interface and does not reform in the lower medium.

Seismic scalar wave equation modeling by a convolutional differentiator

Abstract By means of the inverse Fourier transform, we introduce an easy way to design convolutional differentiators. Such convolutional spatial differentiators may be used for the pseudo-spectral solution of the scalar wave equation. The tapered differentiators have the same form as a high-order finite difference (FD) operator, and share some of the advantages of both the conventional FD and Fourier methods. The differentiators have been implemented for 2-D zero-offset and common-source seismic modeling. Results using filters with as few as five points are acceptable; longer operators (e.g., 15 points) yield results of comparable quality to seismograms obtained with the Fourier method, but with considerable savings in computer time. Accuracy of the convolutional differentiator approach is appreciably better than that of the fourth-order FD method. The method can readily be extended to elastic wave field modeling.

Positivity Properties and Stability of Solitary–Wave Solutions of Model Equations For Long Waves

(1992). Positivity Properties and Stability of Solitary–Wave Solutions of Model Equations For Long Waves. Communications in Partial Differential Equations: Vol. 17, No. 1-2, pp. 1-22.

Nonlinear progressive wave equation model for transient and steady-state sound beams

NPE is a nonlinear progressive wave equation and associated computer code that yields a time domain solution for propagation in an acoustic waveguide. In the present study, the NPE equation is modified to describe axisymmetric sound beams in the paraxial approximation. The modified version of NPE is employed to describe three cases of radiation from a baffled piston: linear transient propagation, linear cw propagation, and nonlinear cw propagation. Alternative schemes to initialize the moving window that is convected by NPE are discussed for each type of problem. The linear transient signal or linear cw signal evaluated by NPE is compared to the direct prediction of the transient or steady-state Rayleigh integral. The nonlinear signal evaluated by NPE is compared to experimental data in the near and far field. The results show that NPE gives good results for all three propagation problems, in some cases at close regions where the paraxial approximation was previously believed to be inaccurate.

Monitoring of dynamic compaction by deceleration measurements : Comput Geotech V10, N3, 1990, P189–209

Abstract A method of estimating the degree and depth of improvement in the dynamic compaction of loose granular soil by matching the computed decelerations of the pounder from a numerical model to the measured decelerations is presented. This method uses a simple one-dimensional wave equation model in which the soil beneath the pounder is represented by an elastic laterally confined soil column while the surrounding soil is represented by a series of springs and dashpots. The spring simulates the dynamic soil stiffness while the dashpot accounts for the radiation damping effect. Despite the simplifying assumptions in the model, the computed results for the degree and depth of improvement show an encouraging measure of agreement with available data from a laboratory test and field measurements from a dynamic compaction site. The proposed method is potentially useful for the monitoring of dynamic compaction of loose granular soil. And like any new technique, further verifications of the approach are necessary to develop it into a reliable method.

Comparisons between model equations for long waves

Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.

Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves

Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves are proved under certain conditions. Some particular solutions of the Ito equation are obtained as an illustrative application of the obtained nonlinear superposition formula.

Finite element modeling of shallow water wave equations

Abstract The purpose of this paper is to introduce the major potentials and perspectives of applications of finite element analysis in solving the problems of shallow water wave equations. One‐dimensional and two‐dimensional shallow water wave equations will both be incorporated into the modeling procedures. For one‐dimensional flows, the models will cover the typical single channels, confluence channels system, division channels system, and natural river systems. As far as two‐dimensional flows are concerned, the overland flows are investigated. The simulation results are compared with the data obtained by physical modeling and field observation and with the results of other existing literature. The models were found to be very feasible in modeling the complex flow fields of shallow water wave equation problems.

Nonlinear model equations for waves in a stratified fluid

The present work is a theoretical study of the combined propagation of acoustic and internal waves of finite amplitude in a stratified fluid. The motion is described within the framework of a thermoviscous fluid, although relaxation effects can be readily accounted for. Model equations are derived under the assumption of weak nonlinearity. The nonlinear terms combine to form source terms in two coupled, governing equations in the pressure and vertical particle velocity or displacement. Examples are given that show effects of nonlinear coupling between acoustic and internal waves. Generation of vorticity and steady flow (acoustic streaming), and their interaction with the sound field, is also discussed briefly. [Work supported by The Norwegian Research Council for Science and Humanities (NAVF) and VISTA/STATOIL, Norway.]

Properties of Conservation Laws of Nonlinear Evolution Equations

A systematic method is developed for evaluating conservation laws of nonlinear evolution equations (NEE's) by employing the time parts of the Backlund transformations. The properties of conservation laws are then investigated in detail to obtain the informations about the structure of NEE's themselves. In particular, we focus our attention on the independence of conservation laws. The NEE's considered in this paper are the Boussinesq equation, a model equation for shallow-water waves due to Hirota and Satsuma, the Sawada-Kotera and the Kaup equations and the Ito equation. For all these equations, the inverse scattering transform problems have not been fully solved since the associated isospectral equations become higher-order ones in comparison with the usual Schrodinger equation.

Interaction of weak shocks with a simple fluid‐fluid interface: Numerical simulation

The nonlinear progressive wave equation (NPE) model [J. Acoust. Soc. Am. 81, 1406 (1987)] has been used to simulate low grazing angle, steady‐state weak shock interaction with a planar interface defined by a discontinuity in sound speed across a fluid‐fluid interface. Model simplification results from taking the density to be continuous. This condition may be relaxed when early results have been diagnosed. The theory includes a term due to distortion of the interface by the wave, although it is estimated to be insignificant in the cases considered here. For a shock incident on a fluid of higher sound speed, it is possible to give a simple theoretical estimate for a critical grazing angle below which finite amplitude wave energy does not penetrate to great distances. When the grazing angle is above this value, numerical results indicate a finite amplitude head wave, with some reradiation of energy back across the interface, and a deep penetration shock offset in space from the incident shock. [Work supported by NRL and Sandia National Lab.]

Monitoring of dynamic compaction by deceleration measurements

A method of estimating the degree and depth of improvement in the dynamic compaction of loose granular soil by matching the computed decelerations of the pounder from a numerical model to the measured decelerations is presented. This method uses a simple one-dimensional wave equation model in which the soil beneath the pounder is represented by an elastic laterally confined soil column while the surrounding soil is represented by a series of springs and dashpots. The spring simulates the dynamic soil stiffness while the dashpot accounts for the radiation damping effect. Despite the simplifying assumptions in the model, the computed results for the degree and depth of improvement show an encouraging measure of agreement with available data from a laboratory test and field measurements from a dynamic compaction site. The proposed method is potentially useful for the monitoring of dynamic compaction of loose granular soil. And like any new technique, further verifications of the approach are necessary to develop it into a reliable method.

A model equation for steady surface waves over a bump

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F>1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F<1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.

A finite element study of tidal flow data for the North Sea and English Channel

An implicit wave equation model has been applied to a data base for two-dimensional tidal flow in the North Sea and English Channel. The simulation was carried out for 190 days to allow for a direct comparison with field measurements over a one-day period and for decomposition of the tide into its fifty-one principal components. No attempt was made to tune the model to the data. The purpose of the study is to facilitate an intercomparison of numerical models of the data set as well as to demonstrate the ability of the current model to capture the flow physics.

Exact and explicit solitary wave solutions to a model equation for water waves

Exact and explicit solitary wave solutions are obtained for the higher order Korteweg-d Vries equaion, which has been obtained recently in a formal asymptotic approximation for water waves with surface tension.

Closure to “ Rational Wave Equation Model for Pile‐Driving Analysis ” by S. L. Lee, Y. K. Chow, G. P. Karunarantne, and K. Y. Wong (March, 1988, Vol. 114, No. 3)

Closure to “ <i>Rational Wave Equation Model for Pile‐Driving Analysis</i> ” by S. L. Lee, Y. K. Chow, G. P. Karunarantne, and K. Y. Wong (March, 1988, Vol. 114, No. 3)

Discussion of “ Rational Wave Equation Model for Pile‐Driving Analysis ” by S. L. Lee, Y. K. Chow, G. P. Karunarantne, and K. Y. Wong (March, 1988, Vol. 114, No. 3)

Discussion of “ <i>Rational Wave Equation Model for Pile‐Driving Analysis</i> ” by S. L. Lee, Y. K. Chow, G. P. Karunarantne, and K. Y. Wong (March, 1988, Vol. 114, No. 3)

New Type of Algebraic Solitons Expressed in Terms of Pfaffians

The algebraic N -soliton solution of a model equation for deep water waves presented by the author (J. Math. Phys. 29 (1988) 49) is shown to be expressed in terms of Pfaffians. It exhibits a new type of solution which has quite different characteristics in comparison with those of the well-known algebraic N -soliton solution of the Benjamin-Ono equation. It is also verified that the N -soliton solution satisfies certain quadratic identity for Pfaffians and this provides a direct proof of the solution without recource to mathematical induction.

Large-Scale, Low-Frequency Response on the Continental Shelf Due to Localized Atmospheric Forcing Systems

Abstract The first-order wave equation model of Gill and Schumann, which describes the flow induced on the continental shelf by the longshore wind stress, evaluated at the coast, is extended in three ways. First, we relax the nondivergent approximation and hence are able to estimate the response in the Kelvin wave mode. Second, we include the full wind stress forcing, and allow for an offshore variation in the wind stress. We also include a representation of pressure forcing. Thus we are able to describe the response due to localized forcing systems. Third, we consider the coupling of the deep-ocean region to the shelf region across a continental slope region, modeled as a discontinuity in depth at the shelf break. The results are applied to the response on the shelf due to a localized forcing system, either traveling longshore parallel to the coast, or traveling offshore normal to the coast.