Wave Equation Model Research Articles

Modeling of the Acoustic Wave Equation by Staggered‐Grid Finite‐Difference Schemes with High‐Order Temporal and Spatial Accuracy

Modeling of the Acoustic Wave Equation by Staggered‐Grid Finite‐Difference Schemes with High‐Order Temporal and Spatial Accuracy

An asymptotic model for the propagation of oceanic internal tides through quasi-geostrophic flow

We derive a time-averaged ‘hydrostatic wave equation’ from the hydrostatic Boussinesq equations that describes the propagation of inertia–gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave–wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents.

A nonlinear wave equation with fractional perturbation

We study a $d$-dimensional wave equation model ($2\leq d\leq 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\ldots,H_d) \in (0,1)^{d+1}$ of the noise: if $\sum_{i=0}^d H_i > d-\frac12$, then the equation can be treated directly, while in the case $d-\frac34<\sum_{i=0}^d H_i\leq d-\frac12$, the model must be interpreted in the Wick sense, through a renormalization procedure. Our arguments essentially rely on a fractional extension of the considerations of \cite{gubinelli-koch-oh} for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.

Thermodynamically viable wave equations for power law attenuation in viscoelastic media

Many complex media of great practical interest, such as in medical ultrasound, display an attenuation that increases with a power law as a function of frequency. Usually, measurements can only be taken over a limited frequency range, while wave equations often model attenuation over all frequencies. There is therefore some freedom in how the models behave outside of this limited interval, and many different wave equations have been proposed, in particular, fractional ones. In addition, it is desirable that a wave equation models physically viable media and for that two conditions have to be satisfied. The first is causality, and the second is a criterion that comes from thermodynamic considerations and implies that the relaxation modulus is a completely monotonic function. The latter implies that attenuation asymptotically cannot rise faster than frequency raised to the first power. These criteria will be explained and used to evaluate several of the ordinary and fractional wave equations that exist.

Comparison of artificial absorbing boundaries for acoustic wave equation modelling

Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.

A numerical study of rotation effect on the propagation of nonlinear internal solitary waves in the northern South China Sea

In the paper, the rotation effect on the propagation of internal solitary waves (ISWs) in the northern South China Sea (SCS) is studied by applying non-dimensional analysis and a modified two-dimensional regularized long wave equation model. First, the contribution of each term in the model equation with rotation effect is estimated. It is shown that, the rotation effect is important once the scale of the propagating ISWs is larger than 100 km. Then, the model equation is differentiated and set up for the numerical simulation of ISWs in the northern SCS. Several numerical experiments are designed, and it is found that, the rotation effect mainly affects the symmetry of the ISW crestline, the amplitudes of the leading ISW and the north/south-ward extension crestline. With the rotation effect, the ISW crestline is not symmetrical, and the ISW amplitude decreases faster. For an incident depression internal soliton with an amplitude larger than 60 m, after the leading ISW propagates a distance of 330 km, the length of ISW crestline gets larger than 150 km, the contribution of rotation effect on the variation of leading ISW amplitude can reach more than 8%; the rotation effect and the bottom topographic effect on the variation of ISW amplitudes are opposite. The contribution of rotation effect on the variation of leading ISW amplitude decreases with an increase in amplitude of incident soliton, but it increases with the extension distance from the center of ISW crestline for the northern/southern crestlines of propagating ISWs.

Parametric Structural Schemes of Piezoactuators for Nano-and Micrometric Movements at a Longitudinal Piezoeffect

Solutions to the wave equation and structural-parametric models of piezoactuators at a longitudinal piezoeffect were obtained. Effects of the geometric and physical parameters of the piezoactuators and external load on its static and dynamic characteristics were determined. The parametric structural schemes for piezoactuators for nano- and micrometric movements at a longitudinal pie-zoeffect were obtained. The transfer functions were determined. The parametric structural schemes and the transfer functions of the piezoactuators were obtained for calculation of the automatic control systems for nano- and micrometric movements with piezoac-tuators at a longitudinal piezoeffect. The static and dynamic characteristics of the piezoactuators were determined. Application of the piezoactuators solves the problems of precise matching in microelectronics and nanotechnology, compensation for the temperature and gravitational deformations, atmospheric turbulence by a wave front correction. By solving the wave equation with allowance for the corresponding equations of the piezoelectric effect, the boundary conditions on the loaded working surfaces of a piezoactuator, the strains along the coordinate axes, it is possible to construct a structural parametric model of a piezoactuator. The transfer functions and the parametric structural schemes of the piezoelectric actuator were obtained from the equations describing the corresponding structural parametric models of the piezoactuators for nano- and micrometric movements at a longitudinal piezoeffect.

Convective wave breaking in the KdV equation

The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface boundary condition. The condition for breaking can be conveniently formulated as a convective breaking criterion based on the local Froude number at the wave crest. This breaking criterion can also be applied to time-dependent situations, and one case of interest is the development of an undular bore created by an influx at a lateral boundary. It is shown that this boundary forcing leads to wave breaking in the leading wave behind the bore if a certain threshold is surpassed.

The Effect of Stress Wave Dispersion on the Drivability Analysis of Large-Diameter Monopiles

Due to the increasing need for energy from renewable resources, a large number of offshore wind farms are planned to be constructed in the near future. Despite the plethora of available foundation concepts for offshore wind turbines, the monopile foundation is the most widely adopted concept in practice. To predict the installation process for a monopile a so-called drivability study is performed. Such a study allows one to decide on a number of key parameters for the installation process, such as, the appropriate size of the hydraulic hammer, the number of hammer blows and energy input needed to reach the final penetration depth, and the induced stresses in the system. The latter is important for the prediction of the fatigue life of the pile.Currently, drivability studies are based on one-dimensional wave equation models as first proposed by Smith in the 1950s. These models are valid as long as the diameter of the pile is small compared to the excited wavelengths in the structure due to the hammer impact. For large-diameter monopiles that are currently being used in the offshore wind industry, the latter condition is not met and the effect of stress wave dispersion can no longer be neglected.In this paper the classical wave equation model is amended by an extra term which accounts for the lateral inertia of the cross-section, resulting in the so-called Rayleigh-Love rod theory. With this new model, a parametric study is performed in which the effect of stress wave dispersion on the induced stresses and the number of hammer blows needed to reach the final penetration depth are assessed. A comparison with the results obtained from the classical model is also included in order to define the applicability range of the models. It is shown that the effect of stress wave dispersion can not be neglected for a drivability study of large-diameter monopiles.

Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.

Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations

The traditional wave equation models wave propagation in an ideal conducting medium. For characterizing the wave propagation in inhomogeneous media with frequency dependent power-law attenuation, the space-time fractional wave equation appears; further incorporating the additive white Gaussian noise coming from many natural sources leads to the stochastic space-time fractional wave equation. This paper discusses the Galerkin finite element approximations for the stochastic space-time fractional wave equation forced by an additive space-time white noise. We firstly discretize the space-time additive noise, which introduces a modeling error and results in a regularized stochastic space-time fractional wave equation; then the regularity of the regularized equation is analyzed. For the discretization in space, the finite element approximation is used and the definition of the discrete fractional Laplacian is introduced. We derive the mean-squared $L^2$-norm priori estimates for the modeling error and for the approximation error to the solution of the regularized problem; and the numerical experiments are performed to confirm the estimates. For the time-stepping, we calculate the analytically obtained Mittag-Leffler type function.

Local discontinuous Galerkin schemes for a nonlinear variational wave equation modelling liquid crystals

We consider a nonlinear variational wave equation that models the dynamics of nematic liquid crystals. Discontinuous Galerkin schemes that either conserve or dissipate a discrete version of the energy associated with these equations are designed. Numerical experiments illustrating the stability and efficiency of the schemes are presented. An interesting feature of these schemes is their ability to approximate two distinct weak solutions of the underlying system.

Quasi-periodic wave solutions and two-wave solutions of the KdV–Sawada–Kotera–Ramani equation

The two-wave solutions of the KdV–Sawada–Kotera–Ramani equation are studied in this paper. By reducing this high-order wave equation into two associated solvable ordinary differential equations, we derive the two-wave solutions in the form \(u(x,t)=U(x-c_1t)+V(x-c_2t)\) which includes solitary wave solutions, periodic solutions and quasi-periodic wave solutions by letting \(c_1=c_2\). We obtain a family of new exact two-wave solutions combined by a solitary wave and a periodic wave with two different wave speeds. These new exact two-wave solutions are neither periodic nor quasi-periodic wave solutions but approximating periodic wave solutions as time tends to infinity. The process of translation of the two-wave solution combined by two solitary wave solutions is illustrated by simulation. The approach presented in this work might be applied to study the bifurcation of multi-wave solutions of some important high-order nonlinear wave model equations.

Splitting algorithms for the high-order compact finite-difference schemes in wave-equation modeling

We have developed efficient splitting algorithms for high-order compact finite-difference methods to approximate second-order space derivatives. In general, the methods’ high-order compact finite-difference schemes require the inversion of a multidiagonal matrix that is commonly less efficient. To solve this problem, we used ideas from splitting algorithms in one-way wave-equation migration that work by decomposing the multidiagonal matrix into a series of tridiagonal matrices and then subsequently solving the tridiagonal matrices. This approach results in more efficient algorithms with little loss of accuracy. The splitting algorithms can be implemented in three different ways. Our computational complexity analysis verifies that our methods can reduce the calculation burden from exponential to linear growth. Numerical experiments demonstrate the correctness and effectiveness of our algorithms.

Optimal staggered-grid finite-difference schemes by combining Taylor-series expansion and sampling approximation for wave equation modeling

High-order staggered-grid finite-difference (SFD) schemes have been universally used to improve the accuracy of wave equation modeling. However, the high-order SFD coefficients on spatial derivatives are usually determined by the Taylor-series expansion (TE) method, which just leads to great accuracy at small wavenumbers for wave equation modeling. Some conventional optimization methods can achieve high accuracy at large wavenumbers, but they hardly guarantee the small numerical dispersion error at small wavenumbers. In this paper, we develop new optimal explicit SFD (ESFD) and implicit SFD (ISFD) schemes for wave equation modeling. We first derive the optimal ESFD and ISFD coefficients for the first-order spatial derivatives by applying the combination of the TE and the sampling approximation to the dispersion relation, and then analyze their numerical accuracy. Finally, we perform elastic wave modeling with the ESFD and ISFD schemes based on the TE method and the optimal method, respectively. When the appropriate number and interval for the sampling points are chosen, these optimal schemes have extremely high accuracy at small wavenumbers, and can also guarantee small numerical dispersion error at large wavenumbers. Numerical accuracy analyses and modeling results demonstrate the optimal ESFD and ISFD schemes can efficiently suppress the numerical dispersion and significantly improve the modeling accuracy compared to the TE-based ESFD and ISFD schemes.

Mitigation of destructive fluid force on buildings due to trapping of floating debris by coastal forest during the Great East Japan tsunami

The effectiveness of coastal or inland forests in trapping floating debris produced by destructive tsunamis has long been discussed but never clarified quantitatively. To estimate the effectiveness, especially in reducing the damage to wooden buildings (houses) behind the forest, a nonlinear long wave equation model that includes not only destruction of the coastal forest but also a trap mechanism has been developed, and a simulation was conducted where trapping was observed during the 2011 Great East Japan tsunami. A post-tsunami survey of Sendai City showed that some houses just behind the coastal forest remained, although the inland-side houses were washed out. Based on the damage to houses, fragility curves as a function of moment by fluid force were analyzed. Although only the fringe of a coastal forest increases the fluid force due to a tsunami current concentration where a coastal forest is damaged, analyses using the model with and without the trapping function suggest that the debris-trapping function of a coastal forest greatly mitigates the damage observed in the post-tsunami survey, not only just behind the forest, but further inland, at around 1300 m from the coast. By including the trapping effect, the inundation height can be well reproduced, especially at around 2000–3000 m. The fluid force reduction was increased inland because some houses there remained standing in the tsunami and maintained resistance to the current.

Efficient Unsplit Perfectly Matched Layers for Finite-Element Time-Domain Modeling of Elastodynamics

AbstractThe perfectly matched layer (PML) is a highly efficient absorbing boundary used for the numerical modeling of an elastic wave equation on an unbounded domain. In this work, the authors are concerned with a second-order unsplit PML for transient elastodymanic problems in a semi-plane medium with finite-element approximations. First, based on the concept of stretched coordinates, an efficient unsplit PML formulation is proposed without higher derivatives. Then a finite-element time-domain scheme of a second-order PML in a displacement formulation is developed, in which the Galerkin method is used in space discretization and a Newmark-type scheme is employed for time stepping. Inside the absorbing layer, only one auxiliary vector is required. Hence, the scheme is cheap to implement and easily coupled with standard finite-element methods. Finally, the accuracy and efficiency of the present unsplit PML is demonstrated in numerical examples with a finite-element time-domain scheme.

Electromagneticwaves caused by tectonic earth movement

Prior to large earthquakes the Earth sends out transient signals, sometimes strong, more often subtle and fleeting. These signals may consist of local magnetic field variations, electromagnetic emissions over a wide range of frequencies, a variety of atmospheric and ionospheric phenomena. Great uncertainty exists as to the nature of the processes that could produce such signals, both inside the Earth’s crust and at the surface..When rocks are stressed, peroxy links break, releasing electronic charge carriers, h_, known as positive holes. The positive holes are highly mobile and can flow out of the stressed subvolume. The situation is similar to that in a battery. The h_ outflow is possible when the battery circuit closes. The h_ outflow constitutes an electric current, which generates magnetic field variations and low frequency EM emissions. The study of chemical processes based on a model equation for electromagnetic waves is presented. Keywords : Electromagnetic waves; pre earthquake; tectonic.

Electromagneticwaves caused by tectonic earth movement

Prior to large earthquakes the Earth sends out transient signals, sometimes strong, more often subtle and fleeting. These signals may consist of local magnetic field variations, electromagnetic emissions over a wide range of frequencies, a variety of atmospheric and ionospheric phenomena. Great uncertainty exists as to the nature of the processes that could produce such signals, both inside the Earth’s crust and at the surface..When rocks are stressed, peroxy links break, releasing electronic charge carriers, h_, known as positive holes. The positive holes are highly mobile and can flow out of the stressed subvolume. The situation is similar to that in a battery. The h_ outflow is possible when the battery circuit closes. The h_ outflow constitutes an electric current, which generates magnetic field variations and low frequency EM emissions. The study of chemical processes based on a model equation for electromagnetic waves is presented.

Comparison between model equations for long waves and blow-up phenomena

Recently, to describe the unidirectional propagation of water waves, Bona et al. [7] introduced a fifth order KdV–BBM type model(0.1)ηt+ηx−16ηxxt+δ1ηxxxxt+δ2ηxxxxx+34(η2)x+γ(η2)xxx−112(ηx2)x−14(η3)x=0, where η=η(x,t) is a real-valued function, and δ1>0, δ2,γ∈R. In this work, we plan to compare solution of the initial value problem (IVP) associated to the fifth-order KDV–BBM type model (0.1) to that of the IVP associated to the fifth-order KdV model(0.2)ut+δ3uxxxxx+c1uxuxx+c2uuxxx+c3u2ux=0, where u=u(x,t) is a real-valued function and δ3, c1, c2 and c3 are real constants with δ3≠0. This later model (0.2) was proposed by Benney in [4] to describe the interaction of long and short waves. Also, we will study the possibility of blow-up phenomenon of the fifth-order KDV–BBM type model under certain restrictions on the coefficients.