Nonlinear Transient Waves Research Articles

A boundary integral approach to linear and nonlinear transient wave scattering

In this work, we introduce a method for solving linear and nonlinear scattering problems for wave equations using a new hybrid approach. This new approach consists of a reformulation of the governing equations into a form that can be solved by a combination of a domain-based method and a boundary-integral method. Our reformulation is aimed at a situation in which we have a collection of compact scattering objects located in an otherwise homogeneous unbounded space.The domain-based method is used to propagate the equations governing the wave field inside the scattering objects forward in time. The boundary integral method is used to supply the domain-based method with the required boundary values for the wave field.In this way, the best features of both methods come into play. The response inside the scattering objects, which can be caused by both material inhomogeneity and nonlinearities, is easily considered using the domain-based method, and the boundary conditions supplied by the boundary integral method makes it possible to confine the domain-method to the inside of each scattering object.

A 3D nonlinear Maxwell’s equations solver based on a hybrid numerical method

In this paper, we explore the possibility of solving 3D Maxwell’s equations in the presence of a nonlinear and/or inhomogeneous material response. We propose using a hybrid approach, which combines a boundary integral representation with a domain-based method. This hybrid approach has previously been successfully applied to 1D linear and nonlinear transient wave scattering problems. The basic idea of the approach is to propagate Maxwell’s equations inside the scattering objects forward in time using a domain-based method, while a boundary integral representation of the electromagnetic field is used to supply the domain-based method with the Required surface values. Thus, no grids outside the scattering objects are needed, and this greatly reduces the computational cost and complexity.

Weakly nonlinear transient waves on a shear current: ring waves and skewed Langmuir rolls

We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. The shear field makes this problem three-dimensional and rotational in nature, but an analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960s and 70s for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second-order dynamics of initial value problems. To second order, irrotational wave dynamics interacts with the background vorticity field in a way that creates new vortex structures. A notable example is the large parallel vortices which drive Langmuir circulation as oblique plane waves interact with an ocean current. We also investigate the effect on wave pairs which are misaligned with the shear current to find that similar, but skewed, vortex structures are generated in every case except when the mean wave direction is precisely perpendicular to the direction of the current. This is in contrast to a conjecture by Leibovich (Annu. Rev. Fluid Mech., vol. 15, 1983, pp. 391–427). Similar nonlinear wave–shear interactions are found to also generate near-field vortex structures in the Cauchy–Poisson problem with an initial surface elevation. These interactions create further groups of dispersive ring waves in addition to those present in linear theory. The second-order solution is derived in a general manner which accommodates any initial condition through mode coupling over a continuous wave spectrum. It is therefore applicable to a range of problems including special cases of resonance. As a by-product of the general theory, a simple expression for the Stokes drift due to a monochromatic wave propagating at oblique angle with a current of uniform vorticity is derived, for the first time to our knowledge.

Development and Application of Two-Dimensional Numerical Tank using Desingularized Indirect Boundary Integral Equation Method

In this study, a two-dimensional fully nonlinear transient wave numerical tank was developed using a desingularized indirect boundary integral equation method. The desingularized indirect boundary integral equation method is simpler and faster than the conventional boundary element method because special treatment is not required to compute the boundary integral. Numerical simulations were carried out in the time domain using the fourth order Runge-Kutta method. A mixed Eulerian-Lagrangian approach was adapted to reconstruct the free surface at each time step. A numerical damping zone was used to minimize the reflective wave in the downstream region. The interpolating method of a Gaussian radial basis function-type artificial neural network was used to calculate the gradient of the free surface elevation without element connectivity. The desingularized indirect boundary integral equation using an isolated point source and radial basis function has no need for information about the element connectivity and is a meshless method that is numerically more flexible. In order to validate the accuracy of the numerical wave tank based on the desingularized indirect boundary integral equation method and meshless technique, several numerical simulations were carried out. First, a comparison with numerical results according to the type of desingularized source was carried out and confirmed that continuous line sources can be replaced by simply isolated sources. In addition, a propagation simulation of a 2nd-order Stokes wave was carried out and compared with an analytical solution. Finally, simulations of propagating waves in shallow water and propagating waves over a submerged bar were also carried and compared with published data.

Nonlinear transient waves in coupled phase oscillators with inertia.

Like the inertia of a physical body describes its tendency to resist changes of its state of motion, inertia of an oscillator describes its tendency to resist changes of its frequency. Here, we show that finite inertia of individual oscillators enables nonlinear phase waves in spatially extended coupled systems. Using a discrete model of coupled phase oscillators with inertia, we investigate these wave phenomena numerically, complemented by a continuum approximation that permits the analytical description of the key features of wave propagation in the long-wavelength limit. The ability to exhibit traveling waves is a generic feature of systems with finite inertia and is independent of the details of the coupling function.

Nonlinear transient wave–body interactions in steady uniform currents

Fully nonlinear monochromatic wave three-dimensional body interactions are studied with and without the presence of steady uniform currents. The mixed initial boundary value problem is set-up in a numerical wave tank (NWT) and solved by an indirect desingularized boundary integral equation method (DBIEM). The Laplace equation is solved at each time-step and time-dependent nonlinear free surface boundary conditions are simultaneously integrated by a mixed Eulerian–Lagrangian (MEL) method. A piston-type wave maker is used for generating the incident waves and a steady current is imposed everywhere in the computational domain at t=0 +. A spatially varying sponge layer on the free surface is devised to dissipate the outgoing waves. The resulting influence coefficient matrix is divided into sub-matrices by domain decomposition technique and solved simultaneously by block line Jacobi method (BJM). A specially devised preconditioning matrix is used to accelerate the convergence rate of the matrix equation by obtaining the minimized condition number of the influence coefficient matrix. Computations are performed for the nonlinear diffractions of steep monochromatic waves by a truncated vertical cylinder both without currents and in the presence of uniform coplanar or adverse currents. A φ n -type adaptive beach is developed and its performance compared with some beaches in the literature. Results of NWT simulations are compared with the first-order potential theory (Buchmann et al., Proceedings of the Seventh International Offshore and Polar Engineering Conference, Honolulu, Hawaii, USA, 1997), second-order diffraction theory [Kim and Yue, J. Fluid Mech. 200 (1989) 235–264], modified marker and cell (MAC) method (Park et al., International Journal for Numerical Methods in Fluids 29 (1999) 685–703) and the experimental and second-order potential theory results of Mercier and Niedzwecki [Proceedings of the Seventh International Conference on Behavior of Offshore Structures, vol. 2, TX, USA, 1994, pp. 265–287].

A Two-Dimensional Numerical Wave Flume—Part 1: Nonlinear Wave Generation, Propagation, and Absorption

A numerical model is developed to simulate fully nonlinear transient waves in a semi-infinite, two-dimensional wave tank. A mixed Eulerian-Lagrangian formulation is adopted and a high-order boundary element method is used to solve for the fluid motion at each time step. Input wave characteristics are specified at the upstream boundary of the computational domain using an appropriate wave theory. At the downstream boundary, a damping region is used in conjunction with a radiation condition to prevent wave reflections back into the computational domain. The convergence characteristics of the numerical model are studied and the numerical results are validated through a comparison with previous published data.

On the theory of pressure and temperature nonlinear waves in compressible fluid-saturated porous rocks

Thermo-poro-elastic equations describing fluid migration through fluid-saturated porous media at depth in the crust are analyzed theoretically following recent formulations of Rice and Cleary (1976), McTigue (1986) and Bonafede (1991). In this study these ideas are applied to a rather general model, namely to a deep hot and pressurized reservoir of fluid, which suddenly enters into contact with an overlaying large colder fluid-saturated layer. In a one-dimensional idealization this system can be described by two nonlinear differential heat-like equations on the matrix-fluid temperature and on the fluid overpressure over the hydrostatic value. The nonlinear couplings are due to Darcy thermal advection and to the mechanical work rate. Here we first sketch nonlinear solutions corresponding to Burgers' “solitary shock waves”, which have recently been found valid for rocks with very low fluid diffusivity. Subsequently other nonlinear transient waves are discussed, such as “thermal” and “compensated” waves, which are found to exist for every value of the parameters present in the equations involved. One interesting aspect of these mechanisms is that the resulting time-scales are particularly small. Moreover, in order to figure out the system time-evolution and the role played by the fluid diffusivity/thermal diffusivity ratio, a mechanical similitude is proposed, which we treat both analytically and numerically. Although for realistic systems these solutions are somewhat idealized, they allow one to gain fundamental insight into fluid migration mechanisms in volcanic areas and in fault regions under strong frictional heating. As already discussed by McTigue, the theory is also of interest in studying areas of nuclear waste disposal. Furthermore such a theoretical study allows one to investigate the site at depth at which such nonlinear waves are generated.

Nonlinear transient wave motions in base-excited rectangular tanks

A nonlinear, dispersive, dissipative shallow water theory in two horizontal dimensions is developed to study the waves produced by the bidirectional, translational, oscillatory motion of a rectangular tank partially filled with liquid. An efficient finite difference technique is used to solve the resulting system of equations. Numerical results are presented which illustrate the form of the generated wave profile and its development in time for a range of geometric and excitation parameters. The computed results show the influence of the direction, duration and frequency of excitation, liquid depth and dispersive effects on the wave motions.

Linear and nonlinear propagation of water wave groups

Specific waveforms with known analytical group shapes were generated, and their evolution observed, in a wave tank in the form of both transient wave groups and the cnoidal (cn) and dnoidal (dn) wave trains as derived from the nonlinear Schrödinger equation. Low‐amplitude transients behaved as predicted by linear theory. The cn and dn wave trains of moderate steepness behaved almost as predicted by the nonlinear Schrödinger equation. There is no adequate theory for the higher nonlinear transient wave groups. The functions of time at successive wave staffs were analyzed in terms of calculations of Fourier integral spectra to interpret the nonlinear behavior of these groups. Dispersed wave groups that were less nonlinear at the wave maker and that became highly nonlinear as they traveled along and coalesced provided an unusual data set. The effects of sum and difference frequencies increased. The apparent phase and group velocities increased for the higher frequencies. The Fourier integral spectra changed shape from one wave staff to the next over the entire range of frequencies that could be analyzed as the waves coalesced. In general, the spectra broadened, shifting energy to both lower and higher frequencies. This experimental exploration of the properties and evolution of transients is motivated by the possibility that the chance occurrence of steep transient wave groups on the ocean may be an important aspect of the evolution of a wind‐driven sea.

A fast algorithm for full non-linear transient waves

The transient surface wave motion is studied by a method which transfers the computational region into a unit circle. In place of the Green's formula of harmonic function the Taylor's series of an analytical function is used. The coefficients of the series are determined by FFT. The numerical filtration is accomplished in the Fourier domain.

Estimate of the time of occurrence of discontinuities in the solution of a boundary value problem for a second order quasilinear hyperbolic system: PMM vol. 36, n≗3, 1972, pp. 528–532

An estimate is obtained for the time of occurrence of a discontinuity in the solutions of a hyperbolic system of quasilinear differential equations with zero initial and continuous boundary conditions. Examples are presented for one-dimensional gasdynamics and geometrically nonlinear elasticity theory problems. Problems of the analysis of nonlinear transient waves in gasdynamics and elasticity theory result in the integration of quasilinear hyperbolic systems of differential equations. Problems of the occurrence of discontinuities in the solutions of such systems under continuous initial conditions (the Cauchy problem), as well as under continuous boundary conditions (the boundary value problem) have been examined in [1– 6], The most general result has been obtained in [4] for the Cauchy problem, for which upper and lower bounds for the time of occurrence of discontinuities in the solution of a second order system have been established by using a congruence theorem. An analogous result is obtained herein for the boundary value problem by using the Jeffrey method [4].