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1.
Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order.  相似文献   

2.
This scattering is considered for the relativistic case with allowance for radiation corrections by a method previously described [1–3] for solving radial equations for the nonrelativistic case, the basis being an approximate method [4] employing trial wave functions whose parameters are found without resort to variational methods. Detailed formulas are deduced. The scattering of Klein-Gordon particles is also considered.  相似文献   

3.
The variational method in a reformulated Hamiltonian formalism of Quantum Field Theory (QFT) is used to derive relativistic n-body wave equations for scalar particles (bosons) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic n-body wave equations. The equations are shown to have the Schrödinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Some examples of approximate ground state solutions of the n-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields.  相似文献   

4.
The method of wave function expansion is adopted to study the scattering of a plane harmonic acoustic wave incident at an arbitrary angle upon an arbitrarily thick cylindrically orthotropic homogeneous cylindrical shell submerged in and filled with compressible ideal fluids. A laminate approximate model and the so-called state space formulation in conjunction with the classical transfer matrix (T-matrix) approach are employed to present an analytical solution based on the three-dimensional exact equations of anisotropic elasticity. The solution is used to correlate the perturbation in the material elastic constants of an air-filled and water-submerged aluminium cylindrical shell to the sensitivity of resonances associated with various modes of wave propagation appearing in the backscattered amplitude spectrum (i.e., axially guided, Lamb, Rayleigh and Whispering Gallery waves). The effects of shell wall thickness as well as inner fluid loading on the frequency response of the shell are also examined. A limiting case is considered and good agreement with the solution available in the literature is obtained.  相似文献   

5.
This paper proposes and validates a new formulation to investigate the dynamic response of dam-reservoir systems with upstream transmitting boundary conditions (TBCs). The mathematical derivations are provided for the new formulation as well as for exact and various approximate TBCs. The developed analytical equations can be solved numerically to assess the accuracy of a given TBC and determine the associated error independently of FEM or BEM modeling of the reservoir. The method is first validated in the case of semi-infinite reservoirs and an excellent agreement is obtained against classical techniques. The paper presents a fundamental understanding of the behavior of various TBCs and a systematic identification of their influence on the system's dynamic response, considering: (i) dam flexibility, (ii) water compressibility, (iii) reservoir bottom wave absorption, (iv) reservoir truncation length, and (v) excitation frequency. The new method is used to obtain exact error estimators to evaluate the effects of various TBCs on the dam-reservoir first resonant frequency and hydrodynamic forces acting on the dam upstream face. The proposed formulation can be programmed easily and used efficiently for rigorous assessment of classical or newly developed TBCs for vibrating dam-reservoir systems or similar fluid-structure problems.  相似文献   

6.
The propagation of a two-component laser pulse in an optically uniaxial medium is investigated under the conditions of the Zakharov-Benney resonance (viz., resonance of long and short waves). The short-wave ordinary component of the pulse, which is in resonance with the atomic subsystem, effectively generates a video pulse of the extraordinary wave (long-wave component). The latter dynamically detunes the ordinary pulse from the resonance and causes its phase modulation due to nonzero diagonal matrix elements of the dipole moment. An approximate operator approach is proposed for solving constitutive equations for the density matrix, which is equivalent to the asymptotic WKB method and makes it possible to reduce the analysis to solving a system of nonlinear wave equations for both components of the pulse. The possibility an extraordinary wave video pulse being generated with the help of a quasimonochromatic ordinary pulse with a longer wave-length. It is shown that, when the ordinary component dominates, the self-induced transparency mode is realized; in the opposite limit, the effect known as extraordinary transparency takes place. Solitary pulses corresponding to the latter case experience a decrease in the velocity of propagation, which is similar to that observed for self-induced transparency and practically do not change the population of quantum levels. Physical situations reducing the initial system of constituent and wave equations to familiar integrable models are analyzed.  相似文献   

7.
Two very efficient methods for obtaining approximate solutions to nonlinear acoustics equations are discussed. I proposed these methods earlier, but they are still little known. The first method is based on expanding an unknown function into a Taylor series with respect to the coordinate (evolution variable) and on approximate summation of the terms of this series in all orders up to the infinite order. This series can be summed completely only in particular cases, e.g., for a simple wave. It has been noted that the partial summation technique is implemented more easily if all the terms of the series are represented as corresponding topological diagrams. The second method is based on introducing a “nonlinear” phase delay (proportional to the wave amplitude) for the temporal variable in linear solutions of the problem. The application technique of these methods is illustrated by obtaining approximate solutions of the Burgers equation.  相似文献   

8.
A new Eulerian variational principle is presented for the Vlasov-Maxwell equations. This principle uses constrained variations for the Vlasov distribution in eight-dimensional extended phase space. The standard energy-momentum conservation law is then derived explicitly by the Noether method. This new variational principle can be applied to various reduced Vlasov-Maxwell equations in which fast time scales have been asymptotically eliminated (e.g., low-frequency gyrokinetic theory).  相似文献   

9.
This article presents a comparison between two approaches for implementing a variational method when calculating excited states of atoms, namely a numerical approach in which the equations arising from the requirement of an extremum of the variational functional (the Hartree—Fock equations) are solved, and an analytical approach in which the energy functional expressed in terms of analytical test functions is minimized. Both approaches are analyzed from the point of view of the approximations used to ensure that the conditions are satisfied for the complete wave function of the excited state being sought to be orthogonal to all wave functions of lower-lying energy states having the same symmetry. The well-known ATOM package is used for numerically solving the Hartree—Fock equations and the MINMAX package is used for the analytical variational calculations. It is shown that the analytical approach based on the minimax method possesses greater possibilities for taking account of relaxation effects. A comparison is made between single-electron wave functions, the matrix elements, and the energies of dipole transitions for a number of excited states of the Ne atom, as calculated using both approaches. State Pedagogical University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–128, July, 1998.  相似文献   

10.
11.
莫嘉琪  张伟江  陈贤峰 《物理学报》2009,58(11):7397-7401
研究了一类强非线性发展方程. 利用变分迭代方法,首先构造了相应的变分;其次选取了适当的初始近似;再用迭代方法得到了孤波的任意次精度的近似解. 关键词: 发展方程 非线性 孤波 近似方法.  相似文献   

12.
This paper presents a variational formulation which treats initial value problems and boundary problems in a unified manner. The basic ingredients of this theory are (1) adjoint variable and (2) unconstrained variations. It is an extension of the finite element unconstrained variational formulation used previously in solving several non-conservative stability problems. The technique which makes this extension possible is described. This formulation thus enables one to adapt such numerical techniques as the finite element method, which has had great success and popularity for solution of boundary value problems, for solutions of initial value problems as well. These formulations are given here for a forced vibration problem, a heat (mass) transfer problem and a wave propagation problem. Numerical calculations in conjunction with finite elements for two specific examples are obtained and compared with known exact solutions.  相似文献   

13.
14.
This paper investigates a novel approximate Bayesian inference procedure for numerically solving inverse problems. A hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted. The framework is of variational type, and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated. It approximates the posteriori probability distribution by separable distributions based on Kullback–Leibler divergence. Two approximations are derived within the framework, and some theoretical properties, e.g. variance estimate and consistency, are also provided. Algorithms for their efficient numerical realization are described, and their convergence properties are also discussed. Extensions to nonquadratic regularization/nonlinear forward models are also briefly studied. Numerical results for linear and nonlinear Cauchy-type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method, and compared with that by Markov chain Monte Carlo. The results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way.  相似文献   

15.
An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.  相似文献   

16.
We present two flexible stochastic models for 2D and 3D ocean waves with potential to reproduce severe and non-homogeneous sea conditions. The first family consists of generalized Lagrange models for the movements of individual water particles. These models can generate crest-trough and front-back statistically asymmetric waves, with the same degree of asymmetry as measured ocean waves. They are still in the Gaussian family and it is possible to calculate different slope distributions exactly from a wave energy spectrum. The second model is a random field model that is generated by a system of nested stochastic partial differential equations. This model can be adapted to spatially non-homogeneous sea conditions and it can approximate standard wave spectra. One advantage with this model is that Hilbert space approximations can be used to obtain computationally efficient representations with Markov-type properties that facilitate the use of sparse matrix techniques in simulation and estimation.  相似文献   

17.
石兰芳  周先春  莫嘉琪 《物理学报》2013,62(23):230202-230202
本文研究了一类非线性浅水波系统,首先构造了相应的泛函. 其次选取Lagrange乘子,再用改进的广义变分迭代方法,得到了相应模型的行波近似解析解. 关键词: 浅水波 非线性 变分迭代 近似解  相似文献   

18.
Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.  相似文献   

19.
The multiple soliton solutions of the approximate equations for long water waves and soliton-like solutions for the dispersive long-wave equations in 2+1 dimensions are constructed by using an extended homogeneous balance method. Solitary wave solutions are shown to be a special case of the present results. This method is simple and has a wide-ranging practicability, and can solve a lot of nonlinear partial differential equations.  相似文献   

20.
陈春丽  张近  李翊神 《中国物理》2007,16(8):2167-2179
An extended Boussinesq equation that models weakly nonlinear and weakly dispersive waves on a uniform layer of water is studied in this paper. The results show that the equation is not Painlev\'e-integrable in general. Some particular exact travelling wave solutions are obtained by using a function expansion method. An approximate solitary wave solution with physical significance is obtained by using a perturbation method. We find that the extended Boussinesq equation with a depth parameter of $1/\sqrt 2$ is able to match the Laitone's (1960) second order solitary wave solution of the Euler equations.  相似文献   

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