A note on reflection of spherical waves

In 1909 Sommerfeld gave an exact solution for the reflection of a spherical wave from a plane surface in terms of an oscillatory integral and also presented an asymptotic solution for the case where both source and receiver are at the boundary. Weyl (1919) presented an alternative solution and also an asymptotic solution for the case where the source is at the boundary. It is known that the general case is solved if a general solution for the case where the source is at the boundary is known. Here it is demonstrated that it is sufficient to have the general solution for the case where both source and receiver are at the boundary. This is mainly of theoretical interest, but may have practical applications. As an example it is demonstrated that Sommerfeld's approximate solution gives Ingard's (1951) approximate solution which is valid for arbitrary source and receiver heights.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

Reference-wave solutions for the high-frequency fields in inhomogeneous-background random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. For computations of the statistical measures it is therefore desirable to have a solution for the high-frequency field propagating along an isolated ray trajectory. A new reference wave is applied to obtain an analytic solution of the parabolic wave equation that describes propagation along the ray trajectory of the deterministic-background medium. The methodology is based on defining a paired-field measure as a product of an unknown field propagating in a disturbed medium and the complex-conjugate component propagating in a medium without random fluctuations. When a solution of the equation for the paired-field measure is obtained, the solution of the deterministic component can be extracted from the paired solution to determine the solution of the unknown field in an explicit form.     

Bäcklund transformations and magnetic monopoles

We show that a recent solution of an SU(2) gauge theory, claimed to be a 3-monopole solution, is in fact an infinite-energy solution.     

一类非线性方程类孤波的近似解法

采用了一个简单而有效的技巧,研究了一类扰动发展方程.首先引入求解一个相应典型方程的类孤波近似解,然后利用泛函映射方法得到了原扰动发展方程的近似解,指出了近似解级数的收敛性,并用解析方法,讨论了近似解的精度.     

Bifurcation resulting in self-focusing of the electric field near the poles of a conducting microgranule

The experimentally found phenomenon of self-focusing of an electric field and the concentration of ions near the poles of an ion-selective granule in an electrolyte solution under the action of an external electric field is theoretically justified. An asymptotic solution of the problem is obtained, and it is shown that the solution found loses stability at high field strengths and, after the corresponding bifurcation, passes into a new solution in which there is a sharp local amplification of the electric field and the ion concentration.     

各向异性介质中的椭圆空间光孤子特性

对矩形铅玻璃中椭圆孤子的形成进行了理论研究,在理论模型中引入各向异性衍射效应.采用变分法,得到了强非局域线性各向异性椭圆孤子的变分解.结果表明,各向异性衍射效应对椭圆孤子的形成有很大的影响.为了验证变分解的正确性,采用牛顿迭代法算出强非局域线性各向异性椭圆孤子的数值解,变分解和数值解符合得很好.     

Exterior and interior metrics with quadrupole moment

We present the Ernst potential and the line element of an exact solution of Einstein’s vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This opens the possibility of considering the quadrupole moment as an additional physical degree of freedom that could be used to search for a realistic exact solution, representing both the interior and exterior gravitational field generated by a self-gravitating axisymmetric distribution of mass of perfect fluid in stationary rotation.     

Improved solution for the vortical and acoustical mode coupling inside a two-dimensional cavity with porous walls

This work presents an improved solution to a former study that analyzes the oscillatory motion of gases prescribed by vortico-acoustical mode coupling inside a two-dimensional porous cavity. The physical problem arises in the context of an oscillating gas inside a rectangular enclosure with wall transpiration, sublimation, or sweating. Previously, a multiple-scale solution was derived for the temporal field. The asymptotic formulation was based on an unconventional choice of scales. Its accuracy was also commensurate with the size of epsilon, a parameter that captured the effect of small viscosity. Currently, an exact solution is derived and compared to the previous formulation. A simple WKBJ solution is also constructed for validation purposes. Unlike both asymptotic formulations, the exact solution remains accurate regardless of the range of physical parameters.     

A radiating Kerr-Newman solution

A non-static exact solution of Einstein’s equations corresponding to a field of flowing null radiation plus an electromagnetic field is presented. The geometry of the solution is described by the Kerr-Schild metric. The solution admits a shear-free, geodetic null congruence. It has the symmetry of the Kerr-Newman solution and when a certain parameter is put equal to zero the solution becomes static and reduces to the Kerr-Newman solution.     

Vibrations of double-string complex system under moving forces. Closed solutions

In this paper the dynamic response of a double-string system traversed by a constant or a harmonically oscillating moving force is considered. The force is moving with a constant velocity on the top string. The strings are identical, parallel, one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a double-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations.     

Gravitation of the Klein-Gordon scalar field

This paper presents an exact solution of the Einstein-Klein-Gordon equations in the static and spherically symmetric case and points out the differences between it and Yilmaz's solution. In addition, the essential difference between the exact solution and the post-Newtonian approximate solution is also shown.     

Solution of the mean spherical approximation for hard ions and dipoles of arbitrary size

The general solution of the mean spherical approximation (MSA) for an arbitrary mixture of hard spherical ions and dipoles, in which the ions can be of different size, is found. This solution is given in terms of three parameters that are calculated by solving an algebraic equation. Two of these parameters are scaling parameters required to satisfy the general symmetry of the pair correlation functions, and are similar to the one introduced in the solution of the MSA for an ionic mixture in earlier work. For equal size and low ionic concentration, we get a rather explicit solution of the MSA, which is formally similar to the Waisman-Lebowitz solution of the restricted primitive model, but with a concentration-dependent dielectric constant.Supported in part through NSF grant 77-04597.     

On multimeron solutions of the classical Yang-Mills equation

The double-meron solution and the solution indicated by Protogenov [1] of the SU(2) Yang-Mills equation in four-dimensional euclidean space are generalised. It is noted that the double-meron solution has an analogous solution in Minkowski space which has a finite energy integral.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

An elastodynamic solution of finite long orthotropic hollow cylinder under torsion impact

The paper presents an analytical method to solve the elastodynamic problem of a finite-length orthotropic hollow cylinder subjected to a torsion impact often occurring in engineering fields. The elastodynamic solution is composed of a quasi-static solution of homogeneous equation satisfied with the non-homogeneous boundary condition and a dynamic solution of non-homogeneous equation satisfied with homogeneous boundary condition. The quasi-static solution can be obtained by directly solving the quasi-static equation satisfied with the non-homogeneous boundary condition. The solution of a non-homogeneous dynamic equation is obtained by means of a finite Hankel transform to a radial variable r, Laplace transform to a time variable t and finite Fourier transform to an axial variable z. Thus, the elastodynamic solution of the finite length of an orthotropic hollow cylinder subjected to a torsion impact is obtained. On the other hand, a dynamic finite element for the same problem is also carried out by applying the ANSYS finite-element analysis system. Comparing the theoretical solution with finite-element solution, it can be found that two kinds of results obtained by making use of two different solving methods are suitably approached. Therefore, it is further concluded that the methods and computing processes of the theoretical solution are effective and accurate.     

Unified approach to the conservation laws and soliton solution for sine-gordon system

It is possible to generate an infinite number of conserved quantities and the most general soliton solution in an arbitrary background with the help of the Darboux-Backlund transformation and an expansion of the Lax eigenfunction in the eigenvalue parameter. Use is not made of the Riccati form of the Lax equation which is used in the usual derivation of the conserved quantities. It is shown that for zero seed solution one retrieves the usual one-soliton solution.     

Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method

In this study,by means of homotopy perturbation method(HPM) an approximate solution of the magnetohydrodynamic(MHD) boundary layer flow is obtained.The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve.HPM produces analytical expressions for the solution to nonlinear differential equations.The obtained analytic solution is in the form of an infinite power series.In this work,the analytical solution obtained by using only two terms from HPM soluti...     

The virtual mass of an oblate-ellipsoidal bubble

A semi-analytical solution is presented for the virtual mass coefficient of an oblate-ellipsoidal bubble rising in liquid. The solution was found to be a function of Weber number. The present solution compared well with the numerical solution of the lattice Boltzmann method and the analytical solutions of other investigators.