Pseudo interface waves observed at the fluid/porous-medium interface. A comparison of two methods

At the fluid/porous-medium interface the pseudo-Rayleigh (pR) and pseudo-Stoneley (pSt) waves exist. The relation with the corresponding poles in the slowness plane is not unambiguous, depending on the choice of branch cuts. For a point-force excitation, the far-field Green's functions are computed using vertical branch cuts (method I) implying that the pR- and pSt-poles obey the radiation condition. Then, a separate pseudo interface wave is entirely captured by the corresponding pole residue because the loop integral along a branch cut contributes to a body wave only. When hyperbolic branch cuts are used (method II) the poles lie on the "principal" Riemann sheet. Then, also the loop integrals necessarily contribute to the pR-wave because the pR-pole is different from that in method I. They do not contribute to the pSt-wave when the pSt-pole lies on the principal Riemann sheet because the pole is identical to that in method I. When the pSt-pole has migrated to another Riemann sheet, however, the pSt-wave is fully captured by the loop integrals. In conclusion, the phase velocity and attenuation of a separate pseudo interface wave can be computed from the pole location in method I, but should be extracted from the full response in method II.     

A physically important class of integrals expressed as a parameter derivative

The paper deals with a class of integrals, the integrands of which contain the square of a solution of a second-order linear, ordinary differential equation. Such integrals often arise in quantum mechanics as normalization integrals or expectation values. A generalized, unified procedure for rewriting such an integral, associated with a differential equation of the Sturm-Liouville type with unspecified boundary conditions, as a parameter derivative is presented. The formula thus obtained can be used for the evaluation of various integrals of physical interest. As an application we present a simplified derivation of a formula given by de Alfaro and Regge, in which the quantal normalization integral is expressed in terms of the Jost function. Other applications to integrals involving special functions and to integrals associated with the one-dimensional Schrödinger equation are also presented. Furthermore, it is explained why an approximate formula for expectation values is much more accurate than one can expect from the usual, crude derivation of it, and why certain attempts to improve that derivation have failed.     

Influence of a magnetically permeable surface layer on transient fields for a thin circular loop antenna

The transient fields, in the time-domain, of a thin circular loop antenna, on a two-layered earth’s model are reexamined when the usually neglected magnetic permeability contrast is considered. It is shown that for a two-layered earth model, where the upper layer is permeable, the transient fields are modified over the nonpermeable case. The fields in the time domain are obtained as the inverse Laplace transforms of derived full wave time-harmonic solution. These time-domain solutions are obtained as a summation of waveguide modes plus contributions from branch cuts in the complex plane of the longitudinal wave number. The results should be useful for interpreting airborne electromagnetic systems and in cases where super-paramagnetic mineral constant is present.     

Numerical integration of an inhomogeneous boundary value problem

A numerical method is developped for the integration of stiff inhomogeneous coupled ordinary differential equations. It is shown how to compute integrals with integrands containing the solution of the differential equations. The method is stable and avoids storing of large amounts of intermediate results. The present method can also be applied to problems involving solutions of homogeneous coupled equations.     

An integral representation of the Green function for a linear array of acoustic point sources

We present a new algorithm for the evaluation of the quasi-periodic Green function for a linear array of acoustic point sources such as those arising in the analysis of line array loudspeakers. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate these acoustic fields. However as we show, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedentedly large frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with several classical methods both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance.     

Cylindrical wave diffraction at a half-plane in an anisotropic plasma

The problem of cylindrical wave diffraction at a half-plane located in a cold anisotropic plasma is considered. The exact solution is shown to be expressible in terms of Fresnel integrals.The author thanks M. S. Bobrovnikov for suggesting the problem and his interest.     

Color-kinematics duality for Sudakov form factor in non-supersymmetric pure Yang–Mills theory

We study the duality between color and kinematics for the Sudakov form factors of $\mathrm{tr}({F}^{2})$ in non-supersymmetric pure Yang-Mills theory. We construct the integrands that manifest the color-kinematics duality up to two loops. The resulting numerators are given in terms of Lorentz products of momenta and polarization vectors, which have the same powers of loop momenta as that from the Feynman rules. The integrands are checked by d-dimensional unitarity cuts and are valid in any dimension. We find that massless-bubble and tadpole topologies are needed at two loops to realize the color-kinematics duality. Interestingly, the two-loop solution contains a large number of free parameters suggesting the duality may hold at higher loop orders.     

On the acoustic field in a Pekeris waveguide with attenuation in the bottom half-space

The acoustic field in a Pekeris channel with an attenuating basement is critically examined, based on contour integrations of the wave number integrals for the two domains. In both regions, the field consists of a finite sum of proper (square integrable) normal modes plus a branch line integral around a hyperbolic cut. For low bottom attenuation, only "trapped" modes exist but as the loss increases additional "dissipation" modes contribute to the mode sum. A Newton-Raphson iterative procedure is introduced for finding the complex eigenvalues of the modes and a new expression is derived which shows that the total number of proper (trapped plus dissipation) modes supported by the waveguide increases essentially linearly with rising bottom attenuation. Approximations for the complex shape functions of the modes in the water column and the basement are developed and compared with the exact shape functions determined from the Newton-Raphson procedure. An expression derived for the modal attenuation coefficient scales in proportion to the square of the mode number and inversely with the square of the frequency. Stationary-phase approximations for the branch line integrals for both domains are developed, which serve to illustrate the asymptotic range dependence of the lateral wave but otherwise have little utility.     

Travelling wave solutions to nonlinear physical models by means of the first integral method

This paper presents the first integral method to carry out the integration of nonlinear partial differential equations in terms of travelling wave solutions. For illustration, three important equations of mathematical physics are analytically investigated. Through the established first integrals, exact solutions are successfully constructed for the equations considered.     

Calculation of the Coulomb electrostatic potential created by a uniformly charged square on its plane: Exact mathematical formulas

We calculate exactly the Coulomb electrostatic potential created by a uniformly charged square on its plane. The chosen solution method leads to certain types of integral expressions which involve combinations of products of error functions and power functions. These type of integrals that should be solved are not standard ones, therefore, the integration formulas we derive in the process may be deemed of mathematical interest on their own right. Use of these formulas enables us to derive an exact analytic expression for the Coulomb electrostatic potential created by a uniformly charged square at an arbitrary point on its plane.     

孔隙地层震电测井波场分波分析

由于孔道的双电层结构和流体-固体耦合运动, 弹性波在孔隙地层中传播时会引起电磁场. 前人提出了基于这种耦合效应的震电测井方法, 但对井中震电波的特性缺少细致的分析. 本文对孔隙地层充流体井孔中点声源激发的震电耦合波的分波进行分析, 证明了井孔流体电磁波波数不是井内震电波场支点, 计算了地层电磁波支点、横波支点和快纵波支点对应的侧面波. 在利用辐角原理求出震电波场函数的复极点之后, 针对典型砂岩地层计算了震电波场模式波(包括泄漏模式)的频散曲线和声压、电场激发曲线. 计算了各个侧面波和模式波的电场激发强度与声压激发强度的比值(电声激发比), 发现纵波的电声激发比相对于横波和模式波更高, 而Stoneley波电声激发比的辐角对渗透率敏感, 其敏感性会随频率增大而增加, 随孔隙度增大而降低.     

Nonperturbative model of Liouville gravity

We formulate nonperturbative 2D gravity in the framework of Liouville theory. In particular, we express the specific heat of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. We recognize the relevant divisors on moduli spaces and write the integrands in terms of the Liouville action. We evaluate the integrals (rational intersections) and show that satisfies the Painlevé I.     

基于倾斜相位的抗振动干涉面形测量

环境振动会在干涉测量过程中产生随机倾斜、移相误差,导致测量精度下降.为了降低环境振动对移相干涉测量的影响,提出了一种基于倾斜相位的抗振动干涉面形测量方法.首先,利用Fourier变换将干涉图变换到频域;然后,利用频域细分操作对峰值坐标进行亚像素精度定位,求解出振动倾斜平面;最后,利用最小二乘法计算出待测面的相位分布.实...     

Analysis of Radiation Phenomena in a Wedge Ended Dielectric Slab Waveguide

Radiation phenomena observed in a wedge shape ended dielectric slab waveguide are analyzed using mode matching technique. The case of transverse electric polarization (TE) being parallel to dielectric slab waveguide is assumed. In order to describe the fields in the wedge region, a stack of dielectric plates is assumed and in each layer the fields are expanded in terms of the mixed spectrum of guided and radiated modes. A similar expansion is used in the constant thickness slab waveguide while in free space medium a continuous-radiation mode expansion is used. Then a mode matching approach is applied, incorporating the orthogonality properties of mixed spectrum modes, in order to compute the wave fields inside the dielectric slab waveguide and wedge medium. Mode matching is achieved by discretizing the continuous radiation mode spectrum leading into a numerically stable solution provided a sufficient large number of points are used to convert integrals into finite summations. Numerical computations are carried out for various wedge geometries and shapes including linear and exponential profiles.     

A new algorithm for anisotropic solutions

We establish a new algorithm that generates a new solution to the Einstein field equations, with an anisotropic matter distribution, from a seed isotropic solution. The new solution is expressed in terms of integrals of an isotropic gravitational potential; and the integration can be completed exactly for particular isotropic seed metrics. A good feature of our approach is that the anisotropic solutions necessarily have an isotropic limit. We find two examples of anisotropic solutions which generalise the isothermal sphere and the Schwarzschild interior sphere. Both examples are expressed in closed form involving elementary functions only.     

水平层状介质中电磁场并矢Green函数的一种快速新算法

本文利用提取直射波并结合自适应数字滤波等技术提出一种计算水平层状介质中电磁场并矢Green函数的快速算法. 首先将谱域Green函数中表征均匀介质作用的直射波提取出来并对其积分进行解析计算,这种处理降低了谱域Green函数的奇异性,可在很大程度上缩短其积分收敛区间. 然后在将谱域Green函数剩余部分对应积分转化为三个快速下降积分的基础上,引入一种自适应数字滤波算法对其进行快速求解. 最后通过具体算例验证了本文所述算法的有效性. 关键词： 并矢Green函数 快速算法 水平层状介质     

Line-integral solution for the stress and displacement fields of an arbitrary dislocation segment in isotropic bi-materials in 3D space

The solutions for the stress and displacement fields due to an arbitrary dislocation segment in an isotropic bi-material medium consisting of joined three-dimensional (3D) half spaces are derived and expressed in terms of line integrals, integrands of which are given in an exact analytical form that, in turn, can also be integrated to yield analytical expressions for the stress–displacement field. The solution is constructed by employing a general solution derived by Walpole [Int. J. Eng. Sci. 34 (1996) p.629] for any elastic singularity in joined isotropic half space, and combining it with Mura's integral formula for the displacement gradient of an arbitrary dislocation segment in homogeneous medium. The resulting new solution provides a framework for deriving analytical expressions for stress and displacement fields of dislocation curves of arbitrary shapes and orientations. The benefit of the method developed, as compared with other methods found in the literature, is that the new solution presented is naturally divided into two components, a homogenous component representing the field of a dislocation in an infinitely homogenous medium, and an image component. This makes it easy and straightforward to modify existing dislocation dynamics codes that already include the homogenous part. To illustrate the accuracy of the method, the stress field expressions of an edge dislocation with Burgers vector perpendicular to the bi-material interface are derived as a degenerate case of the general result. It is shown that our solution is identical to that found in the literature for this case.     

Technique for determining the channel expansion rate at the stage of electrical breakdown using a grounded intercepting ring

A method is described that allows one to study the conductivity dynamics of a channel produced by explosion of a wire at the stage of electrical breakdown. Experimental data have been presented for the expansion rate of the conductivity channel in extended (up to 1.9 m long) arbitrarily shaped gapes that were produced by an exploding copper wire 90 µm in diameter. The initial stored energy and applied voltage were, respectively, 2.7–3.7 kJ and 8–10 kV. It has been shown that the expansion rate of the conductivity channel coincides with the propagation rate of a shock wave and is inversely proportional to the square root of its radius and propagation time. The radius of the shock wave is a linear function of the square root of its propagation rate. Experimental data are in satisfactory agreement with the calculated results obtained by Lin [18] in terms of the model of an intense shock wave. It has been shown that the diameter of the conductivity channel depends on the position of the trailing edge of the shock wave.     

Study of ground states of <Superscript>3,4,6</Superscript>He nuclides by Feynman’s continual integrals method

The energy and the square modulus of the wave function for the ground states of helium nuclides ^3,4,6He has been calculated by Feynman’s continual integrals method. A new parameterization of the shell model for light nuclei is proposed.     

A twistorial description of the dynamics of complexified dirac fields

A method that allows the construction of a twistorial least-action principle for complexified Klein-Gordon fields is adapted to the context of the two-component theory of Dirac fields. It appears that the viability of the adaptation procedures rests crucially upon the freedom in the selection of spinor kernels for the integrands of the universal twistor contour integrals for massive fields carrying arbitrary spin. An intrinsic ambiguity borne by the inner structure of the twistor representation of the dynamics of the spinning fields taken into consideration is then brought out. This feature appears to be reflected by the fact that many Poincaré-invariant Lagrangian densities can be regarded as candidates for entering the new variational principle. It is remarked that all the corresponding equations of motion agree with a single twistor translation of the pertinent field equations. The dynamical expressions that result from the implementation of the procedures turn out to be related to one another through a simple interchange scheme. A set of wave equations for the system allowed for is also obtained. This work was supported financially by the Brazilian agency CNPq.