A k-space Green's function solution for acoustic initial value problems in homogeneous media with power law absorption

An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.     

Regularity of Solutions to Vorticity Navier–Stokes System on ℝ<Superscript>2</Superscript>

The Cauchy problem for the Navier–Stokes system for vorticity on plane is considered. If the Fourier transform of the initial data decays as a power at infinity, then at any positive time the Fourier transform of the solution decays exponentially, i.e. the solution is analytic.     

Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades

We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.     

The operational solution of fractional-order differential equations,as well as Black–Scholes and heat-conduction equations

Operational solutions to fractional-order ordinary differential equations and to partial differential equations of the Black–Scholes and of Fourier heat conduction type are presented. Inverse differential operators, integral transforms, and generalized forms of Hermite and Laguerre polynomials with several variables and indices are used for their solution. Examples of the solution of ordinary differential equations and extended forms of the Fourier, Schrödinger, Black–Scholes, etc. type partial differential equations using the operational method are given. Equations that contain the Laguerre derivative are considered. The application of the operational method for the solution of a number of physical problems connected with charge dynamics in the framework of quantum mechanics and heat propagation is demonstrated.     

Free vibration analysis of circular cylindrical shells

A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.     

SRLW方程的多辛Fourier谱格式及其守恒律

通过引进正则动量,将对称正则长波方程（简称SRLW方程）转化成多辛形式的方程组,它具有多辛守恒律;介绍了空间方向满足周期边界条件的函数的Fourier谱方法;对SRLW方程的多辛方程组在空间方向利用Fourer谱方法,时间方向上应用Euler中点格式离散,得到其多辛Fourier拟谱格式;证明此格式的一些离散守恒律．用此格式模拟了SRLW方程的单个孤立波,还模拟了多个孤立波的追赶、碰撞和分离过程．     

A generalized spherical harmonics solution for radiative transfer models that include polarization effects

A generalized spherical harmonics solution for all components (m ⩾ 0) in a Fourier representation of the Stokes vector basic to the scattering of polarized light is developed, and computational aspects of the solution are discussed in detail. The established solution is used in regard to two test problems to obtain numerical results, accurate, in general, to five significant figures, for the four Stokes parameters.     

Analytical approach for solving the radiative transfer equation in two-dimensional layered media

This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.     

Unfolding and Aggregation of Myoglobin Can Be Induced by Three Hours’ Exposure to Mobile Phone Microwaves: A FTIR Spectroscopy Study

The effects of 3 hours’ exposure to mobile phone microwaves at 1765 MHz at the power density around 800 mW/m² on the secondary structure of myoglobin in D₂O solution were investigated by Fourier transform infrared spectroscopy. A significant shift to lower frequencies of the amide I vibration was observed after exposure. Furthermore, a significant increasing of the β-sheet components with respect to the α-helix content after exposure was highlighted after applying Fourier self-deconvolution analysis in the amide I region. These results led to the conclusion that mobile phone microwaves induce unfolding of myoglobin and formation of aggregates.     

Fourier transforms method for measuring thermal lens induced in diluted liquid samples

Drawing on interferometry and Fourier analysis, this paper describes the use of a two-beam thermal lens technique for measuring thermo-optical properties in optical materials. The procedure consists of yield interference patterns deformed by a localized photothermal effect. The photothermal phase shift is locally induced by the pump beam focused on a tested sample located on an on-axis probe beam, which is the first arm of a Mach-Zehnder interferometer. The plane where the effect is localized is imaged onto a CCD camera. Then two interferograms are recorded: one without effect and the other one with the induced photothermal phase. Fourier analysis performed on these interferograms allow us to plot the thermal lens map and, therefore, to estimate thermo-optic constant of Malachite Green in water solution. The method is applied to measure low linear absorptions of a diluted sample of Rhodamine B in water solution at 633 nm, showing that the proposed technique allows to measure photothermal phase shift as low as 3.1 mrad at 8 mW of input power in diluted materials.     

A new perturbative approach to the classical anharmonic oscillator

The periodic motion of the classical anharmonic oscillator characterized by the potentialV(x)=1/2x ²+λ/2k x ^2k is considered. The period is first determined to all orders inλ in a perturbative series. Making use of this, the solution of the nonlinear equation of motion is then expressed in the form of a Fourier series. The Fourier coefficients are obtained by solving simple algebraic relations. Secular terms are inherently absent in this perturbative scheme. Explicit solution is presented for generalk up to the second order, from which the Duffing and the sextic oscillator results follow as special cases.     

On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame

Analytic solution for unsteady magnetohydrodynamic (MHD) flow is constructed in a rotating non-Newtonian fluid through a porous medium. Constitutive equations for a Maxwell fluid have been taken into consideration. The hydromagnetic flow in the uniformly rotating fluid is generated by a suddenly moved infinite plate in its own plane. Analytic solution of the governing flow problem is obtained by means of the Fourier sine transform. It is shown that the obtained solution satisfies both the associate partial differential equation and the initial and boundary conditions. The solution for a Navier-Stokes fluid is recovered if λ→0. The steady state solution is also obtained for t→∞.     

Solutions of the space-time fractional Cattaneo diffusion equation

The object of this paper is to present the exact solution of the fractional Cattaneo equation for describing anomalous diffusion. The classical Cattaneo model has been generalised to the space-time fractional Cattaneo model. The method of the joint Laplace and Fourier transform is used in deriving the solution. The solutions of the fractional Cattaneo equation are obtained under integral and series forms in terms of the H-functions. Finally, the fractional order moments are also investigated.     

A fast wave superposition spectral method with complex radius vector combined with two-dimensional fast Fourier transform algorithm for acoustic radiation of axisymmetric bodies

Based on the two-dimensional fast Fourier transform (2D FFT) algorithm, a wave superposition spectral method with complex radius vector has been proposed to efficiently analyze the acoustic radiation from an axisymmetric body. First, the complex Fourier series are used along both circumferential and meridian directions, to expand the integral kernel function and unknown source strength density distributed function. Then, by means of the rectangular integral formula, the radiation sound pressure is described in the form of two-dimensional discrete Fourier transform and generalized through 2D FFT algorithm. Finally, several numerical examples are performed to verify the accuracy and efficiency of the present method. Comparing with the other existing analysis ways, the present method has three different characteristics: (1) there is no singular integral in the numerical computation; (2) the unique solution can be given for all eigen wavenumbers owing to the application of the virtual boundary technology with complex radius vector; and (3) the computational efficiency is improved remarkably because all Fourier terms are calculated simultaneously through 2D FFT algorithm.     

Fourier analysis of the parametric resonance in neutrino oscillations

Parametric enhancement of the appearance probability of the neutrino oscillation under the inhomogeneous matter is studied. Fourier expansion of the matter density profile leads to a simple resonance condition and manifests that each Fourier mode modifies the energy spectrum of oscillation probability at around the corresponding energy; below the MSW resonance energy, a large-scale variation modifies the spectrum in high energies while a small-scale one does in low energies. In contrast to the simple parametric resonance, the enhancement of the oscillation probability is itself an slow oscillation as demonstrated by a numerical analysis with a single Fourier mode of the matter density. We derive an analytic solution to the evolution equation on the resonance energy, including the expression of frequency of the slow oscillation.     

傅里叶变换法求解两类简单的薛定谔方程

自由粒子和一维无限深势阱的薛定谔方程的求解是量子力学较为基础的内容.本文采用傅里叶变换对这两类简单的薛定谔方程进行了求解讨论.通过偏微分方程的傅里叶变换解法和偏微分方程作分离变数成常微分方程后的傅里叶变换解法的深入讨论,均得到与有关教材一致的结果,并讨论了这两种方法之间的差别和联系.     

Exponential convergence of a spectral projection of the KdV equation

It is shown that a spectral approximation of the Korteweg–de Vries equation converges exponentially fast to the true solution if the Fourier basis is used and if the solution is analytic in a fixed strip about the real axis. Computations are carried out which show that the exponential convergence rate can be achieved in practice.     

Near-field ultrasound tomography

In this paper, a near-field tomographic solution is introduced to solve the imaging problem of fluid objects assumed to be weakly heterogeneous (Born approximation) and excited by spherical waves. The solution to the forward problem is based on the Huygens-Fresnel principle which describes the scattered field as the result of the interference scheme of all the secondary spherical waves. From the derivation of the scattered field, a new Fourier transform that has been called the elliptical Fourier transform is defined: It differs from the standard Fourier transform in that instead of a plane wave decomposition, a harmonic ellipsoidal wave decomposition is obtained. Based on this spectral analysis, a near-field Radon transform is designed that complements the "far-field tools" published in diffraction tomography literature. Then, assuming that the measuring distance is greater than one wavelength, the feasibility of reconstructing either the impedance or the velocity maps of an acoustical (perfect fluid) model is demonstrated. Numerical simulations were performed which confirmed the validity of the theory presented here; a theory which has many potential applications in future wave theory research.     

三能级钾原子气体三维傅里叶变换频谱的解析解

利用投影切片定理、傅里叶位移定理和误差函数给出三能级钾原子气体三维傅里叶变换频谱在T=0界面的解析解.固定均匀线宽,非均匀展宽和对角线相关系数可以定量地识别,通过在适当方向上拟合三维傅里叶变换频谱谱峰的切片来确定.结果表明,非均匀展宽增大,频谱图沿着对角线方向延伸,对角线相关系数增大,频谱图逐渐变圆,振幅也逐渐变小.     

A fourier transform solution for radiative transfer in a slab with isotropic scattering and boundary reflection

The Fourier transform method is used to formulate an exact solution for radiative heat transfer in an isotropically scattering, absorbing and emitting plane-parallel slab with specularly reflecting boundaries. The analysis leads to a solution involving expansion coefficients which can be determined from a matrix equation to the desired degree of accuracy. The numerical results obtained for the hemispherical reflectivity and transmissivity show that even the lowest order solution gives good accuracy.