半球面上的电荷量有多大?

在球坐标下,带电上下半球面保持固定相反电势,所产生的全空间电势表达式是含有勒让德函数的无穷求和.给出了半球面上的电荷密度并计算了半球面上的电荷量.计算发现,这个电荷量是无穷大.     

半无界空间上函数的傅里叶-贝塞尔积分展开

柱坐标系中,本征函数族贝塞尔函数构成完备正交系,因此可作为广义傅里叶级数展开的基.本文从定义在有限区间[0,ρ0]上函数的广义傅里叶级数展开出发,利用贝塞尔函数的渐近展开公式以及贝塞尔函数零点的近似公式,讨论了半无界空间上函数的傅里叶-贝塞尔积分展开问题,得到了本征函数模方的近似表达式.当ρ0趋于无穷时,不连续参量变成连续参量,得到了函数的傅里叶-贝塞尔积分及其展开系数公式.     

一种新的阶跃折射率光纤本征函数表达形式

为了阐明阶跃折射率光纤的模式特性,根据电磁波辐射的能量守恒定律和经过狄拉克函数奇异性修正的亥姆霍兹方程,通过数学推导和证明得出:柱面径向行波场的本征函数是经过狄拉克函数修正的整数阶汉克尔函数,阶跃折射率光纤模式场的本征函数是零、一阶贝塞尔函数经过狄拉克函数修正的零、一阶诺埃曼函数和虚参量汉克尔函数.该结论揭示了光纤芯层和包层模式场分别是径向驻波场和倏逝波场的本质,并基此推导出新的光纤模式特征方程,模式存在条件,模式数目和符合光纤实际的基模归一化截止频率的理论值.     

统计力学中的勒让德变换

勒让德变换是理论物理学中一个重要的工具,多变量函数的勒让德变换有明显的对称性.系综是统计力学教学中的核心内容.利用勒让德变换的无量纲性,从微正则系综的特性函数熵的勒让德变换出发,类比的定义了配分函数的勒让德变换.同时得到各种系综的特性函数和配分函数以及它们之间的关系和相应的热力学公式.     

一种提高时间平均数字全息测振条纹模式对比度的方法

针对基于时间平均数字全息法的微振动原位检测,提出了一种有效提高振动条纹模式对比度的方法,进而可准确获得待测面振幅分布.首先针对实际振动过程中物体存在纵向微位移而引起的相位误差进行校正;然后去除记录过程中物光场散斑引起的相位噪音,有效提高了被第一类零阶贝塞尔函数调制的条纹对比度;最后利用第一类零阶贝塞尔函数的变化规律和条纹级数,定量估计全场的振幅分布.搭建了以声波作为激励源的时间平均全息测振实验系统,采用频率分别为0、1 100、2 000和9 000 Hz的正弦声波激励簧片进行实验验证.结果表明,以振动频率2 000Hz为例,本方法能够有效提高振动调制条纹质量,并计算得到三级零点处的振幅值分别为109.4nm、244.2nm、356.4nm,与理论值相比较,误差分别为6.31%、3.34%和3.78%,在允许范围之内,证明了该方法的有效性.     

辐射传输中的相函数展开方法研究

从辐射传输方程中相函数的展开方法出发，分别介绍了δ-M展开法和δ-fit展开法，并对这两种相函数展开方法进行了比较。结果表明，在相函数展开中，当所取勒让德多项式的阶数相同时，δ-M法的图像始终在真实相函数附近波动，始终无法逼近真实的相函数的值，而δ-fit法符合较好。     

一维变系数耗散系统Lagrange函数和Hamilton函数的新构造方法

提出构造二阶微分方程的Lagrange函数和Hamilton函数的新路径. 将二阶方程写成一阶方程组并构造出对应的一阶Lagrange函数后,直接从一阶Lagrange函数导出二阶Lagrange函数和Hamilton函数. 利用上述方法得到若干耗散和类耗散系统的一阶和二阶Lagrange函数以及Hamilton函数;讨论了这种方法的优点. 举例说明所得结果的应用. 关键词： 逆问题 耗散系统 Lagrange函数 Hamilton函数     

改进的三维分离变量表象方法计算振动光谱

用改进的三维分离变量表象(DVR3D)方法研究了基态电子态H2S和H2O的振动光谱。哈密顿形式在Jacobi坐标下给出,每个坐标都用一组DVR基函数表示.角度部分的DVR基组选择勒让德多项式形式,而径向坐标采用正弦函数形式.为了降低需要对角化的最终哈密顿矩阵的维数,采用连续对角化截断方法.用H2S和H2O振动能级的计算验证算法的正确性.     

基于Simulink S-函数的物理实验动画设计

结合单摆振动仿真实例介绍了用Simulink对物理实验进行仿真建模以及利用Simulink S-函数创建动画的方法.给出了位移、角速度等振动曲线以及单摆摆动动画.体现出Simulink S-函数仿真物理实验的科学性和优越性.     

圆阵的水平方向性

本文给出了阵元具有任意指向性的圆阵水平方向性函数的计算公式。对单面辐射阵元组成的圆阵进行了深入的讨论,给出了各种参数对阵方向性影响的规律性。对点元组成的圆阵,本文得出如下规律:使用奇数个阵元比用偶数个阵元更易于得到均匀的方向性;阵半径和波数的乘积接近零阶贝塞尔函数的零点,方向性的不均匀度明显增加,而且阵的接收灵敏度(或发射响应)显著降低。     

基于格林函数法的奇型Mathieu-Gaussian光束

根据光束传播的独立性和叠加性原理,引入了一组能够产生第一类(2n+2阶)奇型Mathieu-Gaussian光束的虚光源点.利用虚源点技术和格林函数法,计算得到第一类奇型Mathieu-Gaussian光束的严格解析积分表达式.利用该表达式得到了轴上光场分布的积分解析解.以三阶非旁轴修正为例,得到了第一类奇型Mathieu-Gaussian光束保留到三阶非旁轴修正项的轴上光场分布精确解.     

Estimates of radiation over clouds and dust aerosols: Optimized number of terms in phase function expansion

The bulk-scattering properties of dust aerosols and clouds are computed for the community radiative transfer model (CRTM) that is a flagship effort of the Joint Center for Satellite Data Assimilation (JCSDA). The delta-fit method is employed to truncate the forward peaks of the scattering phase functions and to compute the Legendre expansion coefficients for re-constructing the truncated phase function. Use of more terms in the expansion gives more accurate re-construction of the phase function, but the issue remains as to how many terms are necessary for different applications. To explore this issue further, the bidirectional reflectances associated with dust aerosols, water clouds, and ice clouds are simulated with various numbers of Legendre expansion terms. To have relative numerical errors smaller than 5%, the present analyses indicate that, in the visible spectrum, 16 Legendre polynomials should be used for dust aerosols, while 32 Legendre expansion terms should be used for both water and ice clouds. In the infrared spectrum, the brightness temperatures at the top of the atmosphere are computed by using the scattering properties of dust aerosols, water clouds and ice clouds. Although small differences of brightness temperatures compared with the counterparts computed with 4, 8, 128 expansion terms are observed at large viewing angles for each layer, it is shown that 4 terms of Legendre polynomials are sufficient in the radiative transfer computation at infrared wavelengths for practical applications.     

Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering

Exact integral equations are derived describing the source function and radiative flux in a two-dimensional, radially infinite cylindrical medium which scatters anisotropically. The problem is two-dimensional and cylindrical because of axisymmetric loading. Radially varying collimated radiation is incident normal to the upper surface while the lower boundary has no radiation incident upon it. The scattering phase function is represented by a spike in the forward direction plus a series of Legendre polynomials. The two-dimensional integral equations are reduced to a one-dimensional form by separating variables for the case when the radial variation of the incident radiation is a Bessel function. The one-dimensional form consists of a system of linear, singular Fredholm integral equations of second kind. Other more complex boundary conditions are shown to be solvable by a superposition of this basic Bessel function case. Diffusely incident radiation is also considered.     

水下任意刚性散射体对Bessel波的散射特性分析

本文利用Bessel波的谐波展开式, 采用T矩阵方法的推导思路, 建立了水下任意刚性散射体在Bessel波照射下的声散射场计算公式. 以水下刚性椭球体和两端附连半球的刚性圆柱体为例, 计算了在不同波锥角β 下的反向散射形态函数, 同时, 依据镜反射波和绕行波的干涉物理模型, 给出了预报Bessel波照射下的反向散射形态函数峰峰间隔值的计算模型. 仿真结果表明本文提出的Bessel波照射下反向散射形态函数峰峰间隔值预报方法是准确有效的, 同时也说明, 本文建立的基于T矩阵法计算水下任意刚性散射体在Bessel波束下的声散射场方法是有效的, 这拓展了T矩阵法的应用领域.     

Dirac-delta function approximations to the scattering phase function

Dirac-delta function approximations are used to represent the single scattering phase function of large spherical particles or voids. The phase function for a spherical particle or void can be represented by a series of Legendre polynomials; however, as the diameter is increased, forward scattering becomes dominant and the number of terms in the series becomes very large. A Dirac-delta function approximation consists of a Dirac-delta function in the forward direction plus a finite series of Legendre polynomials. The Dirac-delta function accounts for strong forward scattering. Particular attention is given to large ice spheres and spherical voids in ice. The Dirac-delta function is shown effective in reducing the number of terms needed to describe the phase function.     

Computation of Green's function for radiative transfer

In an accompanying paper, we develop the computational expressions for the higher order perturbation of the radiative transfer equation, and present some numerical results for typical cases. In this article, we discuss a number of issues regarding the implementation of the HOP computation: obtaining the Green's function, its expansion as a double series of Legendre polynomials, and obtaining the adjoint radiance of more general sources such as those for the fluxes at arbitrary altitudes. Examples of Green's function and its expansion coefficients are presented.     

Two-dimensional rayleigh scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium with second order Legendre phase function scattering. The incident radiation is collimated, normal to the top surface, and is dependent only on the radial coordinate. Boundary conditions which vary as a Bessel function and as a Gaussian distribution are investigated. The Gaussian distribution approximates a laser beam. Numerical results are presented in graphical and tabular forms for a Rayleigh scattering medium. The results are compared with those of isotropic scattering.     

求解贝塞尔类方程的试探函数法

借助数学软件,给出了求贝塞尔类方程线性独立解的一种直接方法——试探函数法.     

点列图能量分布曲线和几何光学传递函数

本文对光学系统的点列图和能量分布曲线以及几何光学传递函数给出数值计算方法,这些方法节省很大计算量。