Wide-Angle Parabolic Approximation for Modeling High-Intensity Fields from Strongly Focused Ultrasound Transducers

A novel numerical algorithm based on the wide-angle parabolic approximation is developed for modeling linear and nonlinear fields generated by axially symmetric ultrasound transducers. An example of a strongly focused single-element transducer is used to compare the results of ultrasound field simulations based on the Westervelt equation, Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with differently modified boundary condition, and nonlinear wide-angle parabolic equation. It is demonstrated that having a computational speed comparable to modeling the KZK equation, the use of wide-angle parabolic approximation makes it possible to obtain solutions for highly focused ultrasound beams that are closer in accuracy to solutions based on the Westervelt equation.     

Spheroidal analysis of the generalized MIC-Kepler system

This paper deals with the dynamical system that generalizes the MIC-Kepler system. It is shown that the Schrödinger equation for this generalized MIC-Kepler system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic, and spheroidal) are studied in detail. It is found that the coefficients for this expansion of the parabolic basis in terms of the spherical basis, and vice versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the prolate spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations.     

Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods

A nonlinear conformable time-fractional parabolic equation with exponential nonlinearity is explored, in this article. First, under the specific transformations, the time-fractional parabolic equation is changed into a nonlinear ODE of integer order, and then, the reduced equation is solved using two lately established techniques called the \({ \exp }\left( { - \varphi \left( \varepsilon \right)} \right)\)-expansion and modified Kudryashov methods. Several exact solutions in various wave forms for the nonlinear conformable time-fractional parabolic equation with exponential nonlinearity are formally constructed.     

Generalization of the rotated parabolic equation to variable slopes

The rotated parabolic equation [J. Acoust. Soc. Am. 87, 1035-1037 (1990)] is generalized to problems involving ocean-sediment interfaces of variable slope. The approach is based on approximating a variable slope in terms of a series of constant slope regions. The original rotated parabolic equation algorithm is used to march the field through each region. An interpolation-extrapolation approach is used to generate a starting field at the beginning of each region beyond the one containing the source. For the elastic case, a series of operators is applied to rotate the dependent variable vector along with the coordinate system. The variable rotated parabolic equation should provide accurate solutions to a large class of range-dependent seismo-acoustics problems. For the fluid case, the accuracy of the approach is confirmed through comparisons with reference solutions. For the elastic case, variable rotated parabolic equation solutions are compared with energy-conserving and mapping solutions.     

A perturbation approach to the propagation characteristics of inhomogeneous optical fibres with deformed core boundaries

The propagation characteristics of inhomogeneous optical fibres with different types of deformed core boundaries are analysed by a perturbative approach based on the scalar wave equation. The influence of the core deformation on the propagation constants and group delays is studied in detail for the truncated parabolic profile. We can conclude from this analysis that the difference in group delays due to deviation from circularity does not cause any difficult tolerance problems for realistic deformations and is generally negligible for practical optical fibres.     

基于JASMIN框架的抛物方程有限差分解法并行计算及其应用

抛物方程有限差分解法的网格步长严格受波长限制,在求解城市小区电波传播问题时,计算速度明显变慢,为此,基于JASMIN框架研究了抛物方程有限差分解法的并行方法,通过将同一步进面划分成多个网格片,并分配到不同的处理器进行运算,实现了抛物方程有限差分解法的并行计算。与解析解的对比验证了并行程序的正确性,同时通过实例分析了并行程序的高效性,算例表明,抛物方程有限差分解法的求解效率得到了有效的提高。最后,模拟和分析了某一电信基站天线在包含9栋规则建筑物的城市小区环境中的电磁特性,结果表明,该方法能够得到基站在空间各处的信号覆盖强弱,可以为基站选址提供参考。     

基于JASMIN框架的抛物方程有限差分解法并行计算及其应用

抛物方程有限差分解法的网格步长严格受波长限制,在求解城市小区电波传播问题时,计算速度明显变慢,为此,基于JASMIN框架研究了抛物方程有限差分解法的并行方法,通过将同一步进面划分成多个网格片,并分配到不同的处理器进行运算,实现了抛物方程有限差分解法的并行计算。与解析解的对比验证了并行程序的正确性,同时通过实例分析了并行程序的高效性,算例表明,抛物方程有限差分解法的求解效率得到了有效的提高。最后,模拟和分析了某一电信基站天线在包含9栋规则建筑物的城市小区环境中的电磁特性,结果表明,该方法能够得到基站在空间各处的信号覆盖强弱,可以为基站选址提供参考。     

An extended parabolic equation and its application to the calculation of transverse and longitudinal mutual coherence functions in atmospheric turbulence

An extended, wide forward angle scattering version of the parabolic equation is considered and an operator expression for the solution of the generalized nmth moment of the electromagnetic wave field is obtained. Here, 'generalized' connotes the consideration of both the transverse as well as the longitudinal spatial moments of the wave field. A unified solution for the generalized second-order moment, i.e. the mutual coherence function (MCF), is found. The solution is applied to the case of Kolmogorov turbulent fluctuations within the atmosphere. In addition to demonstrating an interesting decaying oscillatory behaviour of the longitudinal MCF in atmospheric turbulence, it is found that the use of the extended parabolic equation yields negligible corrections to the transverse MCF, as calculated from the parabolic equation in the paraxial approximation.     

Energy levels and states of parabolic cylindrical lens shaped quantum dots

The parabolic cylindrical lens shaped quantum dot is investigated theoretically. The Schrǒdinger equation for an electron confined in this structure is solved in the parabolic cylindrical coordinate system. The wavefunctions for the electron are presented in terms of confluent hypergeometric functions, and the electron energy spectra are also obtained.     

Dyadic Green's functions for the perfectly conducting parabolic cylinder

Solutions of the scalar wave equation for parabolic cylinder coordinate system are discussed here. Dyadic Green's functions of the magnetic type for free space and for a perfectly conducting parabolic cylinder are developed. These functions are of fundamental importance for the solution of electromagnetic problems developed in the parabolic cylinder coordinate system, particularly those in the presence of perfectly conducting parabolic cylinders such as that of a parabolic cylinder reflector.     

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.     

基于柱坐标系抛物方程和矩量法的电波传播混合算法

针对包含近源障碍物条件下的电波传播问题,提出了一种新颖的电波传播预测混合建模方法:矩量法(MOM)和圆柱坐标系抛物方程法(PEM)混合建模方法(MOM-PEM);MOM用于包含辐射源和近源障碍物的小圆柱区域内的电波传播建模,PEM用于MOM计算空间外的大区域范围内电波传播建模。MOM和PEM的计算过渡区域进行精细化网格剖分处理以避免场强数值传递的不兼容。仿真模拟了三类近源障碍物存在场景下的电波传播问题:有限开窗屏障碍物、立方体障碍物以及包含辐射源的半封闭空间障碍物,并将混合算法计算得到的结果和相同环境下采用全矩量法计算得到的结果进行了数值对比,结果表明混合算法和矩量法在精度上吻合较好。     

抛物方程基于自然边界归化的耦合法

将冯康和余德浩提出的自然边界归化方法^[1~4]应用于求解抛物方程初边值外区域问题,提出一种自然边界元与有限元耦合算法。先将控制方程对时间进行离散化,得到关于时间步长的离散化格式,给出圆外域上的自然积分方程,基于此研究抛物方程无界区域问题的自然边界元与有限元耦合法,最后给出相应的数值例子。     

Parabolic equation solution of seismo-acoustics problems involving variations in bathymetry and sediment thickness

Recent improvements in the parabolic equation method are combined to extend this approach to a larger class of seismo-acoustics problems. The variable rotated parabolic equation [J. Acoust. Soc. Am. 120, 3534-3538 (2006)] handles a sloping fluid-solid interface at the ocean bottom. The single-scattering solution [J. Acoust. Soc. Am. 121, 808-813 (2007)] handles range dependence within elastic sediment layers. When these methods are implemented together, the parabolic equation method can be applied to problems involving variations in bathymetry and the thickness of sediment layers. The accuracy of the approach is demonstrated by comparing with finite-element solutions. The approach is applied to a complex scenario in a realistic environment.     

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ _t  + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.     

Numerical simulation of planar Bragg waveguide bends

An analytical form was found and transparency boundary conditions were numerically approximated for the parabolic wave equation in curvilinear coordinates. It was shown that the solutions obtained by the parabolic equation method are in agreement with the solutions to the spectral problem defining the Bragg waveguide modes. The field amplitude and bending loss were numerically simulated depending on the curvature radius and parameters of the Bragg waveguide.     

Parabolic nondiffracting optical wave fields

We demonstrate the existence of parabolic beams that constitute the last member of the family of fundamental nondiffracting wave fields and determine their associated angular spectrum. Their transverse structure is described by parabolic cylinder functions, and contrary to Bessel or Mathieu beams their eigenvalue spectrum is continuous. Any nondiffracting beam can be constructed as a superposition of parabolic beams, since they form a complete orthogonal set of solutions of the Helmholtz equation. A novel class of traveling parabolic waves is also introduced for the first time.     

The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space H ₁. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic nonlinear equation. We use gauge symmetry breaking to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.     

On the size dependence of the surface tension

The general form of a differential equation that deduces a size dependence of the surface tension is derived. The well-known Gibbs-Tolman-Koenig-Buff equation for the spherical surface is a particular case of the newly derived one. Analytical solutions to this equation for the spherical, cylindrical, parabolic, and conical surfaces are found.