Aerodynamic Sound due to a Point Source near a Half-plane

In this paper we extend Jones's analysis of the diffractionof sound in three dimensions by a semi-infinite plane with aplane vortex sheet attached in the two cases when the wave equationis in the form for still air and when convection is present.It is found that in so far as the moving medium is concernedthe imposition or otherwise of the Kutta-Joukowski conditiondoes not have much influence on the scattered field away fromthe diffracting plane; when the source is near the edge thefield has the same directionality and order of magnitude. Onthe other hand, near the wake the Kutta-Joukowski conditionproduces a much stronger field than elsewhere even when thesource is not near the edge. We also conclude that the samephenomenon occurs for arbitrary sources and not just for theline source discussed by Jones.     

Aerodynamic Sound Emission as a Singular Perturbation Problem

Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ? 1, Λ ? 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment T_ij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ? × ψ{ ω } + ?φ The problem is solved for M ? 1, Λ ? 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if T_ij is approximated by ρ₀υ_iυ_j, with v = ? × ψ. The situation M ? 1, Λ ? 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.     

亚声速热射流声源模型的应用

对亚声速转捩热射流中失稳波相关的噪声产生机制进行了研究,并与冷射流中的结果进行了对比.基于时均大涡模拟(LES)流场,通过求解线性抛物化稳定性方程(LPSE)得到了失稳波的空间演化特性,然后基于LPSE的解与声比拟方法构建了射流的线性及非线性声源模型.LPSE结果表明,加热可以提高失稳波的空间增长率,使其更早达到饱和.由线性模型分析可知,加热会提高高频模态的声压级(SPL).与冷射流相比,热射流中线性模型预测的声压级与大涡模拟结果间的差距更小,表明线性机制在热射流中作用更大.在亚声速冷射流中,非线性模型在之前的研究中已经被证明可以提高声辐射效率.在当前热射流中,发现非线性模型与大涡模拟间的声压级差距被进一步的缩小,且温度相关的声源项在声辐射中发挥更重要的作用.     

半平面中调和函数的积分表示

In this article, we consider the integral representation of harmonic functions. Using a property of the modified Poisson kernel in a half plane, we prove that a harmonic function u（z） in a half plane with its positive part u^＋（z） = max{u（z）, 0} satisfying a slowly growing condition can be represented by its integral of a measure on the boundary of the half plan.     

非均匀半平面问题的弹性力学解

本文使用非均匀平面弹性力学的基本方程,通过富氏积分变换,求得了应力函数通解。在此基础上对弹性模量E(x)=E_oexp[βx]为指数型的非均匀半平面问题,具体求得了当边界上受任意载荷作用的精确解。最后经退化处理,还得到了有名的Boussnesq解,这说明本文的方法是成功的。     

On a Theorem Due to Birkhoff

The manifold M being closed and connected, we prove that every submanifold of T*M that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.     

半平面中有限阶解析函数的因子分解

与经典有限阶整函数的Hadamard因子分解定理和半平面中属于Hardy空间的解析函数的内外函数的因子分解类似,对右半平面中有限阶ρ解析函数f,可以分解为三个解析函数G,eQ和eg的乘积GeQeg,其中G是一个加权Blaschke乘积,Q是一个次数不超过ρ的多项式以及eg是一个加权外函数,log|G|,ReQ和Reg-log|f|在右半平面的边界恒为零．     

Effect of a Floating Elastic Plate/Membrane on the Motion Due to a Ring Source in Water With Porous Bed

The velocity potentials due to the presence of a horizontal circular ring of wave sources of timedependent strength in water of finite constant depth with a floating elastic plate or a floating membrane are determined. The uniform bottom is composed of non-dissipative porous medium. The problems are formulated as the initial value problems and the Laplace transform method is used to solve these. For time-harmonic source strength, the steady-state analysis of the potentials reveals the existence of outgoing progressive waves. Graphs for the surface profiles are presented for different values of the tension parameter for the membrane, flexural rigidity of ice and the porous-effect parameter.     

On Zeros of One Class of Functions Analytic in a Half-Plane

We describe sequences of zeros of functions f 0 analytic in the half-plane and satisfying the condition where : [0; +) (0; +) is an increasing function such that the function ln (r) is convex with respect to ln r on [1; +).     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Estimation over Curves of the Functions Given by Dirichlet Series on a Half-Plane

We study the maximal term of the Hadamard composition of Dirichlet series with real exponents. We obtain a lower estimate for the sum of a Dirichlet series over a curve arbitrarily approaching the convergence line.     

Wave Packet Defocusing Due to a Highly Disordered Bathymetry

Slowly modulated water waves are considered in the presence of a strongly disordered bathymetry. Previous work is extended to the case where the random bottom irregularities are not smooth and are allowed to be of large amplitude. Through the combination of a conformal mapping and a multiple‐scales asymptotic analysis it is shown that large variations of a disordered bathymetry can affect the nonlinearity coefficient of the resulting damped nonlinear Schrödinger equations. In particular it is shown that as the bathymetry fluctuation level increases the critical point (separating the focusing from the defocusing region) moves to the right, hence enlarging the region where the dynamics is of a defocusing character.     

Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.     

Random Vibrations of a Building Due to Kinematic Excitation

Vibration problem of a building, excited by the movement of its base, is considered. The kinematic excitation is regarded as a stream of successive overlapping random signals appearing at random times. It idealizes the base vibrations generated by the trafic or by some machines like hammers, presses etc. To define the excitation process the Poisson process theory is being applied. The solution of the problem is formulated in the correlation theory sphere by using the impulse response function defined for the multi‐degree‐of‐freedom system.     

The Steklov Problem in a Half-Plane: the Dependence of Eigenvalues on a Piecewise-Constant Coefficient

The Steklov problem considered in the paper describes free two-dimensional oscillations of an ideal, incompressible, heavy fluid in a half-plane covered by a rigid dock with two symmetric gaps. Equivalent reduction of the problem to two spectral problems for integral operators allows us to find limits for all eigenfrequencies as the distance between the gaps tends to zero or infinity. For the fundamental eigenfrequency and the corresponding eigenfunction, two terms are found in the asymptotic expansion as the distance tends to infinity. It is proved that all eigenvalues are simple for any distance. Bibliography: 15 titles.Dedicated to the centenary of V. A. Steklovs paper [1]__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 162–190.     

Asymptotics of Markov Additive Chains on a Half-Plane: A Ratio Limit Theorem

Consider a Markov additive chain (V,Z) with a negative horizontal drift on a half-plane. We provide the limiting distribution of Z when V passes a threshold for the first time, as V tends to infinity. Our contribution is to allow the Markovian part of an associated twisted Markov chain to be null recurrent or transient. The positive recurrent case was treated by Kesten [Ann. Probab. 2 (1974), 355–386]. Moreover, a ratio limit will be established for a transition kernel with unbounded jumps.     

Extremal Problems in the Class of Delta-Subharmonic Functions of Finite Order in a Half-Plane

We obtain exact lower bounds of the upper limits of ratios of the Nevanlinna characteristics of a delta-subharmonic function in the upper half-plane.