周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移北大核心CSCD

该文解析研究了有限个周期排列的抛物形沟槽激发的水波Bragg共振.首先,利用变量替换,先将系数为隐函数的修正缓坡方程(MMSE)转化为系数为显函数的显式方程.然后,构造了修正缓坡方程的Frobenius级数解,并给出了级数解的收敛条件.最后,利用质量守恒的耦合条件,建立了反射系数的解析公式.根据反射系数的解析公式,分析了沟槽个数、沟槽深度与宽度对Bragg共振峰值、共振相位和共振带宽的影响.当沟槽深度和宽度固定而沟槽个数增加时,共振峰值逐渐增大并趋向于1,而共振带宽则逐渐变窄并趋于固定值.当沟槽个数和宽度固定时,Bragg共振峰值随沟槽深度增加而增加.当沟槽个数和深度固定时,Bragg共振反射峰值随沟槽宽度增加而先增后减,预示了沟槽存在某个宽度使得共振峰值达到最大,为Bragg共振反射针对沟槽宽度的优化奠定了理论基础.特别地,前不久在有限个周期排列旋轮线形沟槽上刚刚观察到的Bragg共振反射峰值相位的上移现象,再次在该文考虑的抛物形沟槽上得到确认,表明针对有限周期排列的沟槽地形,Bragg共振反射峰值的相位上移是一个普遍现象.也因此说明,凡是正弦沙纹和周期人工沙坝所激发的Bragg共振反射,其主振相位将会下移,而凡是周期系列沟槽所激发的Bragg共振反射,无论沟槽形状如何,其主振相位都将上移.另外,我们从Bragg共振的原始定义出发,定量地解释了相位上移发生的数学机理.     

Propagation of surface gravitational waves over two parallel bottom inhomogeneities

On the basis of the method of power series we construct a solution of the two-dimensional problem of propagation of surface gravitational waves in water in the presence of two bottom trenches or projections. The problem is studied in the context of the shallow water model. We carry out an analysis of the effect of the trench parameters and their location on the amplitude of the reflected wave as functions of the wave number. Four figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 126–132.     

Ocean-Atmosphere Interaction in the Lifecycle of ENSO： The Coupled Wave Oscillator

To explain the oscillatory nature of E1 Nino/Southern Oscillation （ENSO）, many ENSO theories emphasize the free oceanic equatorial waves propagating/reflecting within the Pacific Ocean, or the discharge/recharge of Pacific-basin-averaged ocean heat content. ENSO signals in the Indian and Atlantic oceans are often considered as remote response to the Pacific SST anomaly through atmospheric teleconnections. This study investigates the ENSO life cycle near the equator using long-term observational datasets. Space-time spectral analysis is used to identify and isolate the dominant interannual oceanic and atmospheric wave modes associated with ENSO. Nino3 SST anomaly is utilized as the ENSO index, and lag-correlation/regression are used to construct the composite ENSO life cycle. The propagation, structure and feedback mechanisms of the dominant wave modes are studied in detail. The results show that the dominant oceanic equatorial wave modes associated with ENSO are not free waves, but are two ocean-atmosphere coupled waves including a coupled Kelvin wave and waves are not confined only to the Pacific a coupled equatorial Rossby （ER） wave. These Ocean, but are of planetary scale with zonal wavenumbers 1-2, and propagate all the way around the equator in more than three years, leading to the longer than 3-year period of ENSO. When passing the continents, they become uncoupled atmospheric waves. The coupled Kelvin wave has larger variance than the coupled ER wave, making the total signals dominated by eastward propagation. Surface zonal wind stress （x） acts to slow down the waves. The two coupled waves interact with each other through boundary reflection and superposition, and they also interact with an off-equatorial Rossby wave in north Pacific along 15N through boundary reflection and wind stress forcing. The precipitation anomalies of the two coupled waves meet in the eastern Pacific shortly after the SST maximum of ENSO and excite a dry atmospheric Kelvin wave which quickly circles the whole equator and leads to a zonally symmetric signal of troposphere temperature. ENSO signals in the Indian and Atlantic oceans are associated with the two coupled waves as well as the fast atmospheric Kelvin wave. The discharge/recharge of Pacific-basin-averaged ocean heat content is also contributed by the two coupled waves. The above results suggest the presence of an alternative coupled wave oscillator mechanism for the oscillatory nature of ENSO.     

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.     

The configuration of shock wave reflection for the TSD equation

In this paper, we mainly study the nonlinear wave configuration caused by shock wave reflection for the TSD (Transonic Small Disturbance) equation and specify the existence and nonexistence of various nonlinear wave configurations. We give a condition under which the solution of the RR (Regular reflection) for the TSD equation exists. We also prove that there exists no wave configuration of shock wave reflection for the TSD equation which consists of three or four shock waves. In phase space, we prove that the TSD equation has an IR (Irregular reflection) configuration containing a centered simple wave. Furthermore, we also prove the stability of RR configuration and the wave configuration containing a centered simple wave by solving a free boundary value problem of the TSD equation.     

On the resonant reflection of weak,nonlinear sound waves off an entropy wave

We analyze the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave for spatially periodic solutions of the one‐dimensional, nonisentropic gas dynamics equations. The case of an entropy wave with a sawtooth profile is of interest because the oscillations of the reflected sound waves are nondispersive with frequency independent of their wavenumber, leading to an unusual type of nonlinear dynamics. On an appropriate long time scale, we show that a complex amplitude function for the spatial profile of the sound waves satisfies a degenerate quasilinear Schrödinger equation. We present some numerical solutions of this equation that illustrate the generation of small spatial scales by a resonant four‐wave cascade and front propagation in compactly supported solutions.     

Verification of Complex Transonic Phenomena with CFD Codes

Classical analytic models for the theoretical behavior of transonic flows have guided the development of numerical simulation of practically relevant flows. But while operational successfully for the usual applications (conventional configuration high speed aerodynamics), CFD codes fail frequently once detailed information regarding even just inviscid flow structure for freestream Mach number close to unity is needed. In this contribution shock configurations are computed verifying analytical results for both the far field behavior of detached bow waves and the local structure of attaching bow waves. The first example is challenging the ability of a CFD code (DLR‐τ) to adapt its unstructured grid to a weak bow wave detached from the airfoil. Known analytical results for the asymptotic bow wave behavior at decreased Mach number → 1 serve for judging numerical results where practical case studies neither from CFD nor from experiment are available. In the second example the behavior of a bow wave attaching to a wedge for increased Mach number is studied, with grid adaption allowing for a verification of both the singular analytical model when attachment takes place and also the transition to plain supersonic wedge flow.     

Reflection of plane waves from the boundary of a pre-stressed compressible elastic half-space

The effect of pre-stress on the propagation and reflection ofplane waves in an incompressible isotropic elastic half-spacehas been examined recently by the authors (Ogden & Sotiropoulos,1997). In the present paper the corresponding analysis for compressiblematerials is detailed. In the two-dimensional context consideredfor incompressible materials the (homogeneous) plane waves werenecessarily shear waves. By contrast, in the compressible contextpure shear waves can propagate only in specific directions inthe considered principal plane and, in a general direction,a quasi-shear wave may be accompanied by a quasi-longitudinalwave, as is the case in the anisotropic linear theory. The dependenceof the (in-plane) slowness section on the pre-stress (and finitedeformation) and on the choice of constitutive law is elucidated.This information is used to determine the reflection coefficientsfor reflection of either a (quasi-) shear wave or a (quasi-)longitudinal wave from the boundary of the half-space and tocharacterize the different cases which arise depending on thegeometry of the slowness section. The theoretical results are illustrated by numerical calculationsfor the range of possible types of behaviour with referenceforms of strain-energy function and different states of finitedeformation and to the question of stability of the half-space.     

Mathematical simulation of deformation and wave processes in multilayered structures

The deformation and wave processes induced by collisions of an impactor with deformable layered targets of various configurations are analyzed. The numerical solution of such problems is associated with an adequate treatment of wave processes in a continuous medium, which is an especially difficult task in the case of layered targets. To deal with the former problem, it is proposed to use adaptive Lagrangian triangular meshes. Wave processes are simulated using the grid-characteristic method, which can serve as a basis for algorithms that do not fail near the boundary of the computational domain and at numerous material interfaces. Additionally, hybrid and hybridized grid-characteristic schemes are applied that substantially improve numerical solutions with steep gradients (discontinuous solutions). These methods provide an adequate treatment of wave processes in layered targets (wave reflection and refraction at contact surfaces, secondary-wave interaction, changes in the conditions on these boundaries, etc.).     

Weak and Strong Interactions between Internal Solitary Waves

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.     

A short note on the asymptotic stability of an oscillation-free eikonal splitting method

Different beam propagation methods (BPMs) have been fundamental in modern electromagnetical wave simulations. Challenges of the numerical strategy include the computational efficiency and stability, in particular when highly oscillatory optical waves are present. This paper concerns an eikonal splitting BPM scheme for two-dimensional paraxial Helmholtz equations together with transparent boundary conditions in slowly varying envelope approximations of active laser beams. It is shown that the finite difference method investigated is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in highly oscillatory wave applications.     

自由剪切流大尺度结构的二次稳定性*

本文用二次稳定性理论研究自由剪切湍流中周期性基本流空间增长扰动的稳定性。数值结果表明三维亚谐扰动对横向波数有很强的选择性,二维亚谐波的空间增长率最大。与之相反,基本模式的三维扰动在很大的波数范围内存在不稳定性,证明β=0时存在“转移”不稳定性;当KH波的幅值A≥0.06时出现分叉现象。     

Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

多级透水板消波的一个特解

本文对无限长常水深平底渠道中一小振幅入射波经由多个间隔相等、透水性能一致的细孔透水板的反射和透射进行了研究,得到了相邻两板间距l为入射波半波长的倍数时的一个特解.结果表明,当无量纲的孔隙影响参数G₀等于透水板个数的一半时消波效果最佳,入射波能量的50%能被消掉.此时反射波与透射波的振幅相等.     

On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid

A study is made of over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in an isothermal compressible electrically conducting fluid in the presence of an external magnetic field. The reflection and transmission coefficients of hydromagnetic acoustic-gravity waves incident upon magnetic shear layer are calculated. The invariance of wave-action flux is used to investigate the properties of reflection, transmission and absorption of the waves incident upon the shear layer, and then to discuss how these properties depend on the wavelength, length scale of the shear layers, and the ratio of the flow speed and phase speed of the waves. Special attention is given to the relationship between the wave-amplification and critical-level behaviour. It is shown that there exists a critical level within the shear layer and the wave incident upon the shear layer is over-reflected, that is, more energy is reflected back towards the source than was originally emitted. The mechanism of the over-reflection (or wave amplification) is due to the fact that the excess reflected energy is extracted by the wave from the external magnetic field. It is also found that the absence of critical level within the shear layer leads to non-amplification of waves. For the case of very large vertical wavelength of waves, the coefficients of incident, reflected and transmitted energy are calculated. In this limiting situation, the wave is neither amplified nor absorbed by the shear layer. Finally, it is shown that resonance occurs at a particular value of the phase velocity of the wave.     

Convergence analysis of non-conforming Trigonometric Finite Wave Elements

Trigonometric Finite Wave Elements (TFWE) are finite elements for solving problems in computational optics. The solution of those problems consist of highly oscillatory waves. TFWE are designed for obtaining optimal approximation properties for such kinds of waves with a changing wave number k. In this article, we study the convergence properties of 2-dimensional non-conforming TFWE by applying Strang’s Lemma.     

Undercompressive shock waves in Hall‐magnetohydrodynamics

We study the propagation of nonlinear waves in a Hall‐magnetohydrodynamic model. An asymptotic method is used to derive the Gardner‐Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second‐ and third‐order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting—a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave‐fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.     

关于《保辛水波动力学》的一个注记

研究完全非线性水波的数值模拟方法，将《保辛水波动力学》中提出的保辛摄动方法，扩展到非线性水波压强的分析.数值算例表明，该文方法可用于水波的非线性演化分析，也可用于模拟孤立波、尖锐波峰的涌波等非线性水波，并给出水波的压强分布.     

Reflection and refraction of attenuated waves in semi-infinite elastic solid media

In earthquakes it is known that waves of one type starting from the origin inside the earth produce waves of other types after impact at the free surface of the earth. A theoretical discussion is given here dealing with reflection and refraction of waves which have their amplitudes increasing with depth similar in nature to the initial diverging earthquake waves which after reflection produce the apparent surface waves in which the displacement decreases with depth. In one case it has been found that at a free surface a plane wave of SV type, travelling parallel to the surface, is capable of generating surface waves after internal reflection which under a certain state of attenuation are identical with the Rayleigh Waves.