Subharmonic resonance of a trapped wave near a vertical cylinder by narrow-banded random incident waves

It is well-known that near an infinite linear array of periodically spaced cylinders trapped waves of certain eigenfrequencies can exist. If there are only a finite number of cylinders in an infinite sea, trapping is imperfect. Simple harmonic incident waves can excite a nearly trapped wave at one of the eigen frequencies through a linear mechanism. However, the maximum amplification ratio increases monotonically with the number of the cylinders; hence the solution is singular in the limit of infinitely many cylinders. Recently, a nonlinear theory of subharmonic resonance of perfectly trapped waves has been completed. In this article the theory is further extended to random incident waves with a narrow spectrum centered near twice the natural frequency of the trapped wave. The effects of detuning and bandwidth of the spectrum are examined. Dedicated to Professor J. N. Newman on his 70th birthday. We wish to express our profound admiration for Professor Newman’s scientific contributions and leadership in the ship-hydrodynamics discipline. The relation between this article and an early work of his reflects in part his impact on us.     

Trapping and Near-Trapping by Arrays of Cylinders in Waves

In this paper, some recent developments and new results concerning the trapping of waves by arrays of vertical circular cylinders is presented. In particular, the cases are examined when there is a circular arrangement of cylinders and both finite and infinite periodic linear arrays of identical cylinders. Only for the infinite array is there pure trapping of waves – known as Rayleigh–Bloch or edge waves – which, for particular dominant wavenumbers, reduce to the well-known trapped-mode solutions for a cylinder between two parallel walls having either Neumann or Dirichlet conditions upon them. This latter case is considered separately and some new results are presented. In the circular array and finite linear array the concept of near-trapping is introduced where large resonant motions are found to occur at certain frequencies of the incident wave field. In the case of the finite linear array, these near-trapping frequencies are related to the Rayleigh–Bloch trapped-wave frequencies for the infinite array. Finally, the case when there are two or more lines of cylinders in the linear array is examined.     

The trapping of surface waves by multiple submerged horizontal cylinders

The existence of edge waves, or trapped modes, travelling above a single long horizontal submerged cylinder is well established in the linearised theory of water waves. In the present paper, the possibility of wave-trapping by multiple submerged horizontal circular cylinders is considered. The trapped mode solutions are constructed by means of a multipole approach combined with an addition formula for Bessel functions and requires finding the non-trivial solutions of a real infinite system of algebraic equations. The case of a single submerged cylinder is returned to briefly, where results for symmetric trapped modes are reproduced and new numerical results for antisymmetric modes are presented. A large range of results are also presented for multiple cylinders.     

Motion trapping structures in the three-dimensional water-wave problem

Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded fluid with a free surface. It has been known for some time that such modes are supported by certain structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes are also possible for freely floating structures that are able to respond to the hydrodynamic forces acting upon them. For a freely floating structure such a mode is a coupled oscillation of the fluid and the structure that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry. Nick Newman has made many important contributions to the theory of the interaction between water waves and structures, and his panel code WAMIT has been adopted as one of the industry standards for the calculation of wave loading on offshore structures. It gives us great pleasure to celebrate Nick’s achievements through the presentation of this work on a new type of structure that both draws on his theoretical work and uses WAMIT to perform relevant computations.     

On Helmholtz and higher-order resonance of twin floating bodies

The Helmholtz mode and other symmetric modes of resonance of a moonpool between two heaving rectangular floating cylinders are investigated. The hydrodynamic behavior around these resonant modes is examined together with the associated mode shapes in the moonpool region. It is observed that near each of the resonance frequencies, the damping coefficient can vanish. The Helmholtz mode is characterized by a region of modest variation of added-mass value from negative to positive near the Helmholtz frequency. The peaks are, however, bounded with the cross-over point in sign corresponding to a bounded spike in damping. The higher-order resonant modes are characterized by the presence of standing waves in the moonpool, which leads to large spikes in the hydrodynamic behavior near the resonance frequencies. The Helmholtz frequency has a distinct value, while the higher-order resonances occur at fairly regular intervals of the frequency parameter, σ²(w ? b)/g, where w ? b is the moonpool gap. The parametric dependence of the hydrodynamic behavior on frequency and geometry is discussed. With best wishes to my colleague and good friend, Nick Newman, on the occasion of his 70th birthday. A leader and staunch supporter of marine hydrodynamics, Nick has expanded the reach and influence of this field through his insights and publications. His contributions have been wide-ranged and his graciousness to young researchers is exemplary. May he enjoy the best of health in the years to come. R.W. Yeung      

A note on time integrators in water-wave simulations

The importance of a suitable temporal integrator for fully nonlinear simulations of surface gravity waves is emphasized. Via numerical examples, it is demonstrated that constant-step procedures are inefficient. This relates to the practice of energy-conserving symplectic integration, assuming constant time steps, and is compared to direct numerical simulations using Runge–Kutta integrators with variable time-step control. It is concluded that the latter with automatic variable time-step control is the more efficient, and should be applied. The practice and efficiency of a stabilization procedure for the time-step selection is described and illustrated. An important point of the method is that the linear part of the prognostic equations is integrated analytically, which means that this part is obtained to machine presicion for any (large) time step (time interval). The evolution and instabilities of highly nonlinear water waves in three dimensions are exemplified through an accurate and efficient time-integration procedure. We are grateful to Professor J. N. Newman for his longstanding and pioneering contributions to the research field of marine hydrodynamics. His analytical and numerical works on fundamental and industrial problems in relation to water waves and their interaction with floating bodies have inspired us and many fellow scientist world wide over long time.     

On physical and numerical resonances for water wave problems using the dual boundary element method

Scattering problems of water waves impinging on bottom-mounted vertical cylinders are solved by using the dual boundary element method (DBEM). Both resonances due to near-trapped mode (physics) and fictitious frequency (mathematics) are examined. It is found that the near-trapped mode is a physical phenomenon and the fictitious frequency stems from the numerical instability. A trapped mode is associated with a singularity that lies on the real axis of complex wave number. A near-trapped mode means a localized behavior that energy is trapped in a truncated periodical structure. Critical wave number for the near-trapped mode and fictitious frequency of numerical instability are detected in this work. Numerical oscillation of the resultant force near the fictitious frequency is also observed by using the DBEM. Fictitious frequencies depend on the formulation instead of the specified boundary condition. Both the Burton and Miller approach and the CHIEF method are employed to alleviate the problem of irregular frequencies. Highly rank-deficiency matrices for four identical cylinders are numerically examined and the rank is promoted by adding valid CHIEF constraints. Parameter study of spacing and radius of cylinders on the near-trapped mode and fictitious frequency is also addressed. Several examples of water wave interaction by circular and square cylinders are demonstrated to see the validity of the present formulation.     

声子晶体中弹性波带隙与散射

声子带隙产生条件是声子晶体研究的一个重要问题,首先对钢圆柱体嵌入空气基质组成的声子晶体散射截面进行计算,然后用平面波方法计算了弹性波带结构。结果表明声子带隙将出现在两个完全分离的共振态之间。因此,声子晶体产生完全带隙的条件是弹性波在基质材料中的传播被禁止并远离共振。     

柱间有相对运动坐底圆柱群的绕射辐射分析

圆柱群是深海海工建筑物的主要承力结构.目前对圆柱群进行的水动力分析大多考虑将圆柱群作为一个整体的情况,即柱群中各柱无相对运动;对于柱间有相对运动圆柱群的研究较少.为此,采用精确代数法研究柱间有相对运动坐底直立圆柱群的绕射-辐射问题.首先开展了柱群中各圆柱做给定单独运动的辐射分析;之后在有入射波浪的情况下,分析了各柱有相对运动圆柱群的绕射-辐射问题;进而求得各圆柱上的水动力、振动幅值等量.作为验证,先将退化情况的结果与已有文献进行了对比.进一步的计算结果表明,在某些波数下,分析中是否考虑柱间相对运动对结果影响较大.     

Acoustic emission from a surface-breaking crack under cyclic loading

Summary An isotropic, homogeneous, elastic half-space is subjected to a uniform stress system composed of a constant tensile stress with a superimposed cyclic tensile stress, both parallel to the free surface. The cyclic stresses are assumed to generate a surface-breaking crack of length l(t) which propagates normal to the surface. The unloading of the crack faces generates acoustic emission, which is primarily composed of surface waves. The elastodynamic reciprocity theorem for time-harmonic waves is used to determine the radiated system of transient and steady state surface waves. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

On Axisymmetric Longitudinal Wave Propagation in Double-Walled Carbon Nanotubes

An attempt is made into the investigation of longitudinal axisymmetric wave propagation in the DWCNT with the use of the exact equations of motion of the linear theory of elastodynamics. The DWCNT is modeled as concentricallynested two circular hollow cylinders between which there is free space. The difference in the radial displacements of these cylinders is coupled with the van der Waals forces and it is assumed that full slipping conditions occur on the inner surface of the outer tube and on the outer surface of the inner tube. Numerical results on the influence of the problem parameters such as the thickness/radius ratio, the distance between the tubes of the DWCNT and the van der Waals forces on the dispersion curves are presented and discussed. In particular, it is established that new types of modes arise under propagation of axisymmetric longitudinal waves as a result of the van der Waals interaction between the tubes of the DWCNT. The limit values of the wave propagation velocity are also analyzed.     

The influence of inertia forces on steady-state rolling contact

Summary The influence of inertia forces on steady-state rolling contact between two elastic cylinders is investigated. It is assumed that the cylinders consist of the same material. The contact problem is dealt with analytically based on the two-dimensional constitutive relations which are derived by considering dynamic effects of moving loads on the surface of a half-space. Dynamic effects are found to be negligible for contact problems concerning practical engineering applications.With 4 Figures     

Diffraction of hydroelastic waves by multiple vertical circular cylinders

The diffraction problem of hydroelastic waves beneath an ice sheet by multiple bottom-mounted circular cylinders is considered. The elastic thin-plate theory is adopted to model the ice sheet, while the linearized velocity potential theory adopted for the fluid flow. The velocity potential corresponding to each cylinder is expanded into a series of eigenfunctions, and the total potential is expressed as a summation of these expansions over the entire NC number of cylinders. For each cylinder, the Green’s second identity is used outside its domain to obtain a set of linear equations. For each different cylinder, the domain used is different. NC cylinders give NC sets of coupled linear equations. Investigations are made for different arrangements of cylinders, piercing through ice sheets. Results for the wave forces on the cylinders with clamped and free conditions of the ice edge are obtained. Physical phenomena corresponding to cylinders arranged in square, in an array, in a double-array and in a staggered double array are discussed.     

多向不规则波浪作用下群墩结构所受波浪力的实验研究

实际海域中波浪是多向不规则的,波浪的方向分布宽度对作用在结构物上的波浪力具有显著的影响。为了研究方向分布对大尺度群墩结构所受波浪力的影响规律,该文通过物理模型实验模拟了多向不规则波浪与群墩结构的相互作用。通过对波浪荷载的时间过程线进行统计分析得到了作用在墩柱上的正向力、横向力。实验结果表明波浪的方向分布对群墩结构中墩柱所受横向力具有明显的影响。多向波浪作用时,墩柱所受横向力随着方向分布宽度的增大有显著的增大趋势,且后排墩柱受方向分布宽度的影响较前排墩柱更明显。     

Pulse propagation in a linear and nonlinear diatomic periodic chain: effects of acoustic frequency band-gap

One-dimensional nonlinear phononic crystals have been assembled from periodic diatomic chains of stainless steel cylinders alternated with Polytetrafluoroethylene spheres. This system allows dramatic changes of behavior (from linear to strongly nonlinear) by application of compressive forces practically without changes of geometry of the system. The relevance of classical acoustic band-gap, characteristic for chain with linear interaction forces and derived from the dispersion relation of the linearized system, on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes are investigated with numerical calculations and experiments. The limiting frequencies of the acoustic band-gap for investigated system with given precompression force are within the audible frequency range (20–20,000 Hz) and can be tuned by varying the particle’s material properties, mass and initial compression. In the linear elastic chain the presence of the acoustic band-gap was apparent through fast transformation of incoming pulses within very short distances from the chain entrance. It is interesting that pulses with relatively large amplitude (nonlinear elastic chain) exhibit qualitatively similar behavior indicating relevance of the acoustic band gap also for transformation of nonlinear signals. The effects of an in situ band-gap created by a mean dynamic compression are observed in the strongly nonlinear wave regime.     

Trapping of water waves by moored bodies

Certain types of floating bodies are known to support trapped modes, with oscillatory fluid motion near the body and no energy radiation in the far field. Previous work has considered either fixed bodies, where the boundary conditions are homogeneous, or bodies which are freely floating and moving without any exciting force. For a fixed body the existence of a trapped mode implies that there is no unique solution of the boundary-value problem for the velocity potential with a prescribed body motion. For a free body which supports a trapped mode, the solution of the coupled problem for the motions of the fluid and body does not have a unique solution. A more general case is considered here, of a body with a linear restoring force such as an elastic mooring. The limiting cases of a fixed and free body correspond to infinite or zero values of the corresponding spring constant. A variety of body shapes are found including cylinders in two dimensions and axisymmetric bodies in three dimensions, which illustrate this more general case of trapping and provide a connection between the fixed and free cases.     

Axisymmetric vibration analysis of laminated hollow cylinders with ring stiffeners

This paper presents an axisymmetric vibration analysis of laminated hollow cylinders composed of monoclinic layers and stiffened by ring stiffeners. A successive approximation approach, which is based on a transfer matrix and then an equivalent stiffness matrix formulation, is used to enable three-dimensional solutions to be found. It is assumed that the ring stiffeners are attached to the lateral surfaces of the cylinder and only provide elastic supports in radial direction. These constraints are imposed by using Lagrange multipliers to couple the responses of a number of vibration modes of corresponding cylinders without stiffeners. Using this method the natural frequencies of a stiffened cylinder are found to be the eigenvalues of a constraint matrix and the predictions can be arbitrarily close to the exact three-dimensional solutions.     

Elastic constants of particle and fiber reinforced metal matrix composites

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropies between directions of the fiber rich plane and that normal to the plane could be predicted by the model.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.     

On the propagation of waves in an electromagnetic elastic solid

The coupling of electromagnetic and elastic waves is considered from the standpoint of linear elasticity and a linearized electromagnetic theory. The problem of plane waves traveling through a uniform magnetostatic field is considered and couplings of the waves are studied. An investigation of the same problem for a uniform electrostatic field shows that the usual plane waves propagate without any change in their phase velocities but that the mechanical waves are accompanied by small fluctuating electromagnetic fields. The problem of the vibration of a free infinite elastic plate in a large magnetostatic field is examined under the assumption that the resulting electromagnetic fields are quasistationary. Frequency equations are obtained for both symmetric and antisymmetric vibrations and the damping caused by the field for both the first two symmetric and antisymmetric modes is obtained as a linear correction to the usual free plate frequencies.     

Linear and nonlinear forced oscillations of a flow of a bubbly liquid in a deformable pipe (survey)

Data reported in numerous publications is used as a basis for analyzing the possibility of the generation of shock waves in a flow moving inside a pipe. A piston vibrating at one end of a tube having a nozzle attached to its other end was chosen as the source of the oscillations. The equations of flow of a bubbly liquid are given for the case when long waves propagate in the liquid and a transverse velocity exists in it. Different formulations of the boundary conditions on the nozzle are given, and methods are described for solving the boundary-value problem far from resonance frequencies. The theoretical results obtained by passing to the gas-flow limit agree with the information obtained by other authors and experimental data. It is established that there are frequencies at which periodic hydraulic shocks are generated in the pipes that are examined. Their amplitude is much greater than the amplitude of the shock waves generated during resonance in a column of gas. By varying the initial content of gas in the mixture, the acoustic properties of the nozzle, and the elastic properties of the pipe, it is possible to control oscillations in the flow, alter the resonance frequencies, and eliminate or generate hydraulic shocks. When the frequencies of oscillation associated with the shocks are the same as the natural frequency of the pipe, the latter may undergo large displacements that can ultimately lead to fatigue failure of the pipe or its supports.Translated from Problemy Prochnosti, No. 9, pp. 3–29, September, 1994.