Scattering of water waves by thin vertical plate submerged below ice-cover surface

The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functions across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.     

Scattering of water waves by a submerged thin vertical elastic plate

The problem of water wave scattering by a thin vertical elastic plate submerged in infinitely deep water is investigated here assuming linear theory. The boundary condition on the elastic plate is derived from the Bernoulli–Euler equation of motion satisfied by the plate. This is converted into the condition that the normal velocity of the plate is prescribed in terms of an integral involving the difference in velocity potentials (unknown) across the plate multiplied by an appropriate Green’s function. The reflection and transmission coefficients are obtained in terms of integrals involving combinations of the unknown velocity potential on the two sides of the plate and its normal derivative on the plate, which satisfy three simultaneous integral equations, solved numerically. These coefficients are computed numerically for various values of different parameters and are depicted graphically against the wave number for different situations. The energy identity relating these coefficients is also derived analytically by employing Green’s integral theorem. Results for a rigid plate are recovered when the parameters characterizing the elastic plate are chosen negligibly small.     

Scattering of water waves by inclined thin plate submerged in finite-depth water

Summary  The problem of water wave scattering by an inclined thin plate submerged in water of uniform finite depth is investigated here under the assumption of irrotational motion and linear theory. A hypersingular integral equation formulation of the problem is obtained by an appropriate use of Green's integral theorem followed by utilization of the boundary condition on the plate. This hypersingular integral equation involves the discontinuity in the potential function across the plate, which is approximated by a finite series involving Chebyshev polynomials. The coefficients of this finite series are obtained numerically by collocation method. The quantities of physical interest, namely the reflection and transmission coefficients, force and moment acting on the plate per unit width, are then obtained numerically for different values of various parameters, and are depicted graphically against the wavenumber. Effects of finite-depth water, angle of inclination of the plate with the vertical over the deep water and vertical plate results for these quantities are shown. It is observed that the deep-water results effectively hold good if the depth of the mid-point of the submerged plate below the free surface is of the order of one-tenth of the depth of the bottom. Received 30 November 2000; accepted for publication 26 June 2001     

Water wave scattering by a submerged circular-arc-shaped plate

The problem of water wave scattering by a thin circular-arc-shaped plate submerged in infinitely deep water is investigated by linear theory. The circular-arc is not necessarily symmetric about the vertical through its center. The problem is formulated in terms of a hypersingular integral equation for a discontinuity of the potential function across the plate. The integral equation is solved approximately using a finite series involving Chebyshev polynomials of the second kind. The unknown constants in the finite series are determined numerically by using the collocation and the Galerkin methods. Both the methods ultimately produce very accurate numerical estimates for the reflection coefficient. The numerical results are depicted graphically against the wave number for a variety of configurations of the arc. Some results are compared with known results available in the literature and good agreement is achieved. The suitability of using a circular-arc-shaped plate as an element of a water wave lens has also been discussed on the basis of the present numerical results.     

Diffraction of Love waves by a stress-free crack of finite width in the plane interface of a layered composite

A rigorous theory of the diffraction of Love waves by a stress-free crack of finite width in the interface of a layered composite is presented. The incident wave is taken to be either a bulk wave or a Love-wave mode. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing the integral equation method. The unknown quantity is expanded in terms of a complete sequence of expansion functions in which each separate term satisfies the edge condition. This leads to an infinite system of linear, algebraic equations for the coefficients of the expansion functions. This system is solved numerically. The scattering matrix of the crack, which relates the amplitudes of the outgoing waves to the amplitudes of the incident waves, is computed. Several reciprocity and power-flow relations are obtained. Numerical results are presented for a range of material constants and geometrical parameters.     

Hypersingular integral equation for multiple curved cracks problem in plane elasticity

The complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations. These hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor, SIF, for the problem considered. Numerical examples for double circular arc cracks are presented.     

DYNAMIC ANALYSIS OF A BURIED RIGID ELLIPTIC CYLINDER PARTIALLY DEBONDED FROM SURROUNDING MATRIX UNDER SHEAR WAVES

This paper investigates the dynamic behavior of a buried rigid ellipticcylinder partially debonded from surrounding matrix under the action of anti-planeshear waves (SH waves). The debonding region is modeled as an elliptic arc-shapedinterface crack with non-contacting faces. By using the wave function (Mathieufunction) expansion method and introducing the dislocation density function as anunknown variable, the problem is reduced to a singular integral equation which issolved numerically to calculate the near and far fields of the problem. The resonanceof the structure and the effects of various parameters on the resonance are discussed.     

剪切波作用下埋藏刚性椭圆柱与周围介质部分脱胶时的动力分析

本文利用波函数展开法和奇异积分方程技术研究了ＳＨ型反平面剪切波作用下埋藏刚性椭圆柱与周围介质部分脱胶时的动力特性．将脱胶区看作表面不相接触的椭圆弧形界面裂纹，利用波函数（Ｍａｔｈｉｅｕ函数）展开法，并引人裂纹面的位错密度函数为未知量，将问题归结为奇异积分方程，通过数值求解积分方程获得了远场和近场物理参量，并讨论了共振特性和各参数对共振的影响．     

A note on the diffraction of an obliquely incident surface wave by a partially immersed fixed vertical barrier

The problem of the diffraction of surface waves, obliquely incident on a partially immersed fixed vertical barrier in deep water, is solved approximately by reducing it to the solution of an integral equation, for small angle of incidence of the incident wave. The corrections to the reflection and transmission coefficients over their normal incidence values for small angle of incidence are obtained and presented graphically for some intermediate values of wave numbers.     

Scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.     

Modeling the interaction of solitary waves in magnetic tubes

The Cauchy problems of the propagation of a single wave and the interaction of two solitary waves of different amplitude are solved numerically for the case of slow symmetric surface waves in a magnetic tube. It is found that the solitary waves interact in the same way as the solitons of the known soliton equations such as the Korteweg-de Vries and Benjamin-Ono equations, i.e., preserve their shape after interacting. The way in which the solitons decrease at infinity is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 183–186, March–April, 1989.The author wishes to thank M. S. Ruderman for formulating the problem and V. B. Baranov for his interest in the work.     

A Comparison of Nonlinear Water Wave Models

We compare the numerical evolution of one-dimensional gravity waves in response to a traveling surface pressure pulse using a highly accurate boundary integral method and two relatively efficient approximate models (West et al. and Benney–Luke). In both water of finite-depth and in the deep-water limit, the steady state effect of the decaying pressure ramp is to create a profile which approximates a Stokes wave. Moreover, the transient surface profile appears to evolve through a series of Stokes waves of time-varying amplitude. Results show all three models yield similar predictions for lower amplitude waves, while the West et al. and boundary integral predictions differ from the Benney–Luke model at higher amplitudes.     

Scattering of flexural waves by a cracked Mindlin plate of soft ferromagnetic material in a uniform magnetic field

Consider the impingement of time harmonic flexural waves on a through crack in a soft ferromagnetic plate the surface of which is subjected to a uniform magnetic field at normal incidence. Mindlin's plate theory is used to account for the magneto-elastic interaction. For an incident wave that gives rise to moments symmetric about the crack plane, Fourier transforms are applied reducing the mixed boundary value problem to a Fredholm integral equation that can be solved numerically. The dynamic moment intensity factor versus frequency is computed to exhibit the influence of the magnetic field.     

Stress Intensity Factor for the Interaction between a Straight Crack and a Curved Crack in Plane Elasticity

Formulation in terms of hypersingular integral equations for the interaction between straight and curved cracks in plane elasticity is obtained using the complex variable functions method. The curved length coordinate method and a suitable numerical scheme are used to solve such integrals numerically for the unknown function, which are later used to find the stress intensity factor, SIF.     

Transient responses of repeated impact of a beam against a stop

The transient behavior of a cantilever beam, driven by periodic force and repeated impacting against a rod-like stop, is the subject of this investigation. As impact and separation phase take place alternately, the transient waves induced either by impacts or by separations will travel in more complicated ways. Thus the transient responses of both the beam and the rod during repeated impact become an important issue. In both impact phase and separation phase, the transient wave propagations are solved by the expansion of transient wave functions in a series of Eigenfunctions (wave modes). From the solutions, the answer of impact force is derived directly, so that the divergence problem, encountered in solving impact force numerically by a strongly non-linear equation coupled the unknown impact force with motions, has been avoided. Numerical results show the convergence of the time-step size and truncation number of wave modes in the calculations of impact force by the present method. As the transient wave effect is considered, the numerical results can show several transient phenomena involving the propagation of transient impact-induced waves, sub-impact phases, long-term impact motion, chatter, sticking motion, synchronous impact, non-synchronous impact (including asynchronous impact) and impact loss.     

Collapsing and Explosion Waves in Phase Transitions with Metastability,Existence, Stability and Related Riemann Problems

Collapsing waves were observed numerically before and were used to explain the ring formations in dynamic flows involving phase transitions with metastability. In this paper, necessary and sufficient conditions for collapsing type of waves to exist are given. The conditions are that the wave speed of the collapsing wave is not less than a number and is supersonic on both sides of the wave. Existence and non-existence conditions for the explosion waves are also found. The stability of these waves are studied numerically. Although there are infinitely many collapsing (or explosion) waves for a fixed downstream state, the collapsing (or explosion) wave appeared in the solution of Riemann problem is numerically verified to be the one with the slowest speed. Although a Riemann problem in the zero viscosity limit may have two solutions, one with, the other without, a collapsing (or explosion) wave, from the vanishing viscosity point of view, the one with a collapsing (or explosion) wave is numerically verified to be admissible.     

多个纵向环形界面裂纹对P波散射的动应力强度因子

研究多个纵向环形界面裂纹的Ｐ波散射问题。以裂纹面的位错密度函数为未知量,利用Ｆｏｕｒｉｅｒ积分变换,将问题归结为第二类奇异积分方程,然后通过数值求解,获得裂纹尖端的动应力强度因子。最后给出了双裂纹动应力强度因子随入射波频率变化的关系曲线。     

A special solution of wave dissipation by finite porous plates

The reflection and transmission of water waves caused by a small amplitude incident wave through finite fine porous plates with equal spacing and permeability in an infinitely long open channel of constant water depth and zero slope are studied. A special solution is obtained when the distance between the two neighbouring plates is an integral multiple of the half-wavelength of the incident wave. It is found, that when the dimensionless porous-effect parameter G_0 is equal to half the total plate number, the wave dissipation reaches a maximum, and only 50% of the incident wave energy remains in the reflected and transmitted waves. Meanwhile, the reflected and transmitted waves have the same amplitude.     

Effect of a viscous liquid loading on Love wave propagation

This paper describes a theory of surface Love waves propagating in a layered elastic waveguide loaded on its surface by a viscous (Newtonian) liquid. An analytical expression for the complex dispersion equation of Love waves has been established. The real and imaginary parts of the complex dispersion equation were separated and resulting system of nonlinear algebraic equations was solved numerically. The influence of the viscosity of liquid on the dispersion curves of phase velocity, the wave attenuation and the distribution of the Love wave amplitude is analyzed numerically. The propagation loss is produced only by the viscosity of liquids. Elastic layered waveguide is assumed to be loss-less. The numerical solutions show the dependence of the phase velocity change, the wave attenuation and the wave amplitude distribution in terms of the liquid viscosity and the wave frequency. The results of the investigations are fundamental and can be applied in the design and development of liquid viscosity sensors and biosensors, in Non-Destructive Testing (NDT) of materials, in geophysics and seismology.     

NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING

In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then transformed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coeffcients, which will later be used in finding the stress intensity factors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution.