Elastic waves in fibre composite laminates

This paper considers the propagation of elastic waves in an eight-ply quasi-isotropic laminate arising from line sources of dislocation located at each of the seven interfaces in turn. The line source sets up a straight crested wave travelling along the laminate in a direction normal to the load line, and the elastodynamic equations within each layer are solved by taking the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate in the direction of propagation. The resulting system of six first-order differential equations in each layer is solved to obtain the transforms of the displacement and stress components throughout the laminate. The time history of any displacement or stress component at any location may then be recovered by numerical inversion of the double transform. Examples are shown of the time history of the normal displacement of the top surface of the laminate at a distance of 20 plate thicknesses from the plane of action of the sources. The numerical inversion involves a summation over different modes of Rayleigh-Lamb waves in the laminate and contributions to the overall response from some of the individual modes are displayed.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

Sound radiation from fluid loaded orthogonally stiffened plates

The radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners. The stiffeners are assumed to be equally spaced and exert only forces on the plate. The response to a convected harmonic pressure is found by using Fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces. These forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation.     

Excitation of a tube wave in a borehole by an external isotropic point source

The excitation of a tube wave in an infinite fluid-filled borehole by an external isotropic point source is considered. The solution to the problem is obtained in the form of a double integral with respect to the ray parameter (slowness) and frequency. The integral with respect to the slowness is transformed to a contour integral in the complex slowness plane and then reduced to the integral over the edges of the cut of the vertical slowness function and the semiresidues at the poles. An asymptotic expression for the wave field in the borehole is obtained with allowance for the radiation condition at infinity. It is shown that, when a longitudinal spherical wave is incident on the borehole, only one low-frequency Stoneley wave is excited and not two, as was assumed earlier [1].     

A convolution integral for Fourier transforms on the groupSL(2,C)

The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.     

Free vibration analysis of continuous rectangular plates

Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.     

Exact and Asymptotic Solutions for the Field of a Point Source in a Transversely Isotropic Medium

Exact and asymptotic solutions are obtained for the acoustic field generated by an isotropic pulsed point source in an infinite transversely isotropic elastic medium. The exact solution for the displacement field is obtained in the form of a double integral over the horizontal slowness and the frequency by using the method of integral transforms. The calculation of the integral over the horizontal slowness by the method of stationary phase reduces the exact solution to an asymptotic solution that is convenient for numerical calculations. Formulas are given for calculating the spreading factors and the wave fronts of quasi-longitudinal qP-waves and quasi-transverse qSV-waves. With the formulas obtained, the displacement field of a point source is investigated for a particular transversely isotropic medium.     

Depolarization of the electromagnetic field scattered by a three-dimensional large-scale irregularity of the lower ionosphere

This paper develops analytical and numerical methods for the solution of three-dimensional problems of radio wave propagation. We consider a three-dimensional vector problem for the electromagnetic field of a vertical electric dipole in a planar Earth-ionosphere waveguide with a largescale local irregularity of negative characteristics at the anisotropic ionospheric boundary. The field components at the boundary surfaces obey the Leontovich boundary conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr≫1, where r is the distance from the source or receiver to the nearest point of the irregularity, k=2π/λ, and λ is the radio wavelength) integration over the direction perpendicular to the ray path, we transform this system to one-dimensional integral equations where integration contours represent the geometric contour of the irregularity. The system is numerically solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and subsequent iterations. The proposed numerical algorithm reduces the computer time required for the solution of this problem and is applicable for studying both small-scale and large-scale irregularities. We obtained novel estimates for the field components that are not excited by the source but result entirely from scattering by the sample three-dimensional ionospheric irregularity.     

The Rate of Convergence of Fourier Coefficients for Entire Functions of Infinite Order with Application to the Weideman-Cloot Sinh-Mapping for Pseudospectral Computations on an Infinite Interval

We analytically compute the asymptotic Fourier coefficients for several classes of functions to answer two questions. The numerical question is to explain the success of the Weideman-Cloot algorithm for solving differential equations on an infinite interval. Their method combines Fourier expansion with a change-of-coordinate using the hyperbolic sine function. The sinh-mapping transforms a simple function like exp(-z²) into an entire function of infinite order. This raises the second, analytical question: What is the Fourier rate of convergence for entire functions of an infinite order? The answer is: Sometimes even slower than a geometric series. In this case, the Fourier series converge only on the real axis even when the function u (z) being expanded is free of singularities except at infinity. Earlier analysis ignored stationary point contributions to the asymptotic Fourier coefficients when u(z) had singularities off the real z-axis, but we show that sometimes these stationary point terms are more important than residues at the poles of u(z).     

Diffraction of cylindrical waves by an ideally conducting wedge in an anisotropic plasma

The problem of diffraction of cylindrical waves by an ideally conducting wedge in an anisotropic plasma is formulated and solved. The integral equations for the field are reduced to function equations, which are solved with the aid of a special function that is introduced. The properties of this function are studied. The general solution is represented as a double contour integral in the plane of a complex variable. The radiation field and surface waves for a number of special cases are analyzed: a source of cylindrical waves on an edge; at infinity; etc. Diffraction in a half-plane is studied separately.     

The non-stationary freefield response for a moving load with a random amplitude

This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.     

A robust spatial filtering technique for multisource localization and geoacoustic inversion

Geoacoustic inversion and source localization using beamformed data from a ship of opportunity has been demonstrated with a bottom-mounted array. An alternative approach, which lies within a class referred to as spatial filtering, transforms element level data into beam data, applies a bearing filter, and transforms back to element level data prior to performing inversions. Automation of this filtering approach is facilitated for broadband applications by restricting the inverse transform to the degrees of freedom of the array, i.e., the effective number of elements, for frequencies near or below the design frequency. A procedure is described for nonuniformly spaced elements that guarantees filter stability well above the design frequency. Monitoring energy conservation with respect to filter output confirms filter stability. Filter performance with both uniformly spaced and nonuniformly spaced array elements is discussed. Vertical (range and depth) and horizontal (range and bearing) ambiguity surfaces are constructed to examine filter performance. Examples that demonstrate this filtering technique with both synthetic data and real data are presented along with comparisons to inversion results using beamformed data. Examinations of cost functions calculated within a simulated annealing algorithm reveal the efficacy of the approach.     

Scattering of SH wave by a semi-cylindrical salient near vertical interface in the bi-material half space

With the aid of the Green function method and image method, the problem of scattering of SH-wave by a semi-cylindrical salient near vertical interface in bi-material half-space is considered to obtain its steady state response. Firstly, by the means of the image method, Green function which is the essential solution of displacement field is constructed to satisfy the stress-free condition on the horizontal boundary in a right-angle space including a semi-cylindrical salient and bearing a harmonic out-of-plane line source force at any point on the vertical boundary. Secondly, the bi-material is separated into two parts along the vertical interface, then unknown anti-plane forces are applied on the vertical interface, and according to the continuity condition, the first kind of Fredholm integral equations is established to determine unknown anti-plane forces by “the conjunction method”, then the integral equations are reduced to the linear algebraic equations by effective truncation. Finally, the dynamic stress concentration factor (DSCF) around the edge of semi-cylindrical salient is calculated, and the influences of incident wave number, incident angle, effect of interface and different combination of material parameters, etc. on DSCF are discussed.     

The flexural vibration of a line-stiffened plate with fluid loading,part I: Analysis

This paper considers the vibration of a symmetrical system consisting of an infinite fluid loaded plate bearing a finite number of parallel stiffeners. The system is driven at the central stiffener by a travelling wave line force. Formal solutions for the equations of motion are found in terms of cosine transforms. Manipulation of the equations allows the problem to be reduced to the solution of a set of linear algebraic equations in the vibration amplitudes at the stiffeners. The coefficients in these equations depend in a simple way upon the stiffener parameters, and upon particular values of the cosine transform of a function which depends only on the plate and fluid parameters, and the stiffener positions.     

Pulse propagation over smooth irregular terrain

The source of the electromagnetic field is assumed to be a vertical electric dipole at height ξ above the surface of the plane earth with arbitrary time varying moment. The problem of finding the transient field of this dipole when the earth is allowed to be slightly rough is solved by means of a perturbation analysis, repeated application of integral transforms and their inversion on the base of Cagniard's method with the modification of de Hoop.     

Two-dimensional radiative transfer in a cylindrical geometry with anisotropic scattering

Exact integral equations are derived describing the source function and radiative flux in a two-dimensional, radially infinite cylindrical medium which scatters anisotropically. The problem is two-dimensional and cylindrical because of axisymmetric loading. Radially varying collimated radiation is incident normal to the upper surface while the lower boundary has no radiation incident upon it. The scattering phase function is represented by a spike in the forward direction plus a series of Legendre polynomials. The two-dimensional integral equations are reduced to a one-dimensional form by separating variables for the case when the radial variation of the incident radiation is a Bessel function. The one-dimensional form consists of a system of linear, singular Fredholm integral equations of second kind. Other more complex boundary conditions are shown to be solvable by a superposition of this basic Bessel function case. Diffusely incident radiation is also considered.     

The numerical approximation of a delta function with application to level set methods

It is shown that a discrete delta function can be constructed using a technique developed by Anita Mayo [The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Comput. 21 (1984) 285–299] for the numerical solution of elliptic equations with discontinuous source terms. This delta function is concentrated on the zero level set of a continuous function. In two space dimensions, this corresponds to a line and a surface in three space dimensions. Delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function. The numerical implementation of these delta functions achieves the expected order of accuracy.     

Dynamic response of a prestressed,orthotropic thick plate strip to a moving line load

This paper is a study of the steady state response of an orthotropic plate strip to a moving line load. The plate is of infinite length and subjected to initial in-plane stresses parallel and perpendicular to the edges. The solution is obtained on the basis of a thick plate theory which takes into account the effects of shear deformation and rotatory inertia. The critical speed of the load which brings about a resonance effect in the system is determined. Further, the bending moment in the plate is calculated for several values of the load speed and the initial stress parameters and shown graphically as a function of the space variable moving with the load.     

Nonlinear and active two-dimensional cochlear models: time-domain solution

A numerical solution method for two-dimensional (2-D) cochlear models in the time domain is presented. The method has particularly been designed for models with a cochlear partition having nonlinear and active mechanical properties. The 2-D cochlear model equations are reformulated as an integral equation for the acceleration of the basilar membrane (BM). This integral equation is discretized with respect to the spatial variable to yield a system of ordinary differential equations in the time variable. To solve this system, the variable step-size, fourth-order Runge-Kutta method described in Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] is used. This method is robust and computationally efficient. The incorporation of a simple middle-ear model can be handled by this method. The method can also be extended to models in which the cochlear partition at each point along its length is represented by more than one degree of freedom.     

Shells and plates loaded by dynamic moments with special attention to rotating point moments

The response of thin shells to line or point moment excitation is formulated by way of distributed moment fields. Twisting moments in the tangent plane are part of this formulation. The approach is illustrated by using Love's thin shell theory, but is valid for any other thin shell theory as well. Dirac delta functions are used to describe line and point moments. As a first example, the response of a plate to a rotating moment is evaluated and shown to be identical to the solution obtained by Bolleter and Soedel [1] by a Green function approach. The three-directional response of a circular cylindrical shell to a rotating moment is given as the second example. It is a technically significant case that has not been treated in the literature before.