Lamb's Solution Research Articles

Modeling of Acoustic Emission Wave and Its Parametric Characterization Based on Peak Sensor Sensitivity on an Aluminum Block

In this paper an acoustic emission (AE) signal has been generated analytically for a homogenous material with a surface load. For the validation of the AE theory, a simulation for a developed AE wave has been conducted with computational algorithms. Experimental validation for the AE wave generation and propagation have also been performed under similar conditions. Aluminum metal block has been selected as the homogenous propagation medium for both simulation and the experiment. A function generator has been used for the excitation of a burst sine signal on the surface. A piezoelectric AE resonance sensor receives the vertical response due to the vertical load of the function generator. The algorithm of Lamb's solution has been employed in the generation of a simulated AE wave. As the transfer function, the peak sensor sensitivity of the AE sensor (R15alpha) has been incorporated along with the acoustic impedance of the metal surface and sensor. For both detected and simulated AE waves, the signal parametric characterizations have been performed based on the rise time and the maximum amplitude. The attenuation properties of the maximum amplitude and the increase of rise time with the increment of source to sensor distance has been evaluated to validate the proposed method.

Hybrid Method Combines Transfinite Interpolation with Series Expansion to Simulate the Anti-Plane Response of a Surface Irregularity

AbstractThe responses to an incident plane SH wave on or near a surface irregularity which is embedded in an elastic half-plane are investigated. The surface irregularity represents a canyon, an alluvial valley or a hill. The wave function expansion method has been employed to solve surface irregularities, such as a semi-cylindrical canyon, a semi-cylindrical alluvial valley, or a semi-elliptical canyon and a semi-elliptical alluvial valley. These solutions to the scattering problem of SH wave can be used to test the accuracy of the other numerical methods. But solutions for surface irregularities with arbitrarily shapes cannot be found easily. A hybrid method combines the finite element method with series expansion is applied to solve scattering problems in this study. A subregion encloses the surface irregularity with a semi-circular auxiliary boundary can be meshed by the finite element method. By using the transfinite interpolation (TFI) produces excellent grid mesh on the subregion. The advantage of TFI is the flexibility to facilitate modeling of the subregion. On the other hand, the boundary data can be formulated by using a series representation with unknown coefficients. The Lamb's solution which satisfies the traction free condition and the radiation condition at infinity is implemented to be the basis function. The unknown coefficients can be obtained by satisfying the continuity conditions of the semi-circular auxiliary boundary between the subregion and the half-plane. The hybrid method that combines TFI with series expansion is successfully herein to solve the scattering problem by a surface irregularity. Numerical results in this study for special cases agree well with those in the published literatures. In this study, the steps and skills of hybrid method are described systematically and completely to solve the surface irregularity.

Development of non-contact SIBIE procedure for identifying ungrouted tendon duct

For non-destructive evaluation (NDE) of concrete structures, the impact-echo method has been successfully applied to locate defects and voids in concrete. To compensate the difficulty in selecting the resonant frequency and to visually locate a defect, an imaging procedure named stack imaging of spectral amplitude based on impact-echo (SIBIE) has been developed. Combining the SIBIE analysis with the detection by a laser vibrometer, a non-contact procedure is studied to develop an automated system. Since waveforms detected by the laser vibrometer are undistorted, elastic waves generated due to a steel-ball drop were investigated theoretically as Lamb's problem in elastodynamics. It is confirmed that the first motion of detected waves by the impact-echo method is identical to Lamb's solution for a surface pulse, when reflected waves arrive later than the Rayleigh wave. An applicability of a non-contact SIBIE procedure to identify an ungrouted tendon duct of plastic sheath was examined in the impact-echo tests, as well as the duct of metal sheath. In order to study theoretically the SIBIE procedure for identifying ungrouted tendon ducts, the three-dimensional boundary element method (3D-BEM) was applied to synthesize frequency spectra. It is confirmed that frequency spectra detected in the tests are in good agreement with those synthesized. An applicability of the SIBIE procedure is also confirmed by the synthesized spectra. Results of the experiments show that the presence of the ungrouted duct can be visually identified by the non-contact SIBIE procedure in both the cases of plastic sheath and metal sheath.

Essential properties of Boussinesq equations for internal and surface waves in a two-fluid system

Boussinesq equations describing motions of internal waves in a two-fluid system with the presence of free surface are theoretically derived, and the associated essential properties are examined in this study. Eliminating the dependence on the vertical coordinate from all variables, four equations constitute the Boussinesq model with two flexible parameters, z u and z l, which indicate the specific elevations, respectively, in the upper and lower fluids. Similar to the Boussinesq model for a single-layer fluid, z u and z l are determined by matching the linear dispersion relation with Lamb's solution. This determines the optimal model. In the analysis stage, this problem is classified into two cases, the thicker-upper-layer case and the thicker-lower-case case, to avoid the possible divergence of wave properties as the thickness ratio grows. Since there exist two modes of motions that may be excited, cases of both modes are separately analyzed. Linear characteristics including the amplitude ratios and normalized particle velocities are analyzed. Second-order harmonic waves are examined to validate nonlinear behaviors of present model. Results of linear and nonlinear investigations show that the present model indeed extends the applicable range of traditional Boussinesq equations.

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

A new three-dimensional fundamental solution to the Stokes flow was proposed by transforming the solid harmonic functions in Lamb's solution into expressions in terms of the oblate spheroidal coordinates. These fundamental solutions are advantageous in treating flows past an arbitrary number of arbitrarily positioned and oriented oblate spheroids. The least squares technique was adopted herein so that the convergence difficulties often encountered in solving three-dimensional problems were completely avoided. The examples demonstrate that present approach is highly accurate, consistently stable and computationally efficient.

Transient response of reservoir–dam–soil systems to earthquakes

AbstractA direct time‐domain numerical procedure is proposed to analyse the transient dynamic response of two‐dimensional reservoir–dam–soil systems. The reservoir extends to infinity and the dam is supported by an unbounded soil. The structure with either linear or non‐linear material properties is modelled by the Finite Element Method (FEM). The soil is assumed to be an elastic, isotropic and homogeneous half‐space represented by a boundary condition in the form of generalized impedance determined by the transient Lamb's solution due to a uniformly distributed traction imposed on the free surface, Guan and Novak.1 Moreover, a technique is developed to include the influence of the reservoir on the dam in terms of nodal accelerations along their interface at different time steps. The advantages of the proposed procedure are obvious. For example, it avoids any additional discretization of the boundaries except the soil–dam interface, and the influence matrix of the fluid is obtained explicitly using shape functions defined at the upstream face of the dam without the finite analysis of the reservoir so that it works very efficiently. Numerical results for a system consisting of reservoir, elastic dam and foundation subjected to the San Fernando, 1971 earthquake ground motion are presented.

Loose-part signal properties

Lamb's solution for two-dimensional wave propagation was used to identify plate bending wave characteristics observed in loose part impact signals. The surface displacement wave is shown to have large second derivatives at points that can be used to measure the impact contact time. General features of the initial impact wave shape provide guidance for the use of spectral analysis techniques to identify the contact time and wave arrival time when there is a low signal-to-noise ratio. Improvements in the determination of contact time improve the estimation of loose part mass and energy through the Hertz impact parameters. The shape of the initial impact wave shape can also be used to identify on which side of the plate the impact took place. This information has applications for Loose Part Monitoring System metal impact signal validation and for improved estimation of loose part parameters.

A new type of solution for plate vibrations at low frequencies

A new analytical solution is derived for the out-of-plane motion of an infinite plate. The plate is of uniform thickness and consists of a material that is isotropic, homogeneous and linearly elastic. The excitation is harmonic in time and space. The new solution is obtained as an asymptotic expression of a general solution and is valid for low frequencies down to the static case. However, in contrast to the corresponding solutions of the classical plate equations, it is valid only for relatively large wave numbers. It is shown that the new solution describes the stiffness of the plate in a transitional region. In the case of comparatively small wave numbers, the new solution turns into the solutions of the classical plate equations and in the case of large wave numbers into Lamb's solution for forced Rayleigh waves. The new solution also shows that two sets of free waves exist beside the usual bending and quasi-longitudinal waves. The practical significance is demonstrated by some examples from the field of building acoustics.

Creeping flow about a sphere

We present the general solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions. Exphclt results are obtained in cartesian coordinates. The relation wRh Lamb's method is indicated. We derive expressions for a set of fundamental tensors, introduced by Brenner, describing the perturbed flow. Finally the force density induced on the sphere is evaluated. In this paper we give the solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions for arbitrary incident flow. A method of solution was presented long ago by Lamb l) in terms of solid spherical harmonics, but his solution left something to be desired from a practical point of view. Lamb's solution is described in some detail in the monograph by Happel and Brenner2). The difficulty with Lamb's solution is that application to particular problems often requires writing the harmonics in terms of spherical coordinates, whereby a specific choice of axes, alien to the problem at hand, is made. For many problems an approach using cartesian coordinates is more convenient. In particular for the problem of hydrodynamic interaction of two spheres this approach has recently proved effective34). In the following we persistently use the cartesian point of view and solve the problem by the so-called method of cartesian ansatz. The solution is a direct generalization of earlier results obtained for constant, linear or quadratic incident flow3-7). At the end we show the relation to Lamb's solution. As a benefit we obtain explicit results for a set of tensors introduced by Brenner 8) in his fundamental paper on Stokes resistance. Brenner only gave explicit

Oscillating Cylinders and the Stokes' Paradox

Measurements are reported of the inertia coefficient k and damping coefficient k′ for a circular cylinder (radius a) oscillating at angular frequency ω in a fluid of kinematic viscosity ν. Good agreement with the theory of Stokes is obtained throughout the range 0.286<(2ν/ωa2)1/2<4.13 even when the Reynolds number R and the relative amplitude are not small compared with 1. Stokes' theory is expressed in terms of modified Bessel functions, and values of k and k′ are tabulated for 0.001<(a2ω/4ν)1/2<0.5. The relation to Stokes' paradox is considered by examining the damping force in the limit (ωa2/ν)1/2≪1. The damping force for a cylinder, unlike that for a sphere, does not reduce to the low R drag in this limit. However, comparison with Lamb's solution of the Oseen equations suggests an expression that agrees well with drag measurements up to R = 2.5.

Motion of a Drop Suspended in a Viscous Flow with Arbitrary Velocity Profile

For a single spherical drop suspended in an unbounded viscous fluid streaming with arbitrary velocity distribution, a general method is presented to calculate the external and internal flow fields of the drop and to evaluate deviation of the drop interface from sphericity. Based upon Lamb's solution in terms of solid spherical harmonics for Stokes flow, boundary conditions to be satisfied at the interface can be replaced by some simple relations between the solid harmonics, which, in conjunction with Laplace's condition for surface tension, give an expression for the drop deformation without detailed knowledge of the flow fields. As examples, both two-dimensional and axially symmetric Poiseuille flows are considered.

The Transmission of a Rayleigh Pulse round a corner

Lamb, in his classical paper of 1904, found the form of the Rayleigh pulse that travels over the surface of an elastic half-space, which has been struck along a line. Let such a pulse travel on one face of an elastic quarter-space, the crest-line being parallel to and travelling towards the edge. We enquire what form the Rayleigh pulse will have after passing round the corner. The differential equations and boundary conditions are reduced to operational form by the use of a Fourier transform with respect to time and Laplace transforms with respect to x and y. Difficulties arise from our lack of knowledge of the displacements at the surfaces x = o and y = o, but solution of the problem is reduced to that of two simultaneous integral equations. An iterative solution can be found, and from this we isolate the parts which describe the incident Rayleigh pulse (identical with Lamb's solution) and the pulse transmitted round the corner. It is found that the form of the pulse is greatly changed, in ways depending on the values of the velocities of P, S and Rayleigh waves. Whereas the dispacements u and v on y = o have the shape of Φ(t) and its allied function (Hilbert transform) Φ′(t) respectively, each of u and v on x = o is given by a linear combination of Φ and Φ′ Since Φ′ may differ greatly from Φ in form, the change in shape of each component of displacement when it turns the corner may be very marked.

An Application of Hankel Transforms in Axially Symmetric Potential Flow

In his book on Hydrodynamics, Lamb obtained a solution for the potential flow of an incompressible fluid through a circular hole in a plane wall. More recently Sneddon (Fourier Transforms, New York, 1951) obtained Lamb's solution by an elegant application of Hankel transforms.Since the streamlines in this solution are symmetric about the wall, it is not of particular physical interest. In this note, Sneddon's method is used to give a solution in which the fluid is infinite in extent on one side of the aperture but issues as a jet of finite diameter on the other side.

The Transmission of a Rayleigh Pulse round a corner

Summary Lamb, in his classical paper of 1904, found the form of the Rayleigh pulse that travels over the surface of an elastic half-space, which has been struck along a line. Let such a pulse travel on one face of an elastic quarter-space, the crest-line being parallel to and travelling towards the edge. We enquire what form the Rayleigh pulse will have after passing round the corner. The problem is taken as two-dimensional; the elastic solid fills the positive quadrant xOy and the incident Rayleigh pulse arises from a pressure p(x, t)= - Qδ(x -a)φ(t) applied on the face y= o. The differential equations and boundary conditions are reduced to operational form by the use of a Fourier transform with respect to time and Laplace transforms with respect to x and y. Difficulties arise from our lack of knowledge of the displacements at the surfaces x= o and y= o, but solution of the problem is reduced to that of two simultaneous integral equations. An iterative solution can be found, and from this we isolate the parts which describe the incident Rayleigh pulse (identical with Lamb's solution) and the pulse transmitted round the corner. It is found that the form of the pulse is greatly changed, in ways depending on the values of the velocities of P, S and Rayleigh waves. Whereas the dispacements u and v on y= o have the shape of φ(t) and its allied function (Hilbert transform) φ(t) respectively, each of u and v on x= o is given by a linear combination of φ and φ. Since φ may differ greatly from φ in form, the change in shape of each component of displacement when it turns the corner may be very marked.

Societies and Academies

LONDON.Royal Society, June 14.Sir J. J. Thomson, president, in the chair.Prof. T. H. Ilavelock: Some cases of wave motion due to a submerged obstacle. In this paper Prof. Lamb's solution for a submerged circular cylinder is carried a stage further in the approximation, and the wave resistance is calculated directly from the resultant fluid pressure on the cylinder. Similar methods are then applied to a three-dimensional problem, the waves produced by a submerged sphere.Prof. L. V. King: The propagation of sound in the free atmosphere and the acoustic efficiency of fog-signal machineryH. J. Shannon, F. F. Renwick, and B. V. Storr: The behaviour of scattering media in fully diffused light. The paper deals with the relationships between the. rejectance (proportion of in- cident light rejected), the obstruction (ratio of incident light to transmitted light), light capacity (ratio of accepted light to transmitted light) when a sheet of diffusing medium is illuminated on one side by diffuse light, and also the relative obstruction, and relative density, when, as in various instruments, the source of light is a first sheet of diffusing medium in contact with the 'sheet being examined. The experimental part of the paper discusses the method of using the theoretical equations obtained for determining the con- stants of a specimen of diffusing medium, certain requirements of the instrument used, and precautions to be taken. Examples are given showing the close agreement between observed and calculated valpes up to seven thicknesses of opal for both air and oil con- tact.J. W. T. Walsh: The theory of decay in radio- active luminous compounds. The theory of destruction of “ active centres “ put forward by Ruthrford to account for the decay of luminosity of radio-active luminous compounds leads to a simple exponential relation in the special case of a compound of constant activity. It has been found for radium zinc suiphide compounds that this relation expresses the observed results to a sufficient accuracy over short periods of less than 200 days, but that it fails to do so over longer periods, such as 500 days, the rate of decay of luminosity becoming gradually slower and slower, so that the brightness tends to a limiting value which is not zero. The paper is an attempt to find a luminosity time relation which will allow of the prediction of the ultimate behaviour of compounds of varying composition, and it assumes the operation of some factor acting in a direction opposite to that of the destruction of the active centtes.