On disturbances in a piezoelectric layer permeated by a magnetic field

Summary The methods of transform calculus are made use of in finding the disturbances in a piezoelectric layer with an impulsive force at its free faces in presence of a magnetic field. The problem is solved for time dependent elastic compliance of the material of the layer.     

Mechanical response of a piezoelectric composite transducer subjected to a thermal field

Summary The present paper studies the mechanical response of a piezoelectric composite transducer—with an elastic compliance partly constant and partly variable with time—acted upon by suitable thermal field besides the electrical voltage. The Laplace transform has been applied to solve the problem.     

Comparison of a simple planetary boundary layer model with measurements of a turbulent boundary layer under pack ice

A simple analytical theory which describes the motion in a turbulent planetary boundary layer near a rough sea bed by using a two-layer eddy viscosity model is presented. The vertical structure of the current in the boundary layer is presented, and comparisons are made with data fromMcPhee and Smith (1976, Journal of Physical Oceanography,6, 696–711) obtained from measurements of the turbulent boundary layer under drifting Arctic ice.     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.     

薄圆管压电换能器瞬态响应及其在井中激发声场

针对声波测井中采用的柱状薄圆管压电换能器,提出一种研究换能器响应的解析方法.该方法考虑了轴向与径向振动之间的耦合作用,导出置于耦合液体中压电换能器径向位移满足的微分方程和频域传递函数,比较分析了不同激励源作用下换能器的位移响应特性.利用波数域实轴积分法,计算了换能器在弹性地层井孔中激发的声场,研究了不同地层中的波形差异.与前人在假定声源强度的模型下只能求出波场的相对幅度不同,本文得出在一定激励电压作用下换能器在井孔中激发声场的幅度值.这为说明声波测井响应信号的强弱及仪器的声系设计提供了依据,对估计动电效应测井信号的强弱有参考意义.     

The micromechanics of friction in a granular layer

A grain bridge model is used to provide a physical interpretation of the rate- and state-dependent friction parameters for the simple shear of a granular layer. This model differs from the simpler asperity model in that it recognizes the difference between the fracture of a grain and the fracture of an adhesion between grains, and it explicitly accounts for dilation in the granular layer. The model provides an explanation for the observed differences in the friction of granular layers deformed between rough surfaces and those deformed between smooth surfaces and for the evolution of the friction parameters with displacement. The observed evolution from velocity strengthening to velocity weakening with displacement is interpreted as being due to the change in the micromechanics of strain accommodation from grain crushing to slip between adjacent grains; this change is associated with the observed evolution of a fractal grain structure.     

Measurement of particle rotation in a saltation layer

Two computational methods to measure particle rotations from shadow images of sand particles saltating in a wind tunnel are presented. One method calculates the maximum of the cross‐correlations through multiple angular rotations of an imaged particle. The second method polar transforms both images and then calculates the correlation coefficient for multiple pixel displacements in the θ axis, corresponding to particle rotations. The results from both methods were analysed as a function of height above sand bed (3.7–33.4 mm) and particle size (0.32–0.93 mm equivalent mean diameter). Our results indicate little evidence that particle rotation speeds depend on either their size or height above the sand bed. Though similar results were obtained from both methods, there existed different advantages and disadvantages between the methods. Erroneous results likely arose from particles that were inadequately described by a 2‐D rotation axis, or from poorly imaged particles. At a wind tunnel speed of about 12 m/s, most particles rotated at around 300–400 rev/s. Negative rotations were also found, and their proportion was approximately 15% within the total range of ?450 to 850 rev/s. The ratio of displacement kinetic energy to rotation energy was compared across the various groups and had values between 15 and 40. The quotient showed little dependence on height, though decreased with increasing particle size. Wider applicability of the measurement methodology to study snow particle rotation is also discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Magneto-viscoelastic disturbances in a ‘hereditary layer’

Summary The dynamical response due to the interaction of viscoelastic field and magnetic field in a layer ofVito Volterra's hereditary model is analysed by making use of Laplace-transform.     

Resistance of a soil layer to horizontal vibration of a pile

The resistance of a soil to horizontal harmonic vibration of a vertical, end bearing pile with circular cross-section is theoretically investigated. The soil is considered to be a continuum with material damping of the hysteretic type. The motions of the soil and pile are expressed as series, and a resistance coefficient on each term in the series of the pile motion is found. The resistance is found to depend on the excitation frequency, the pile slenderness and on the material properties of the soil. Finally a study of the influence of the parameters is carried out.     

Cone models for a soil layer on a flexible rock half-space

A modified truncated cone model is used to calculate approximately the dynamic response of a disk on the surface of a soil layer resting on flexible rock. The procedure is analogous to that for a layer on rigid rock, the only modification being that the reflection coefficient —α at the layer–rock interface is no longer equal to ?1. The modified value of α can be determined straightforwardly by considering one-dimensional wave propagation along the cone. The low- and high-frequency limits lead to a frequency-independent α, which allows the dynamic analysis to be performed directly in the familiar time domain. This cone represents a wave pattern with amplitude decay and also incorporates the reflection at the free surface and the reflection-refraction at the layer–rock interface. The results for the static stiffness of the disk are highly accurate for a wide range of geometrical and material properties of the layer and the rock. For the dynamic stiffness the agreement with the exact solution is satisfactory.     

VSP traveltime inversion for anisotropy in a buried layer

We present a method for calculating the anisotropy parameter of a buried layer by inverting the total traveltimes of direct arrivals travelling from a surface source to a well‐bore receiver in a vertical seismic profiling (VSP) geometry. The method assumes two‐dimensional media. The medium above the layer of interest (and separated from it by a horizontal interface) can exhibit both anisotropy and inhomogeneity. Both the depth of the interface as well as the velocity field of the overburden are assumed to be known. We assume the layer of interest to be homogeneous and elliptically anisotropic, with the anisotropy described by a single parameter χ. We solve the function describing the traveltime between source and receiver explicitly for χ. The solution is expressed in terms of known quantities, such as the source and receiver locations, and in terms of quantities expressed as functions of the single argument x_r, which is the horizontal coordinate of the refraction point on the interface. In view of Fermat's principle, the measured traveltime T possesses a stationary value or, considering direct arrivals, a minimum value, . This gives rise to a key result ‐‐ the condition that the actual anisotropy parameter . Owing to the explicit expression , this result allows a direct calculation of in the layer of interest. We perform an error analysis and show this inverse method to be stable. In particular, for horizontally layered media, a traveltime error of one millisecond results in a typical error of about 20% in the anisotropy parameter. This is almost one order of magnitude less than the error inherent in the slowness method, which uses a similar set of experimental data. We conclude by detailing possible extensions to non‐elliptical anisotropy and a non‐planar interface.     

Theoretical Love wave dispersion in a single layer model

Summary From a study of the Love wave dispersion in a single layer model it is shown that the shape of the dispersion curves is very insensitive to the changes in the density ratio. The bilogarithmic grids giving dispersion curves for different shear velocity ratios and a mean density ratio of 1.20 should be sufficient for interpretation by the method of curve matching. Such grids and the numerical data are given both for the phase velocity and the group velocity. An example is given illustrating the use of these grids.     

Near-surface layer traveltime inversion: a synthetic example1

The near-surface layer is modelled as a constant-velocity layer with varying thickness. The base of the layer is described by a B-spline curve. The optimum model is calculated by minimizing, with respect to the model parameters, the difference between traveltimes predicted by the model and those observed in the data. Once a model has been produced, corrections that are dependent on the raypath geometry through the near-surface layer can be calculated. The effect of the near-surface layer is normally considered to be consistent at each shot or geophone station for all traveltimes arriving at that location (the surface-consistent approximation). This assumption linearizes the problem, allowing timeshifts to be calculated and the traveltimes corrected to a chosen datum, representing static corrections. The single correction at each point is an averaged correction, based on an assumption that is particularly inaccurate in the presence of lateral variations of velocity or thickness of the surface layer, in the presence of large surface layer velocities or in the presence of a thick surface layer. The method presented considers the non-linear relationship between data and model explicitly, hence the correction that is dependent on the raypath. Linearization removes this dependence and reduces the problem to a surface-consistent approximation. The method is applied to synthetic data calculated from a model with surface layer variations. Comparisons are made between the corrected data resulting from the method described here and the conventional surface-consistent approach. From these results it becomes apparent that the near-surface layer inversion method presented here can reproduce accurate models and correct for near-surface layer effects in cases where conventional methods encounter difficulties. Additionally the method can be readily extended to 3D.     

Convective plan-form of two-scale dynamos in a plane layer

We study generation of magnetic fields, involving large spatial scales, by convective plan-forms in a horizontal layer. Magnetic modes and their growth rates are expanded in power series in the scale ratio, and the magnetic eddy diffusivity (MED) tensor is derived for flows, symmetric about the vertical axis in a layer. For convective rolls we demonstrate that MED is never below molecular magnetic diffusivity. For cell patterns possessing the symmetries of a rectangle, critical values of molecular magnetic diffusivity for the onset of small- and large-scale magnetic field generation are the same. No instances of negative MED in hexagonal cells have been detected. A family of plan-forms has been found numerically, where MED is negative for molecular magnetic diffusivity over the threshold for the onset of small-scale magnetic field generation. However, the region in the parameter space, where large-scale dynamo action is observed, is small.     

Unsteady boundary layer flows induced by a wavy plate

Summary Unsteady boundary layer flows generated in an incompressible, homogeneous, nonrotating viscous fluid bounded by a rigid wavy plate are studied theoretically. The Laplace transform method is employed to obtain exact solutions of the unsteady boundary layer equations in a wavy plate configuration. The structures of the unsteady velocity distribution and the associated boundary layers are determined explicitly and several particular solutions are recovered as special cases of this analysis. The physical interpretation of the mathematical results are examined.     

Reflected and refracted waves from a linear transition layer

Summary The problem of a transition layer lying between two homogeneous liquid media is discussed. After obtaining the formal solution for a periodic point source lying in the upper layer, the integrand in the expression for the displacement potential of the upper layer is expanded into series of negative powers of exponentials. Some of the terms in the Bromwich expansion are then evaluated along the Sommerfeld loops which give various reflected and refracted waves. The results for the refracted waves are discussed for the two extreme cases, when frequency is extremely low and extremely high. In both the cases it is found that the frequency dependence for refraction arrivals is the same as expected from a sharp boundary, viz., ^–1. And that for high frequency the travel-time of refraction arrivals is the same as expected from geometric ray theory. Both first- order and second-order discontinuities in density and bulk modulus are considered at the boundaries of the transition layer.     

Rocking vibrations of a rigid embedded foundation in a poroelastic soil layer

An approximate analytical method is presented for the dynamic response of a rigid cylindrical foundation embedded in a poroelastic soil layer under the excitation of a time-harmonic rocking moment. The soil underlying the foundation base is represented by a single-layered poroelastic soil based on rigid bedrock while the soil along the side of the foundation is modeled as an independent poroelastic stratum composed of a series of infinitesimally thin layers. The accuracy of the present solution is verified by comparisons with existing solutions obtained from other researchers. Numerical results for the rocking dynamic impedance and dynamic response factor are presented to demonstrate the influence of nondimensional frequency of excitation, poroelastic soil layer thickness, depth ratio of the foundation and internal friction of the poroelastic soil.     

Large-scale convective dynamos in a stratified rotating plane layer

We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ???1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.     

SH waves in a layer with temperature dependent properties

The paper deals with the problem of SH harmonic wave propagation in an elastic layer with temperature dependent properties. The shear modulus and mass density are linearly dependent on temperature. The layer is rested on a rigid foundation and the upper boundary is free of loadings. The boundary planes are kept at different constant temperatures. The wave velocity and amplitude of stresses are analysed.