Seismic Waves in a Laterally Inhomogeneous Layered Medium, Part II: Analysis

1997 ◽  
Vol 64 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Ruichong Zhang ◽  
Liyang Zhang ◽  
Masanobu Shinozuka
Keyword(s):  
Half Space ◽  
Ground Motion ◽  
Scattered Wave ◽  
Wave Number ◽  
Dislocation Source ◽  
Wave Fields ◽  
Seismic Dislocation ◽  
Body Forces ◽  
The Mean ◽  
Layered Half Space

Seismic wave scattering representation for the layered half-space with lateral inhomogeneities subjected to a seismic dislocation source has been formulated in the companion paper with the use of first-order perturbation (Born-type approximation) technique. The total wave field is obtained as a superposition of the mean and the scattered wave fields, which are generated, respectively, by a series of double couples of body forces equivalent to the seismic dislocation source and by fictitious body forces equivalent to the existence of the lateral inhomogeneities in the layered half-space. The responses in both the mean and the scattered wave fields are found with the aid of an integral transform technique and wave propagation analysis. The characteristics of the scattered waves and their effects on the mean waves or corresponding induced ground and/or underground mean responses are investigated in this paper. In particular, coupling phenomena between P-SV and SH waves and wave number shifting effects between the mean and the scattered wave responses are presented in detail. With the lateral inhomogeneities being assumed as a homogeneous random field, a qualitative analysis is provided for estimating the effects of the lateral inhomogeneities on the ground motion, which is related to a fundamental issue: whether a real earth medium can or cannot be approximately considered as a laterally homogeneous layer. The effects of the lateral inhomogeneities on the ground motion time history are also presented as a quantitative analysis. Finally, a numerical example is carried out for illustration purposes.

Seismic Waves in a Laterally Inhomogeneous Layered Medium, Part I: Theory

1997 ◽  
Vol 64 (1) ◽  
pp. 50-58 ◽  
Author(s):  
Ruichong Zhang ◽  
Liyang Zhang ◽  
Masanobu Shinozuka
Keyword(s):  
Wave Field ◽  
Half Space ◽  
Seismic Waves ◽  
Scattered Wave ◽  
Inhomogeneous Layer ◽  
Dislocation Source ◽  
Seismic Dislocation ◽  
Body Forces ◽  
The Mean ◽  
Layered Half Space

Seismic waves in a layered half-space with lateral inhomogeneities, generated by a buried seismic dislocation source, are investigated in these two consecutive papers. In the first paper, the problem is formulated and a corresponding approach to solve the problem is provided. Specifically, the elastic parameters in the laterally inhomogeneous layer, such as P and S wave speeds and density, are separated by the mean and the deviation parts. The mean part is constant while the deviation part, which is much smaller compared to the mean part, is a function of lateral coordinates. Using the first-order perturbation approach, it is shown that the total wave field may be obtained as a superposition of the mean wave field and the scattered wave field. The mean wave field is obtainable as a response solution for a perfectly layered half-space (without lateral inhomogeneities) subjected to a buried seismic dislocation source. The scattered wave field is obtained as a response solution for the same layered half-space as used in the mean wave field, but is subjected to the equivalent fictitious distributed body forces that mathematically replace the lateral inhomogeneities. These fictitious body forces have the same effects as the existence of lateral inhomogeneities and can be evaluated as a function of the inhomogeneity parameters and the mean wave fleld. The explicit expressions for the responses in both the mean and the scattered wave fields are derived with the aid of the integral transform approach and wave propagation analysis.

Elastic Wave Scattering From an Interface Crack in a Layered Half Space

1985 ◽  
Vol 52 (1) ◽  
pp. 42-50 ◽  
Author(s):  
H. J. Yang ◽  
D. B. Bogy
Keyword(s):  
Integral Equations ◽  
Half Space ◽  
Interface Crack ◽  
Scattered Field ◽  
Crack Tip ◽  
Wave Number ◽  
Graphical Form ◽  
Integral Theorem ◽  
Special Cases ◽  
Layered Half Space

Many applications in industry utilize a layered elastic structure in which a relatively thin layer of one material is bonded to a much thicker substrate. Often the fabrication process is imperfect and cracks occur at the interface. This paper is concerned with the plane strain, time-harmonic problem of a single elastic layer of one material on a half space of a different material with a single crack at the interface. Green’s functions for the uncracked medium are used with the appropriate form of Green’s integral theorem to derive the scattered field potentials for arbitrary incident fields in the cracked layered half space. These potentials are used in turn to reduce the problem to a system of singular integral equations for determining the gradients of the crack opening displacements in the scattered field. The integral equations are analyzed to determine the crack tip singularity, which is found, in general, to be oscillatory, as it is in the corresponding static problem of an interface crack. For many material combinations of interest, however, the crack tip singularity in the stress field is one-half power, as in the case of homogeneous materials. In the numerical work presented here attention is restricted to this class of composites and the integral equations are solved numerically to determine the Mode I and Mode II stress intensity factors as a function of a dimensionless wave number for various ratios of crack length to layer depth. The results are presented in graphical form and are compared with previously published analyses for the special cases where such results are available.

Viscoelastic properties of fine-grained incompressible turbulence

1968 ◽  
Vol 33 (1) ◽  
pp. 1-20 ◽  
Author(s):  
S. C. Crow
Keyword(s):  
Shear Flow ◽  
Reynolds Stress ◽  
Memory Function ◽  
Mean Field ◽  
Constitutive Law ◽  
Wave Number ◽  
Fine Grained ◽  
Body Forces ◽  
Non Linear ◽  
The Mean

A number of shear-flow phenomena can be explained qualitatively if turbulence is regarded as a continuous viscoelastic medium with respect to its action on a mean field. Conditions are sought under which the analogy is quantitative, and it is found that the turbulence must be fine-grained and the mean field weak. For geometrical convenience the turbulence is assumed to be nearly homogeneous and isotropic so that body forces are required to maintain it. The turbulence is found to respond initially to an arbitrary deformation as an elastic medium, in which Reynolds stress is linearly proportional to strain. Three processes that cause the resulting Reynolds stress to relax are distinguished: viscous diffusion, body-force agitation and non-linear scrambling. It is argued that, regardless of which process dominates, Reynolds stress evolves in a continuously changing mean field according to a viscoelastic constitutive law, relating stress to deformation history by means of a scalar memory function. The argument is carried through analytically for weak turbulence, in which non-linear scrambling is negligible, and the memory function is computed in terms of the wave-number-frequency spectrum of the background turbulence. In the course of the analysis, a new type of Reynolds stress arises related to the passage of the turbulence through its sustaining environment of body forces. It is found that the mean field must be surprisingly weak for this ‘translation stress’ to be negligible. Applications of the viscoelasticity theory of turbulent shear flow are discussed in which body forces and therefore translation stress are absent.

scholarly journals Mean and variance of the Eulerian and Lagrangian horizontal velocities induced by nonlinear multi-directional irregular water waves

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Mathias Klahn ◽  
Per A. Madsen ◽  
David R. Fuhrman
Keyword(s):  
Water Waves ◽  
Second Order ◽  
Wave Number ◽  
Order Effects ◽  
Wave Fields ◽  
Fully Nonlinear ◽  
Vertical Coordinate ◽  
Irregular Wave ◽  
Mean And Variance ◽  
The Mean

In this paper, we study the mean and variance of the Eulerian and Lagrangian fluid velocities as a function of depth below the surface of directionally spread irregular wave fields given by JONSWAP spectra in deep water. We focus on the behaviour of these quantities in the bulk of the water, and using second-order potential flow theory we derive new simple asymptotic approximations for their decay in the limit of large depth below the surface. Specifically, we show that when the depth is greater than about 1.5 peak wavelengths, the variance of the Eulerian velocity decays in proportion to exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) , and the mean Lagrangian velocity decays in proportion to 1 ( − k p z ) 1 / 6 exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) . Here, k p is the peak wave number and z is the vertical coordinate measured positively upwards from the still water level. We test the accuracy of the second-order formulation against new fully nonlinear simulations of both short crested and long crested irregular wave fields and find a good match, even when the simulations are known to be affected substantially by third-order effects. To our knowledge, this marks the first fully nonlinear investigation of the Eulerian and Lagrangian velocities below the surface in irregular wave fields.

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