THE RESPONSE OF AN ELASTIC PLATE SUBMERGED IN A LIQUID HALF‐SPACE TO EXPLOSIVE SOUND

Geophysics ◽  
1964 ◽  
Vol 29 (3) ◽  
pp. 370-394
Author(s):  
J. H. Rosenbaum
Keyword(s):  
Half Space ◽  
Liquid Layer ◽  
Elastic Plate ◽  
Layered Medium ◽  
Wave Number ◽  
Sound Waves ◽  
Total Transmission ◽  
Complex Modes ◽  
Horizontal Wave ◽  
Shear Motion

A mathematical analysis is presented for the case of a point‐source explosion in the liquid layer above an elastic plate of infinite horizontal extent immersed in a liquid half‐space parallel to the free surface of the liquid. An asymptotic solution, valid for long times after the explosion, is derived; it expresses the pressure response in the liquid layer in terms of characteristic vibrations of the layered medium. Trapped and exponentially decaying modes have been investigated numerically for the Lucite plate in water. Special emphasis is placed on the description of sustained reverberations (singing). This phenomenon is described in terms of complex modes, where some energy travels back radially towards the source. At long times, singing can be described in terms of “standing” waves of nonvanishing horizontal wave number. It is also closely connected with a type of trapped wave in the liquid layer‐plate combination whose horizontal phase velocity is greater than the velocity of sound in the fluid, but which is completely decoupled from the liquid half‐space below the plate. At very long times, however, the strongest signal is associated with an almost completely decoupled shear motion of the plate, and the horizontal wave number approaches zero. A brief discussion of the total transmission of plane harmonic sound waves through a Lucite plate in water is given. The total transmission curves are used to show qualitatively that singing often may not be observed in connection with the above‐mentioned trapped waves.

scholarly journals Surface Wave Propagation in a Microstretch Thermoelastic Diffusion Material under an Inviscid Liquid Layer

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Rajneesh Kumar ◽  
Sanjeev Ahuja ◽  
S. K. Garg
Keyword(s):  
Surface Waves ◽  
Half Space ◽  
Liquid Layer ◽  
Relaxation Times ◽  
Wave Number ◽  
Normal Velocity ◽  
Thermoelastic Diffusion ◽  
Inviscid Liquid ◽  
Surface Wave Propagation ◽  
The Mathematical Model

The present investigation deals with the propagation of Rayleigh type surface waves in an isotropic microstretch thermoelastic diffusion solid half space under a layer of inviscid liquid. The secular equation for surface waves in compact form is derived after developing the mathematical model. The dispersion curves giving the phase velocity and attenuation coefficients with wave number are plotted graphically to depict the effect of an imperfect boundary alongwith the relaxation times in a microstretch thermoelastic diffusion solid half space under a homogeneous inviscid liquid layer for thermally insulated, impermeable boundaries and isothermal, isoconcentrated boundaries, respectively. In addition, normal velocity component is also plotted in the liquid layer. Several cases of interest under different conditions are also deduced and discussed.

Discrete wave-number representation of seismic-source wave fields

1977 ◽  
Vol 67 (2) ◽  
pp. 259-277
Author(s):  
Michel Bouchon ◽  
Keiiti Aki
Keyword(s):  
Layered Medium ◽  
Near Field ◽  
Seismic Source ◽  
Three Dimensions ◽  
Wave Number ◽  
Number Representation ◽  
Wave Fields ◽  
High Acceleration ◽  
Horizontal Wave ◽  
Source Wave

Abstract A method based on a discrete horizontal wave-number representation of seismic-source wave fields is developed and applied to the study of the near-field of a seismic source embedded in a layered medium. The discretization results from a periodicity assumption in the description of the source. The problem is basically two-dimensional but its extension to three dimensions is sometimes feasible. The source is quite general and is represented through its body-force equivalents. Tests of the accuracy of the method are made against Garvin's (1956) analytical solution (a buried line source in a half-space) and against Niazy's (1973) results for a propagating fault in an infinite medium. In both cases, a remarkably good agreement is found. The method is applied to the modeling of the San Fernando earthquake, and to the computation of synthetic seismograms at short distance from a complex source in a layered medium. In particular, we show that the high acceleration-high frequency phase of the Pacoima Dam records is due to the Rayleigh wave from the point of ground breakage. Other high-acceleration phases, predicted by our model, are associated with the shear-wave arrival from the hypocenter or result from changes in the fault orientation.

Dispersion of Love waves in prestressed double-layered medium over a gravitating half-space

2020 ◽  
Vol 13 (13) ◽  
Author(s):  
Bishwanath Prasad ◽  
Santimoy Kundu ◽  
Prakash Chandra Pal ◽  
Parvez Alam
Keyword(s):  
Half Space ◽  
Layered Medium ◽  
Love Waves

Period equation for Rayleigh waves in a layer overlying a half space with a sinusoidal interface

1962 ◽  
Vol 52 (4) ◽  
pp. 807-822 ◽  
Author(s):  
John T. Kuo ◽  
John E. Nafe
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Length ◽  
Linear Equations ◽  
Wave Number ◽  
Interface Plane ◽  
Period Equation ◽  
Phase And Group Velocities

abstract The problem of the Rayleigh wave propagation in a solid layer overlying a solid half space separated by a sinusoidal interface is investigated. The amplitude of the interface is assumed to be small in comparison to the average thickness of the layer or the wave length of the interface. Either by applying Rayleigh's approximate method or by perturbating the boundary conditions at the sinusoidal interface, plane wave solutions for the equations which satisfy the given boundary conditions are found to form a system of linear equations. These equations may be expressed in a determinant form. The period (or characteristic) equations for the first and second approximation of the wave number k are obtained. The phase and group velocities of Rayleigh waves in the present case depend upon both frequency and distance. At a given point on the surface, there is a local phase and local group velocity of Rayleigh waves that is independent of the direction of wave propagation.

Approximation of surface wave velocity in smart composite structure using Wentzel–Kramers–Brillouin method

2018 ◽  
Vol 29 (18) ◽  
pp. 3582-3597 ◽  
Author(s):  
Manoj Kumar Singh ◽  
Sanjeev A Sahu ◽  
Abhinav Singhal ◽  
Soniya Chaudhary
Keyword(s):  
Surface Wave ◽  
Piezoelectric Material ◽  
Half Space ◽  
Composite Structure ◽  
Functionally Graded ◽  
Wave Number ◽  
Practical Utilization ◽  
Functionally Graded Piezoelectric Material ◽  
Varying Coefficients ◽  
Functionally Graded Piezoelectric

In mathematical physics, the Wentzel–Kramers–Brillouin approximation or Wentzel–Kramers–Brillouin method is a technique for finding approximate solutions to linear differential equations with spatially varying coefficients. An attempt has been made to approximate the velocity of surface seismic wave in a piezo-composite structure. In particular, this article studies the dispersion behaviour of Love-type seismic waves in functionally graded piezoelectric material layer bonded between initially stressed piezoelectric layer and pre-stressed piezoelectric half-space. In functionally graded piezoelectric material stratum, theoretical derivations are obtained by the Wentzel–Kramers–Brillouin method where variations in material gradient are taken exponentially. In the upper layer and lower half-space, the displacement components are obtained by employing separation of variables method. Dispersion equations are obtained for both electrically open and short cases. Numerical example and graphical manifestation have been provided to illustrate the effect of influencing parameters on the phase velocity of considered surface wave. Obtained relation has been deduced to some existing results, as particular case of this study. Variation in cut-off frequency and group velocity against the wave number are shown graphically. This study provides a theoretical basis and practical utilization for the development and construction of surface acoustics wave devices.

Gravitational instability of a viscous fluid in a magnetic field

1965 ◽  
Vol 22 (3) ◽  
pp. 579-586 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Magnetic Field ◽  
Viscous Fluid ◽  
Hartmann Number ◽  
Gravitational Instability ◽  
Unstable Mode ◽  
Stability Boundary ◽  
Wave Number ◽  
Symmetric Mode ◽  
Critical Stability ◽  
Horizontal Wave

The instability of a viscous fluid between two infinite vertical plates and heated from below in the presence of a magnetic field perpendicular to the plates is investigated, and the most critical stability boundary in the space of the Rayleigh number R, Hartmann number M, and the horizontal wave number a is determined. It is found that the most unstable mode is a symmetric mode with zero wave-number, and that for any M the fluid is unstable for any non-zero R, however small.

scholarly journals Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

2011 ◽  
Vol 18 (6) ◽  
pp. 827-838 ◽  
Author(s):  
İ. Coşkun ◽  
H. Engin ◽  
A. Özmutlu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Circular Cross Section ◽  
Elastic Half Space ◽  
Least Square ◽  
Polar Coordinates ◽  
Wave Number ◽  
Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

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