EFFECTS OF A LIQUID CORE ON THE PROPAGATION OF SEISMIC WAVES

1959 ◽  
Vol 37 (2) ◽  
pp. 109-128 ◽  
Author(s):  
George Duwalo ◽  
J. A. Jacobs
Keyword(s):  
Seismic Waves ◽  
Wave Length ◽  
Liquid Core ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
Diffracted Waves ◽  
Viscous Compressible Liquid ◽  
Method Of Residues ◽  
Propagation Of Elastic Waves ◽  
The Waves

Effects of a spherical cavity in an infinite, homogeneous, isotropic elastic solid, containing non-viscous compressible liquid, on the propagation of elastic waves are investigated mathematically. The waves emitted by a simple harmonic point source in the solid are of the types known as SH and P in seismology. The discussion is restricted to the case ka » 1 (ka = 2 π cavity radius/wave length). Series solutions are transformed into contour integrals by Watson's method. Evaluation of these by the method of residues results in expressions describing the P and S components of the diffracted waves.

STUDIES ON SEISMIC WAVES: III. PROPAGATION OF ELASTIC WAVES IN THE NEIGHBORHOOD OF A FREE BOUNDARY

Geophysics ◽  
1947 ◽  
Vol 12 (1) ◽  
pp. 57-71 ◽  
Author(s):  
C. Y. Fu
Keyword(s):  
Surface Waves ◽  
Elastic Waves ◽  
Seismic Waves ◽  
Classical Method ◽  
Epicentral Distance ◽  
Harmonic Waves ◽  
Inhomogeneous Waves ◽  
Different Types ◽  
Propagation Of Elastic Waves ◽  
The Waves

Continuous and spherical harmonic waves are generated at an internal point of the medium. By use of the classical method of Sommerfeld, the different modes of propagation near a free surface after the arrival of the waves are examined. From the approximate evaluations of the integrals, it is found that in addition to the ordinary types of body and surface waves, there are also inhomogeneous waves and surface waves which are not of the Rayleigh type. The amplitude factors of these latter waves vary inversely as the square instead of as the square root of the epicentral distance. Altogether, there are not less than five different types of waves and they are obtained from integrations in the neighborhood of the singularities of the integrals.

ON THE PROPAGATION OF RAYLEIGH WAVES ON THE SURFACE OF A VISCO‐ELASTIC SOLID

Geophysics ◽  
1953 ◽  
Vol 18 (1) ◽  
pp. 70-74 ◽  
Author(s):  
C. W. Horton
Keyword(s):  
Rayleigh Waves ◽  
Dimensionless Parameter ◽  
Attenuation Factor ◽  
Wave Length ◽  
Elastic Solid ◽  
Angular Frequency ◽  
Elastic Parameters ◽  
Lissajous Figure ◽  
The Waves ◽  
Horizontal Displacements

The propagation of Rayleigh waves over the surface of a visco‐elastic solid is examined. It is shown that for a Poisson solid (λ=μ), the behavior of the waves can be characterized by a dimensionless parameter δ=ωη/μ which is less than 0.1 for the frequencies and elastic parameters of interest in geophysics. In this expression ω=angular frequency, μ=shear modulus, η=viscosity. For small values of δ it is possible to modify the usual analysis of Rayleigh waves and obtain the new characteristics without much difficulty. It is shown that the motion of a particle on the earth’s surface is changed from an ellipse to a Lissajous’ figure and that the phase angle between the vertical and horizontal displacements is changed from [Formula: see text] to [Formula: see text] radians. The surface wave has an attenuation factor of [Formula: see text] where [Formula: see text] is the wave length of the Rayleigh wave in the absence of internal friction.

MOTION OF A RIGID SPHERE IN AN ACOUSTIC WAVE FIELD

Geophysics ◽  
1945 ◽  
Vol 10 (1) ◽  
pp. 91-109 ◽  
Author(s):  
Alfred Wolf
Keyword(s):  
Acoustic Waves ◽  
Response Curve ◽  
Wave Length ◽  
Elastic Solid ◽  
Rigid Sphere ◽  
Resonance Effects ◽  
First Order ◽  
Solid Media ◽  
The Waves ◽  
Scattering Of Acoustic Waves

A rigid sphere in the field of plane acoustic waves in a fluid or in an elastic solid medium is subjected to harmonic forces in the direction of propagation of the waves, and proportional to their amplitude. The response curve is a function of the ratio of the circumference of the sphere to the wave length, and of the ratio of the mass of the sphere to the mass of the displaced medium. In an elastic solid, Poisson’s ratio must also be included among the variables. The response curve in fluids decreases continuously with decreasing wave length. In elastic solid media, the response curve has a maximum which is due to resonance effects. In general, the greater the mass of the sphere the smaller the response except in the neighborhood of resonance in elastic solid media. The scattering of acoustic waves by a rigid sphere is determined. The potential of scattered waves is developed in a series of spherical harmonics; it is shown that only the first order coefficients are affected by the motion of the sphere.

Comparison of seismic waves generated by different types of source

1963 ◽  
Vol 53 (5) ◽  
pp. 965-978 ◽  
Author(s):  
David E. Willis
Keyword(s):  
Seismic Waves ◽  
Nuclear Explosion ◽  
Test Site ◽  
Frequency Content ◽  
Nevada Test Site ◽  
Different Types ◽  
The Waves ◽  
Source Duration ◽  
Lower Frequencies

Abstract A comparison of the seismic waves generated by a nuclear explosion and an earthquake is discussed. The epicenter of the earthquake was located within the Nevada Test Site. Both events were recorded at the same station with the same type of equipment. The earthquake waves contained slightly lower frequency than the waves generated by the nuclear shot. The early P phases of the shot had larger amplitudes while the phases after Pg for the earthquake were larger. Seismic waves from collapses were generally found to be composed of lower frequencies than the waves from the original shot. Aftershocks of the Hebgen Lake earthquake were found to generate seismic waves whose frequency content was related to the magnitude of the aftershock. Spectral differences in quarry shot recordings that correlate with source duration times are also discussed.

An experimental investigation of the nature and origin of microseisms at St. Louis, Missouri

1940 ◽  
Vol 30 (2) ◽  
pp. 139-178
Author(s):  
J. Emilio Ramirez
Keyword(s):  
Wave Length ◽  
Vertical Axis ◽  
The Other ◽  
Wave Form ◽  
Stationary Waves ◽  
Retrograde Motion ◽  
Time Signals ◽  
Group Form ◽  
Triangular Network ◽  
The Waves

Summary Over a period of six months, from July to December, 1938, an investigation on microseismic waves has been carried out in the Department of Geophysics of St. Louis University. Four electromagnetic seismographs, specially designed for recording microseisms, were installed in the city of St. Louis in the form of a triangular network. Two of these were E-W components, one at the St. Louis University Gymnasium and the other 6.4 km. due west at Washington University. The other two were arranged as N-S components, one at the St. Louis University Gymnasium and one 6.3 km. due south at Maryville College. The speed of the photographic paper was 60 mm/min., and time signals were recorded automatically and simultaneously on each paper from the same clock every minute and at shorter intervals from a special pendulum and “tickler” combination by means of telephone wires. The results have demonstrated beyond doubt that microseismic waves are traveling and not stationary waves. The same waves have been identified at each one of the stations of the network, and also at Florissant, 21.8 km. away from St. Louis University. The speed of microseismic waves at St. Louis was determined from several storms of microseisms and it was found to be 2.67±0.03 km/sec. The direction of microseisms was also established for most of the storms and it was found that about 80 per cent of incoming microseisms at St. Louis were from the northeast quadrant during the interval from July to December, 1938. No microseisms were recorded from the south, west, or southwest. The period of the waves varied between 3.5 and 7.5 sec. The average period was about 5.4 sec. The microseismic wave length was therefore of the order of 14¼ km. A study of the nature of microseismic waves from the three Galitzin-Wilip components of the Florissant station reveals in the waves many of the characteristics of the Rayleigh waves; that is, the particles in the passage of microseismic waves move in elliptical orbits of somewhat larger vertical axis and with retrograde motion. A comparison carried over a period of more than a year between microseisms and microbarometric oscillations recorded by specially designed microbarographs showed no direct relationship between the two phenomena in wave form, group form, period, or duration of storms. The source of microseisms is to be found not over the land, but rather out over the surface of the ocean. The amplitudes of microseisms depend only on the intensity and widespread character of barometric lows traveling over the ocean. Several correlations between the two phenomena seem to make this conclusion rather evident. Special emphasis is laid on the fact that all the determined directions of incoming microseisms at St. Louis point to a deep barometric low over the ocean. The period of microseisms seems to be a function of the distance between the station and the source of microseisms. The exact mechanism by which barometric lows over the ocean water result in the production of microseisms needs further investigation. Large microseisms have been produced without any indication of surf near the coasts, or with winds blowing from the land toward the ocean.

Acoustic energy radiated by nonlinear spherical oscillations of strongly driven bubbles

2010 ◽  
Vol 466 (2118) ◽  
pp. 1829-1847 ◽  
Author(s):  
Joachim Holzfuss
Keyword(s):  
Wave Model ◽  
High Amplitude ◽  
Compressible Liquid ◽  
Acoustic Energy ◽  
Compressible Medium ◽  
Relevant Parameter ◽  
Wafer Cleaning ◽  
Viscous Compressible Liquid ◽  
Shock Wave Model ◽  
Polytropic Exponent

Based on the theory of F. Gilmore ( Gilmore 1952 The growth or collapse of a spherical bubble in a viscous compressible liquid ) for radial oscillations of a bubble in a compressible medium, the sound emission of bubbles in water driven by high-amplitude ultrasound is calculated. The model is augmented to include expressions for a variable polytropic exponent, hardcore and water vapour. Radiated acoustic energies are calculated within a quasi-acoustic approximation and also a shock wave model. Isoenergy lines are shown for driving frequencies of 23.5 kHz and 1 MHz. Together with calculations of stability against surface wave oscillations leading to fragmentation, the physically relevant parameter space for the bubble radii is found. Its upper limit is around 6 μm for the lower frequency driving and 1–3 μm for the higher. The radiated acoustic energy of a single bubble driven in the kilohertz range is calculated to be of the order of 100 nJ per driving period; a bubble driven in the megahertz range reaches two orders of magnitude less. The results for the first have applications in sonoluminescence research. Megahertz frequencies are widely used in wafer cleaning, where radiated sound may be implicated as responsible for the damage of nanometre-sized structures.

scholarly journals The effect of junctions on the propagation of electric waves along conductors

1913 ◽  
Vol 88 (601) ◽  
pp. 103-110
Keyword(s):  
Cylindrical Shell ◽  
Wave Length ◽  
General Character ◽  
Original Form ◽  
Annular Space ◽  
Approximate Methods ◽  
Positive Direction ◽  
Positive Side ◽  
The Waves ◽  
Electric Waves

Some interesting problems in electric wave propagation are suggested by an experiment of Hertz. In its original form waves of the simplest kind travel in the positive direction (fig. 1), outside an infinitely thin conducting cylindrical shell, AA, which comes to an end, say, at the plane z = 0. Co-axial with the cylinder a rod or wire BB (of less diameter) extends to infinity in both directions. The conductors being supposed perfect, it is required to determine the waves propagated onwards beyond the cylinder on the positive side of z , as well as those reflected back outside the cylinder and in the annular space between the cylinder and the rod. So stated, the problem, even if mathematically definite, is probably intractable; but if we modify it by introducing an external co-axial con­ducting sheath CC (fig. 2), extending to infinity in both directions, and if we further suppose that the diameter of this sheath is small in comparison with the wave-length (λ) of the vibrations, we shall bring it within the scope of approximate methods. It is under this limitation that I propose here to consider the present and a few analogous problems. Some considerations of a more general character are prefixed.

Plane Deformations of an Inhomogeneity–Matrix System Incorporating a Compressible Liquid Inhomogeneity and Complete Gurtin–Murdoch Interface Model

2018 ◽  
Vol 85 (12) ◽  
Author(s):  
Ming Dai ◽  
Min Li ◽  
Peter Schiavone
Keyword(s):  
Analytic Solution ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
External Loading ◽  
Interface Model ◽  
Interface Tension ◽  
Soft Solids ◽  
Remote Loading ◽  
Interface Elasticity ◽  
Plane Deformations

We consider the plane deformations of an infinite elastic solid containing an arbitrarily shaped compressible liquid inhomogeneity in the presence of uniform remote in-plane loading. The effects of residual interface tension and interface elasticity are incorporated into the model of deformation via the complete Gurtin–Murdoch (G–M) interface model. The corresponding boundary value problem is reformulated and analyzed in the complex plane. A concise analytical solution describing the entire stress field in the surrounding solid is found in the particular case involving a circular inhomogeneity. Numerical examples are presented to illustrate the analytic solution when the uniform remote loading takes the form of a uniaxial compression. It is shown that using the simplified G–M interface model instead of the complete version may lead to significant errors in predicting the external loading-induced stress concentration in gel-like soft solids containing submicro- (or smaller) liquid inhomogeneities.

scholarly journals Microbubble dynamics in a viscous compressible liquid near a rigid boundary

2019 ◽  
Vol 84 (4) ◽  
pp. 696-711 ◽  
Author(s):  
Qianxi Wang ◽  
WenKe Liu ◽  
David M Leppinen ◽  
A D Walmsley
Keyword(s):  
Potential Flow ◽  
Stokes Equations ◽  
Bubble Surface ◽  
Compressible Liquid ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Viscous Effects ◽  
Rigid Boundary ◽  
Viscous Potential Flow ◽  
Viscous Compressible Liquid

Abstract This paper is concerned with microbubble dynamics in a viscous compressible liquid near a rigid boundary. The compressible effects are modelled using the weakly compressible theory of Wang & Blake (2010, Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech., 730, 245–272), since the Mach number associated is small. The viscous effects are approximated using the viscous potential flow theory of Joseph & Wang (2004, The dissipation approximation and viscous potential flow. J. Fluid Mech., 505, 365–377), because the flow field is characterized as being an irrotational flow in the bulk volume but with a thin viscous boundary layer at the bubble surface. Consequently, the phenomenon is modelled using the boundary integral method, in which the compressible and viscous effects are incorporated into the model through including corresponding additional terms in the far field condition and the dynamic boundary condition at the bubble surface, respectively. The numerical results are shown in good agreement with the Keller–Miksis equation, experiments and computations based on the Navier–Stokes equations. The bubble oscillation, topological transform, jet development and penetration through the bubble and the energy of the bubble system are simulated and analysed in terms of the compressible and viscous effects.

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