The Frequency Dependence of the Perturbation Fields due to Conductivity Inhomogeneities in the Earth

1973 ◽  
Vol 10 (8) ◽  
pp. 1191-1200 ◽  
Author(s):  
F. W. Jones ◽  
B. A. Ainslie
Keyword(s):  
Frequency Dependence ◽  
Field Component ◽  
Finite Depth ◽  
Infinite Depth ◽  
Perturbation Field ◽  
The Earth

The geomagnetic perturbation fields due to conductivity discontinuities in the Earth are investigated. Two models, one in which the discontinuity extends to infinite depth, and a second one which consists of a dike of finite depth are considered. The perturbation fields are studied for several different frequencies of the alternating inducing field for each model. Both the H-polarization and E-polarization cases are considered and the perturbation field component profiles as a function of height above the surface of the conducting region are studied. The perturbation fields are strongly dependent on frequency, and significant differences are exhibited between the H-polarization and E-polarization cases.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

scholarly journals On the propagation of a disturbance in a fluid under gravity

1910 ◽  
Vol 83 (563) ◽  
pp. 347-356 ◽  
Keyword(s):  
Incompressible Fluid ◽  
Gravity Wave ◽  
Integral Form ◽  
Finite Depth ◽  
Infinite Depth ◽  
Poisson Problem

In a recent paper Lord Rayleigh has called attention to the fact of the instantaneous propagation of a limited disturbance over the surface of heavy incompressible fluid. This instantaneity occurs in spite of the fact that the velocity of any simple-harmonic gravity wave is finite, the depth being finite and constant. As the points thereby raised are of some delicacy, and are not completely settled in the paper quoted, some further remarks on the phenomenon may not be superfluous. 2. Solution of the Cauchy-Poisson Problem for Finite Depth . In his treatment of this case Lord Rayleigh used the integral form of solution. There are, however, great difficulties raised in this way on account of lack of convergence at the surface. I proceed to obtain a solution in the form of a series similar in type to the known serial solution of the problem for infinite depth.

The instability theory of drumlin formation applied to Newtonian viscous ice of finite depth

2010 ◽  
Vol 466 (2121) ◽  
pp. 2673-2694 ◽  
Author(s):  
A. C. Fowler
Keyword(s):  
Transfer Functions ◽  
Three Dimensional ◽  
Analytic Form ◽  
Finite Depth ◽  
Pressure Gradients ◽  
Interfacial Instabilities ◽  
Infinite Depth ◽  
Ice Surface ◽  
Instability Theory ◽  
Ice Flows

The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d i . We find that this is the case if the waveforms have a wavelength less than approximately 1.2 d i . Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography.

Exact large amplitude capillary waves on sheets of fluid

1976 ◽  
Vol 77 (2) ◽  
pp. 229-241 ◽  
Author(s):  
William Kinnersley
Keyword(s):  
Exact Solution ◽  
Large Amplitude ◽  
Elliptic Functions ◽  
Finite Depth ◽  
Capillary Waves ◽  
Infinite Depth ◽  
The Waves

We generalize Crapper's exact solution for capillary waves on fluid of infinite depth. We find two finite-depth solutions involving elliptic functions. We show they can also be interpreted as large amplitude symmetrical and antisymmetrical waves on a fluid sheet. Particularly interesting are the waves obtained from our solution in the limit when the fluid sheet is extremely thin.

scholarly journals The theory of ship waves

1926 ◽  
Vol 111 (759) ◽  
pp. 491-529 ◽  
Keyword(s):  
Finite Depth ◽  
Wave Disturbance ◽  
Present Communication ◽  
Ship Waves ◽  
Infinite Depth ◽  
The Subject ◽  
Considerable Detail

The theory of ship waves, when the sea is considered to be of infinite depth, has been the subject of many researches. When the sea is of finite depth the integrals involved are more complicated, but in this case also the theory has been worked out in considerable detail. The main object of the present communication is to add to the number of cases which have been solved, or, to be more precise, which have been exactly formulated, a certain series in which the depth is variable. Of subsidiary interest, but coming under the title of the paper, are some considerations relating to the wave disturbance when the depth is finite. These are dealt with briefly in section 5.

Coupling of airborne sound into the earth: Frequency dependence

1980 ◽  
Vol 67 (5) ◽  
pp. 1502-1506 ◽  
Author(s):  
H. E. Bass ◽  
L. N. Bolen ◽  
Daniel Cress ◽  
Jerry Lundien ◽  
Mark Flohr
Keyword(s):  
Frequency Dependence ◽  
Airborne Sound ◽  
The Earth

A CLASSIFICATION OF ℤ-GRADED LIE SUPERALGEBRAS OF INFINITE DEPTH

2002 ◽  
Vol 01 (04) ◽  
pp. 425-449
Author(s):  
NICOLETTA CANTARINI
Keyword(s):  
Finite Depth ◽  
Lie Superalgebras ◽  
Infinite Depth ◽  
Infinite Dimensional ◽  
Finite Growth

In 1998 Victor Kac classified infinite-dimensional, transitive, irreducible ℤ-graded Lie superalgebras of finite depth. Here we classify bitransitive, irreducible ℤ-graded Lie superalgebras of infinite depth and finite growth which are not contragredient. In particular we show that the growth of every such superalgebra is equal to one.

scholarly journals Small amplitude capillary-gravity waves in a channel of finite depth

1989 ◽  
Vol 31 (2) ◽  
pp. 142-160 ◽  
Author(s):  
M. C. W. Jones
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Unsolved Problem ◽  
Equations Of Motion ◽  
Finite Depth ◽  
Wave Problem ◽  
Infinite Depth ◽  
Existence And Multiplicity

Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

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