On the approximation of finite loop sources by two‐dimensional line sources

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 1027-1029 ◽  
Author(s):  
M. N. Nabighian ◽  
M. L. Oristaglio
Keyword(s):  
Half Space ◽  
Line Source ◽  
Opposite Polarity ◽  
Closed Form Expression ◽  
Late Time ◽  
Two Dimensional ◽  
Form Expression ◽  
Time Domain Electromagnetics ◽  
Receiving Coil ◽  
Finite Loop

An appealing feature of time‐domain electromagnetics is that the transient response simplifies considerably at late time, usually tending to a power‐law or exponential decay. In this note, we point out an interesting discrepancy between the late‐time asymptotics of a finite loop source over a half‐space and its natural two‐dimensional (2-D) approximation, which is two line sources of opposite polarity lying on a half‐space. Expressions for the transient responses of both loop (Wait and Ott, 1972) and line sources (Oristaglio, 1982) have been derived before; they show that at late times the voltage induced in a horizontal receiving coil decays as [Formula: see text] for a loop source and [Formula: see text] for a line source. Here we show that the slower decay for the line source is inherently a 2-D effect. To do this, we derive a closed‐form expression for the transient voltage induced by a finite wire of length 2L on a half‐space—a new result, for which we can separately examine the limits [Formula: see text] and [Formula: see text] Surprisingly, these limits are not interchangeable. First taking L to be infinite and then doing the late‐time asymptotic expansion yields the [Formula: see text] decay of a line source; in contrast, first doing the late‐time expansion gives a decay of [Formula: see text] for the finite wire, which is formally unchanged as the length goes to infinity.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

Shear compliance of two-dimensional pores possessing N -fold axis of rotational symmetry

2008 ◽  
Vol 464 (2091) ◽  
pp. 759-775 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Rotational Symmetry ◽  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Special Cases ◽  
Fourfold Symmetry ◽  
Shear Compliance

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an N -fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4 θ . An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

The influence of a conductive host on two‐dimensional borehole transient electromagnetic responses

Geophysics ◽  
1984 ◽  
Vol 49 (7) ◽  
pp. 861-869 ◽  
Author(s):  
Perry A. Eaton ◽  
Gerald W. Hohmann
Keyword(s):  
Electric Field ◽  
Half Space ◽  
Time Window ◽  
Time Derivative ◽  
Finite Width ◽  
Late Time ◽  
Time Behavior ◽  
Two Dimensional ◽  
Peak Response ◽  
Transient Electromagnetic

We have computed transient borehole electromagnetic (EM) responses of two‐dimensional (2-D) models using a direct and explicit finite‐difference algorithm. The program computes the secondary electric field which is defined as the difference between the total field and the primary (half‐space) field. The time derivative of the vertical magnetic field in a borehole is computed by numerical differentiation of the total electric field. These models consist of a thin horizontal conductor with a finite width, embedded in a conductive half‐space. Dual line sources energized by a step‐function current lie on the surface of the half‐space and simulate the long sides of a large rectangular loop. Numerical results substantiate several important features of the transient impulse response of such models. The peak response of the target is attenuated as the resistivity of the host decreases. A sign reversal in the secondary electric field occurs later in time as the resistivity of the host decreases. The peak response and the onset of late‐time behavior are delayed in time as well. Secondary responses for models with different host resistivities (10–1000 Ω-m) are approximately the same at late time. If the target is less conductive, the effects of the host, i.e., the attenuation and time delay, are less. It is readily apparent that there exists a time window within which the target’s response is at a maximum relative to the half‐space response. At late time the shape of the borehole anomaly due to a thin conductive 2-D target appears to be independent of the conductivity of the host. The late‐time secondary decay of the target is neither exponential nor power law, and a time constant computed from the slope of a log‐linear decay curve at late time may be much larger than the actual value for the same target in free space.

Two‐dimensional modeling of a towed in‐line electric dipole‐dipole sea‐floor electromagnetic system: The optimum time delay or frequency for target resolution

Geophysics ◽  
1988 ◽  
Vol 53 (6) ◽  
pp. 846-853 ◽  
Author(s):  
R. N. Edwards
Keyword(s):  
Time Delay ◽  
Half Space ◽  
Electric Dipole ◽  
Late Time ◽  
Crustal Material ◽  
Two Dimensional ◽  
Optimum Time ◽  
Electromagnetic System ◽  
Sea Floor ◽  
Dipole System

Towed in‐line transient electric dipole‐dipole systems are being used to map the electrical conductivity of the sea floor. The characteristic response of a double half‐space model representing conductive seawater and less conductive crustal material to a dipole‐dipole system located at the interface consists of two distinct parts. As time in the transient measurements progresses, two changes in field strength are observed. The first change is caused by the diffusion of the electromagnetic field through the resistive sea floor; the second is caused by diffusion through the seawater. The characteristic times at which the two events occur are measures of sea‐floor and seawater conductivity, respectively. Entirely equivalent responses are observed in a frequency‐domain measurement as frequency is swept from high to low. The simple double half‐space response is modified when the towed array crosses over a conductivity anomaly. I evaluate the magnitude of the anomalous response as a function of delay time and frequency using a two‐dimensional theory and a vertical, plate‐like target. If the ratio of the conductivity of the seawater to that of the sea floor is greater than unity, then an optimum time delay or frequency can be found which maximizes the response. For large conductivity contrasts, the optimum response is greater than the response at late time or zero frequency by a factor of the order of the conductivity ratio.

Gust Loading on a Thin Aerofoil

1971 ◽  
Vol 22 (3) ◽  
pp. 301-310 ◽  
Author(s):  
B. D. Mugridge
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Incompressible Flow ◽  
Velocity Component ◽  
Closed Form Expression ◽  
Sound Power ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Form Expression ◽  
Final Expression

SummaryA closed-form expression is derived which gives an approximate solution to the lift generated on a two-dimensional thin aerofoil in incompressible flow with a normal velocity component of the form exp [i(ωt–xx+yy)]. The inaccuracy of the solution when compared with other published work is compensated by the simplicity of the final expression, particularly if the result is required for the calculation of the sound power radiated by an aerofoil in a turbulent flow.

Compressibility of two-dimensional pores having n -fold axes of symmetry

2006 ◽  
Vol 462 (2071) ◽  
pp. 1933-1947 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Pore Compressibility ◽  
Special Cases ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry

The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n -fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

Theoretical seismograms for the two welded quarter-planes

1972 ◽  
Vol 62 (2) ◽  
pp. 619-630
Author(s):  
Dan Loewenthal ◽  
Z. Alterman
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Half Space ◽  
Line Source ◽  
Two Dimensional ◽  
Impulsive Force ◽  
Space Model ◽  
Finite Difference Technique

abstract The finite difference technique developed by Alterman and Rotenberg (1969) and Alterman and Loewenthal (1970) is applied to the problem of obtaining the motion of a two-dimensional half-space model, consisting of two homogeneous and isotropic welded quarter-spaces of different materials. Two source conditions are considered: an internal impulsive line source acting inside one of the quarter-planes, and an impulsive force acting perpendicular to the free surface.

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