The generation of waves in an infinite elastic solid by variable body forces

1956 ◽  
Vol 248 (955) ◽  
pp. 575-607 ◽  
Keyword(s):  
General Solution ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Elastic Solid ◽  
Physical Nature ◽  
Graphical Form ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Numerical Work ◽  
Body Forces

This paper is concerned with the determination of the distribution of stress in an infinite elastic solid when time-dependent body forces act upon certain regions of the solid. It is assumed throughout that the strains are small. In §2 a general solution of the equations of motion for any distribution of body forces is derived by the use of four-dimensional Fourier transforms, and from that is derived the general solution for an isotropic solid (§ 3). From the latter solution are deduced the general solution of the statical problem (§4) and the two-dimensional problem (§5). The solution of the equations of motion in the case in which the distribution of body forces is symmetrical about an axis is derived in §6. The remainder of the paper consists in deducing the solution of special problems from these general solutions. In §§7 to 13 some typical two-dimensional problems are considered and exact analytical expressions found for the components of the stress tensor. In §§14 to 16 examples are given of the use of the general non-symmetrical three-dimensional solution derived in § 3, and in §§17 to 19 examples are given to illustrate the use of the general solution of the axially symmetrical problem. A certain amount of numerical work (presented in graphical form) is quoted to give some idea of the physical nature of the solutions.

Diffraction by a half plane perpendicular to the distinguished axis of a gyrotropic medium (oblique incidence)

1981 ◽  
Vol 59 (3) ◽  
pp. 403-424 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Second Order ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Electromagnetic Plane Wave

An exact, closed form solution is found for the following half plane diffraction problem. (I) The medium surrounding the half plane is gyrotropic. (II) The scattering half plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The direction of propagation of the incident electromagnetic plane wave is arbitrary (skew) with respect to the edge of the half plane. The result presented is a generalization of a solution for the same problem with incidence normal to the edge of the half plane (two-dimensional case).The fundamental, distinctive feature of the problem is that it constitutes a boundary value problem for a system of two coupled second order partial differential equations. All previously solved electromagnetic diffraction problems reduced to boundary value problems for either one or two uncoupled second order equations. (Exception: the two-dimensional case of the present problem.) The problem is formulated in terms of the (generalized) scalar Hertz potentials and leads to a set of two coupled Wiener–Hopf equations. This set, previously thought insoluble by quadratures, yields to the Wiener–Hopf–Hilbert method.The three-dimensional solution is synthesized from appropriate solutions to two-dimensional problems. Peculiar waves of ghost potentials, which correspond to zero electromagnetic fields play an essential role in this synthesis. The problem is two-moded: that is, superpositions of both ordinary and extraordinary waves are necessary for the spectral representation of the solution. An important simplifying feature of the problem is that the coupling of the modes is purely due to edge diffraction, there being no reflection coupling. The solution is simple in that the Fourier transforms of the potentials are just algebraic functions. Basic properties of the solution are briefly discussed.

The Two-Dimensional Electrostatic Solutions of Born's new Field Equations

1935 ◽  
Vol 31 (1) ◽  
pp. 50-68 ◽  
Author(s):  
M. H. L. Pryce
Keyword(s):  
General Solution ◽  
General Theory ◽  
Singular Solution ◽  
Equations Of Motion ◽  
Constant Field ◽  
Two Dimensional ◽  
Field Equations ◽  
Special Cases

The general solution of Born's new field equations is found for the two-dimensional electrostatic case, by which the coordinates are expressed as functions of the field vectors, Conditions for inversion are discussed. Special cases are worked out, namely: singnle charge, two charges, charge in a constant field. Expressions are given for forces acting on the charges. A singular solution is also discussed, with reference to the neutron. The implication of the solutions on the general theory and the equations of motion is discussed in the conclusion.

In-Plane Ice Structure Vibration Analysis by Two-Dimensional Elastic Wave Theory

1984 ◽  
Vol 106 (2) ◽  
pp. 160-168 ◽  
Author(s):  
C. H. Luk
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Fourier Transforms ◽  
Ice Sheet ◽  
Elastic Solid ◽  
Radiation Condition ◽  
Wave Theory ◽  
Two Dimensional ◽  
Transverse Waves ◽  
Cylindrical Structure

This paper presents a theoretical analysis of an in-plane ice sheet vibration problem due to a circular cylindrical structure moving in the plane of an infinite ice sheet, and computes the ice forces exerted on the structure as the motion occurs. The basic equations are derived from two-dimensional elastic wave theory for a plane stress or plane strain problem. The ice material is treated as a homogeneous, isotropic and linear elastic solid. The resulting initial and boundary value problems are described by two wave equations. One equation governs the ice motion associated with longitudinal wave propagation, and the other governs propagation of transverse waves. The equations are subject to 1) either a fixed or a frictionless boundary condition at the ice structure interface, and 2) a radiation condition at large distance from the structure to ensure the existence of only outward traveling elastic waves. The governing equations are then solved by 1) Fourier transforms, or 2) Laplace transforms, depending on the problem. Closed-form solutions are obtained in terms of Bessel functions. Plots are provided for estimating the ice added mass, the damping, and the unit function response for a circular cylindrical structure vibrating in the horizontal plane of an infinite ice sheet.

Regularization of the Kelvin–Helmholtz instability by surface tension

2007 ◽  
Vol 365 (1858) ◽  
pp. 2253-2266 ◽  
Author(s):  
David M Ambrose
Keyword(s):  
Surface Tension ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Energy Estimates ◽  
Two Dimensional ◽  
Helmholtz Instability ◽  
Joint Work ◽  
Numerical Work ◽  
Well Posed

The Kelvin–Helmholtz instability is present in the motion of a vortex sheet without surface tension. This can be seen from the linearization of the equations of motion, and there have also been proofs of ill-posedness for the full nonlinear equations. In the presence of surface tension, the linearized equations no longer exhibit an instability, and it has been believed that the full equations should then be well-posed. In this paper, I sketch a proof that the vortex sheet with surface tension is well-posed in the case of both two- and three-dimensional fluids. The proof in the case of three-dimensional fluids is the joint work with Nader Masmoudi. The method is to first reformulate the problem using suitable variables and parametrizations, and then to perform energy estimates. The choice of variables and parametrizations in the two-dimensional case is the same as that of Hou et al . in a prior numerical work.

Heat Transfer From an Annular Fin of Constant Thickness

1959 ◽  
Vol 81 (2) ◽  
pp. 151-156 ◽  
Author(s):  
H. H. Keller ◽  
E. V. Somers
Keyword(s):  
Graphical Form ◽  
Constant Thickness ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Small Height ◽  
Annular Fin ◽  
Inner Radius ◽  
The One ◽  
Group B ◽  
Annular Fins

The two-dimensional solution for annular fins, given in graphical form with the efficiency plotted versus the design-parameter group, (b − a)[(2h)/(kw)]1/2, supplies needed design information for fins of small height-to-thickness ratio and large height-to-inner radius ratio. The one-dimensional solutions previously given for annular fins are accurate for height-to-width ratios of the order of 10 or more, while the two-dimensional results for rectangular fins are useful as approximations to annular fins when curvature is not large. With height-to-width ratios less than 10 and for annular fins with large curvature, design of fins can be computed with the results presented in this paper.

scholarly journals Modeling strong motions produced by earthquakes with two-dimensional numerical codes

1988 ◽  
Vol 78 (1) ◽  
pp. 109-121
Author(s):  
Donald V. Helmberger ◽  
John E. Vidale
Keyword(s):  
Wave Equation ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Cylindrical Coordinates ◽  
Dimensional Theory ◽  
Dimensional Solution ◽  
Source Excitation ◽  
Dislocation Sources

Abstract We present a scheme for generating synthetic point-source seismograms for shear dislocation sources using line source (two-dimensional) theory. It is based on expanding the complete three-dimensional solution of the wave equation expressed in cylindrical coordinates in an asymptotic form which provides for the separation of the motions into SH and P-SV systems. We evaluate the equations of motion with the aid of the Cagniard-de Hoop technique and derive close-formed expressions appropriate for finite-difference source excitation.

Two-dimensional leaps in near-critical flow over isolated orography

2005 ◽  
Vol 461 (2064) ◽  
pp. 3747-3763 ◽  
Author(s):  
E.R Johnson ◽  
G.G Vilenski
Keyword(s):  
Unsteady Flow ◽  
Equations Of Motion ◽  
Fold Increase ◽  
Flow Speed ◽  
Two Dimensional ◽  
Subcritical Flow ◽  
One Dimensional ◽  
Petviashvili Equation ◽  
Lee Wave ◽  
Supercritical Flows

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

EQUIVALENCE OF TWO-DIMENSIONAL GRAVITIES

1990 ◽  
Vol 05 (16) ◽  
pp. 1251-1258 ◽  
Author(s):  
NOUREDDINE MOHAMMEDI
Keyword(s):  
Gauge Theory ◽  
Explicit Solution ◽  
Light Cone ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Light Cone Gauge ◽  
Dimensional Gravity ◽  
The Relationship ◽  
Chern Simons

We find the relationship between the Jackiw-Teitelboim model of two-dimensional gravity and the SL (2, R) induced gravity. These are shown to be related to a two-dimensional gauge theory obtained by dimensionally reducing the Chern-Simons action of the 2+1 dimensional gravity. We present an explicit solution to the equations of motion of the auxiliary field of the Jackiw-Teitelboim model in the light-cone gauge. A renormalization of the cosmological constant is also given.

A general solution method for two-dimensional nonplanar oxidation

1983 ◽  
Vol 30 (9) ◽  
pp. 993-998 ◽  
Author(s):  
Daeje Chin ◽  
Soo-Young Oh ◽  
R.W. Dutton
Keyword(s):  
General Solution ◽  
Solution Method ◽  
Two Dimensional

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

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