In this paper we study the motion of an elastic conducting wire in a magnetic field. The motion of the conductor induces a current in the wire (Faraday's law) which, in turn produces a force on the wire. We consider the linear equation obtained by linearizing the resulting equations of motion about an equilibrium solution. This is a hyperbolic partial differential equation with a non-local term. We prove existence and uniqueness of a weak solution of an initial-boundary value problem for this equation. (C) 1998 B. G. Teubner Stuttgart-John Wiley & Sons, Ltd.

We consider the equilibrium states of elastic strings loaded transversely by their own weight. We are interested in the relation between the equilibrium states of inextensible strings and those of strings which are nonlinearly elastic and slightly extensible. The inextensible string has a unique concave equilibrium state while the concave equilibrium states of a nonlinearly elastic string come in pairs. In this paper we explore the relation between the two models.

In this paper we study the equilibrium states of a conducting rod with small bending stiffness in a magnetic field. The magnetic field is produced by current flowing in a pair of infinitely long parallel wires. The line between the supports of the rod is in the plane of the wires and equidistant from them. The rod is clamped at both ends. We consider planar deformations of the rod. We prove a bifurcation theorem describing the set of equilibrium states. Our analysis of this problem brings together two important theories in modern applied mathematics; bifurcation theory and the theory of singular perturbations for systems of nonlinear ordinary differential equations.

We study the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field, a problem we have considered in several previous papers. We are now able to prove a global bifurcation theorem for this problem. To do this, two difficulties must be overcome. The first is the presence of the rotation group SO(2) as a symmetry group for the problem. The second is that, for some values of certain parameters, the linearized problem is a nonstandard eigenvalue problem. The former difficulty is overcome by applying an idea due to Healey, who observed the existence of an additional symmetry in a related problem first posed by the present author. The latter problem is handled by using some nonstandard tools from functional analysis.

We study the equilibrium states of an elastic column subject to an end thrust.The column is modeled as an inextensible rod with a nonlinear moment-curvature equation.We bring various mathematical theories to bear on this problem; the calculus of variations, bifurcation theory, phase plane analysis, and singular perturbation theory for ordinary differential equations.We study the connections between the various approaches.Many of the results presented here are not new.Our object is to show how this problem can serve to illustrate various abstract theories and to show how the various approaches compliment each other.

In this paper we continue the study of the deformation of a rod with small bending stiffness begun in two earlier papers. In the theory that we have developed problems are posed as singular perturbations of problems in which the rod is replaced by a string having no resistance to bending. In this paper we apply a general theory of singular perturbations for boundary-value problems for nonlinear systems of ordinary differential equations to obtain our result. This result consists of a construction of an approximate solution to a model problem of the deformation of a rod with small bending stiffness, together with a proof that our solution approximates an exact solution to the problem. Our approximate solution consists of an interior approximation which is a solution of the string problem together with boundary-layer correction terms. The most important point of the paper is the construction of these boundary-layer terms. We also consider a case in which our construction fails together with an example which serves to illuminate this situation.

This paper treats the problem of an inextensible hanging cable loaded by its own weight. Our analysis is mathematically exact and completely rigorous. The cable may have variable density and the support points need not be at the same height. We consider both the perfectly flexible cable and the cable with small bending stiffness. In the latter case the moment-curvature relation may be nonlinear. We treat the problem as a singular perturbation of the perfectly flexible case and construct an approximate solution by introducing boundary layer terms. Some recent results enable us to conclude that our approximate solution is asymptotic to an exact solution as the bending stiffness tends to zero.

In this paper we consider a model problem for the deformation of a rod with small bending stiffness. We show that this problem can be considered as a singular perturbation of the problem in which the rod is replaced by a string with no resistance to bending. We construct an approximate solution to this problem. As the bending stiffness tends to zero this solution tends to the solution of the string problem away from the ends of the rod which are assumed to be clamped. However, as one would expect, there is a boundary layer near each end of the rod. The main point of the paper is to show how to construct the boundary layer corrections.

In this paper we study the equilibrium states of a conducting rod in a magnetic field. The rod is assumed to be non-linearly elastic. It may undergo extension and flexure but not shear. The magnetic field is produced by current flowing in a pair of infinitely long parallel wires. The line between the supports of the rod is in the plane of the wires and equidistant from them. The rod is clamped at both ends. We consider inplane deformations. We study the set of equilibrium states by applying the global bifurcation results of Crandall and Rabinowitz.

In this paper we study the equilibrium states of a nonlinearly elastic wire in a magnetic field. The wire is perfectly flexible, is suspended between fixed supports and carries an electric current. We consider two problems. The first in which the magnetic field is constant can be solved exactly. The set of solutions illustrates the phenomenon of “symmetry breaking” which is a chapter in the theory of imperfect bifurcation. The second problem is one in which the magnetic field is produced by current flowing in a pair of infinitely long parallel wires. When the line of supports of the elastic wire is parallel to these and equidistant from them we may apply the global bifurcation results of Crandall and Rabinowitz to study the set of solutions. We also consider perturbations of this case. This is another example of imperfect bifurcation.

We study the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. The rod, which can undergo flexure, torsion, shear and extension, is welded to fixed supports. The rod carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. The fundamental parameter is λ = IB where I is the current in the rod and B is the strength of the magnetic field. For all λ > 0 there exists a trivial state in which the rod is straight and untwisted. Here we show that, under the assumption that the rod is hyperelastic, in certain cases bifurcation occurs and hence there are infinitely many non-trivial states.

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L2-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal “necessary conditions for optimality.” A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.

In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches.

Let Lu be the integral operator defined by (Lkϑ)(x, y) = ∝ s ∝ ϑ(x′, y′)(eikϱϱ) dx′ dy′, (x, y) ϵ S where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ⩾ 0, Im k ⩾ 0, and ϱ2 = (x −x′)2 + (y − y′)2. We define q(x, y) = [dist((x, y), ∂S)]12, (x, y) ϵ S; L2(q, S) = {ƒ : ∝ s ∝ ¦ ƒ(x, y)¦2 q(x, y) dx dy < ∞}; W21(q, S) = {ƒ : ƒ ϵ L2(q, S), ∂ƒ∂x, ∂f∂y ϵ L2(q, S)}, where in the definition of W21(q, S) the derivatives are taken in the sense of distributions. We prove that Lk is a continuous 1-l mapping of L2(q, S) onto W21(q, S).

SynopsisIn this paper we study the wave equation, in particular the propagation of discontinuities. Two problems are considered: diffraction of a normally incident plane pulse by a plane screen and diffraction of a spherical wave by the same screen. It is shown that when an incident wave front strikes the edge of the screen a diffracted wave front is produced. The discontinuities are precisely computed in a neighbourhood of the edge for a small time interval after the arrival of the incident wave front and a theorem of Hörmander on the propagation of singularities is used to obtain a globalresult.

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.

Journal Article DIFFRACTION OF PLANE WAVES BY A SEMICIRCULAR STRIP Get access PETER WOLFE PETER WOLFE Department of Mathematics and Statistics, The University of New MexicoAlbuquerque, New Mexico 87131Department of Mathematics, University of MarylandCollege Park, Maryland 20742 Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mechanics and Applied Mathematics, Volume 28, Issue 3, August 1975, Pages 355–371, https://doi.org/10.1093/qjmam/28.3.355 Published: 01 August 1975 Article history Received: 13 May 1974 Revision received: 30 December 1974 Published: 01 August 1975

Let θ be an open set in the plane which contains the interval I: {(x, y)∣y=0, -1?x?1}. We consider functions u, which are harmonic in θ-I and continuous in θ. Then, without additional smoothness conditions on f(x)=u(x,0),-1?x$le;1, the one sided normal derivatives(?u/?y)+limh?0+u(x,h)-f(x)/h and (?u/?y)-=limh?0-u(x,h)-f(x)/h, may not exist at any point of I. Here we assume only that f(x) is continuous. We show that in this case the normal derivatives will exist in a "Sobolev-like" Space of distributions. Notes This research was supported in part by the National Science Foundations under Grant GP 12838 with the university of Maryland.

In this paper we consider the problem of diffraction of a normally incident plane pulse by a strip. We use the method of Laplace transformation. By applying results on the asymptotic behavior of solutions of the reduced wave equation, we are able to establish the rate of decay as t → ∞ of the solution of our problem.