Peter Wolfe

Eigenfunctions of the Integral Equation for the Potential of the Charged Disk

The integral equation f(r,θ)= ∫ 02π ∫ 01u(r′,θ′)[r2+r′2−2rr′ cos (θ−θ′)]12·r′(1−r′2)12dr′dθ′ arises in connection with the problem of the electrostatic potential due to a charged disk. We solve the equation by computing a complete set of eigenfunctions and eigenvalues for the integral operator. The eigenfunctions have the form Fm,n±(r,θ)=Pnm[(1−r2)12]e±imθ, 0 ≤ m ≤ n, m + n even. Here Pnm(x) is the associated Legendre function of the first kind.

Equilibrium states of an elastic conducting rod in a magnetic field

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.

The effect of bending stiffness on inextensible cables

Abstract 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.

Bifurcation theory of an elastic conducting wire subject to magnetic forces

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.

Hanging cables with small bending stiffness

IN THIS paper we study the problem of the equilibrium configuration of a cable suspended between two fixed supports. The only load on the cable is its own weight. We model the cable as a Cosserut rod, and assume the cable may undergo extension and flexure but not shear. We wish to study cables having small bending stiffness, which we do by introducing a small parameter E into the constituative equations (stress-strain laws) for the cable. When E = 0 the equilibrium equations reduce to those which arise when the cable is modeled as a string with no resistance to bending. These equations have been analyzed by Antman [l]. He shows that there is always exactly one solution of this problem in which the cable is in tension. The problem for small positive E is a singular perturbation of the string problem. For our problem, based on the rod model, in addition to stipulating the position of the ends of the cable, we must impose additional boundary conditions. We impose the condition that the cable is clamped at both ends. The solution to the string model will not satisfy these additional conditions. Thus we would expect that as E tends to zero the solution of the rod model will tend to the solution of the string model in the interior of the cable. However, in order to satisfy the additional boundary conditions boundary layer corrections must be included to obtain an approximate solution to the rod problem for small E. In a previous paper [2] we considered a similar problem, that of a conducting wire in a magnetic field. This problem was chosen as a starting point for a theory of rods with small bending stiffness because the corresponding reduced (string) problem can be solved explicitly and the solution has a simple form. The main point of that paper was to show how to compute the boundary layer corrections starting from an ad hoc ansatz. There was no claim that the solution constructed there was actually asymptotic to an exact solution as E tends to zero. After completing this paper we became aware of the work of Schmeiser [3] and Schmeiser and Weiss [4]. These papers (and the references cited therein) deal with a general theory of singular perturbations for systems of nonlinear ordinary differential equations. In these papers it is proven that, in fact, the approximate solutions constructed are asymptotic to an exact solution. It became clear that our work could easily be put into the framework of this theory. We also realized that our technique was far more general than originally thought and could be applied to problems more complicated than our model problem addressed in [2]. In Section 2 we will describe the rod model and our concept of small bending stiffness. In Section 3 we will describe the string model and its solution. In Section 4 we will state the general theorem on singular perturbations for nonlinear systems of ordinary differential equations and in Section 5 we will apply this theorem to our problem. Section 6 consists of some concluding remarks.

An integral operator connected with the Helmholtz equation

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).

Diffraction of a plane wave by a circular disk

Abstract 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.

Bifurcation theory of a conducting rod subject to magnetic forces

Abstract 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.

Some distribution solutions of integral equations and their application to partial differential equations

We consider two integral operators, L and L k defined by Let L 2(p)(L 2(q)) be the space of functions defined on [−1, 1] and integrable with respect to the weight function (1−x 2)−½((1−x 2)½) . Let W2 1(q) be the space of functions, f, absolutely continuous on [−1,1] with f ∈ L 2(q) and W 2 −1(q) be its dual. It has previously been shown that L and L k are one to one, continuous maps of L 2(q) onto W 2 l(q). Here we show that these mappings can be extended to mappings L and L k which are one-to-one continuous maps of W 2 −1(q) onto L2(p). These results are applied to the problem of solving the two dimensional Laplace and Helmholtz equations with boundary data given on the interval [−1,1] of the x axis.

Multiple equilibria of elastic strings under central forces: Highly singular nonlinear boundary value problems of the Bernoullis

In this paper we study the multiplicity of equilibrium states of nonlinearly elastic strings under central forces. This problem generalizes that studied by Joh. Bernoulli in 1728 for an inextensible string. (The theory of elastic strings was developed by Jas. Bernoulli in 1691-1704; cf. [ 111.) It is technically more difficult than the related problem of the elastic catenary studied in [ 2, 4, 81 because it does not have a convenient set of integrals and because the central force field may well be infinite at its center. The complications due to this singularity are magnified by the requirement that the compressive contact force in the string must become infinite where the local ratio of deformed to natural length becomes 0. We find that these problems have a multiplicity of both regular and singular solutions, with the analysis of the latter requiring a careful extension of the governing laws of mechanics to handle infinite forces. In the rest of this section we formulate the governing equations. Here we pay special attention to questions of regularity, which cannot be handled routinely in virtue of the singularities in the equations. For a large class of problems we obtain detailed global information about the qualitative behavior of all regular solutions. In Section 2, we give a full characterization of singular solutions. In Section 3 we study purely radial solutions (by means