Tangential Derivatives Research Articles

Variational method for prediction of acoustic radiation from vibrating bodies

Integral equation formulations such as discussed by Chertock, Copley, Schenck, Meyer, Bell, Zinn, Stallybrass, and many others, typically have nonsymmetric kernels and are not easily rephrased as a variational principle. However, a symmetric kernel does result from the normal derivative of the Kirchhoff‐Helmholtz integral, with the exterior point subsequently allowed to approach the surface, but the integrand then has unmanageable singularities. A method due to Maue and Stallybrass allows this integral equation to be recast into a usable form (involving tangential derivatives of the unknown surface pressure) and this version in turn leads to a variational principle that has considerable promise for systematic approximate solutions of radiation problems in the low to moderate ka regimes. Validity of this formulation is substantiated by derivation of known results for vibrating spheres and disks. Numerical results for other geometries are expected to yield higher accuracy than previously because one can incorporate one's best “physical insight” in initial choices for the class of trial functions for surface pressure. If computations now in progress are completed, the example of an oscillating airfoil will be discussed, with some expected clarification of the anomalous results of the Brooks' experiment.

A Weak Formulation of the Shallow-Water Equations for a Rotating Basin

Abstract A weak formulation of the equation for the elevation field arising from the shallow-water equations for a rotating inviscid fluid has been developed. The difficulties of this problem, which are due chiefly to the fact that the normal velocity along the contour is given by a linear combination of normal and tangential derivatives of the elevation field, are overcome. As a result, many problems, until now treated using the elevation and velocity component variables, might be solved dealing only with the elevation.

An integral bound for the strain energy in nonlinear elasticity in terms of the boundary displacements

For the displacement boundary value problem in nonlinear elastostatics with zero body force, an integral bound for the strain energy is obtained in terms of theL2-norms of the given boundary displacements and their tangential derivatives (assumed sufficiently small). The constants involved depend upon the strain energy density function and upon the geometry of the domain.

A Harnack inequality for degenerate elliptic equations on minimal surfaces

Harnack inequalities are known to be of great importance in the theory of quasilinear elliptic partial differential equations. In the case of such equations defined over a domain Ω in Rn, inequalities of this type have been proved for solutions of second-order equations in divergence form which are of either elliptic or degenerate elliptic structure. More recently Bombieri and Giusti (2) have proved a Harnack inequality for solutions of linear elliptic equations on a minimal surface in Rn+1. The equations are of the formwhere summation over i, j = 1, …, n+l is understood, and δ = (δ1, …, δn+1) is the tangential derivative on S. In (2), the inequality is used to give much simplified proofs of some classical results on minimal surfaces, and to generalise some more recent ones.

Kupradze's functional equation for the torsion problem of prismatic bars — Part 1

St. Venant's torsion problem for a prismatic bar is reduced to an integral equation of the Fredholm type with the help of Kupradze's method: i.e. for a given load distribution on the lateral surface of the cylinder (“actual body K + ”) imbedded in an elastic medium (“basic body K 0 ”), a distribution of screw dislocations has to be determined on the lateral surface in such a manner that the external region of the infinite medium (“complementary body K − ”) will be free of tension. The unknown function is the tangential derivative of the warping function. Part 1 demonstrates the method for a smooth boundary. In part 2 the influence of the corners on the convergence of the numerical solution of the integral equation is investigated. Prismatic bars with multiconnected cross sections as well as cross sections with corners can be treated with this method.

Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter

We consider solutions of inhomogeneous, reduced hyperbolic equations of the second order, with a large parameter multiplying the unknown function. These solutions are defined on the m-dimensional region outside a star-shaped body. They satisfy an “outgoing” radiation condition at infinity and a Dirichlet boundary condition. We obtain a priori estimates for these solutions, at every point outside or on the surface of a two- or three-dimensional star-shaped body, that hold for sufficiently large values of the parameter. We prove that each solution is bounded by a linear combination of (i) the maximum norm of its prescribed boundary values, (ii) the L 2 norm of the prescribed values of its tangential derivative, (iii) an L 2 norm of the source term. This result is based on similar inequalities that we first obtain for a certain L 2 norm of the gradient, and of the normal derivative on the boundary, of each solution defined outside an m-dimensional star-shaped body. For the special case of the reduced wave equation, Morawetz and Ludwig [1] have obtained similar estimates. Just as their results have been used in [3] to confirm the geometrical theory of diffraction, the estimates obtained in this paper can be used to establish the validity of certain formal asymptotic solutions of reduced hyperbolic equations.

A Priori Bounds for the Displacement Problem in Viscoelasticity

A priori estimates are obtained for solutions of the displacement problem in isotropic linear viscoelasticity theory under the assumption of monotone nonincreasing positive relaxation moduli. In these estimates, integral measures of the displacement and the displacement gradient are bounded by surface integrals involving the Dirichlet data and a tangential derivative of the Dirichlet data and by volume integrals of the body force. Coefficients appearing in these estimates are explicitly determined. Integral estimates are then used to derive a pointwise bound, in the case of zero body force.

Continuity of tangential derivatives of a thermal potential

Continuity of tangential derivatives of a thermal potential

On Diffusion in a Fractured Medium

A “narrow fracture” approximation is applied to the diffusion equation on an inhomogeneous region in order to convert the parabolic interface problem into a boundary value problem with tangential derivatives on part of the boundary. This boundary value problem is shown to have a unique weak solution. Finally, a particular steady state model is solved, essentially in closed form, with a “Galerkin by lines” technique.

Gravitational Field of Shells and Disks in General Relativity

The problem of obtaining the gravitational field of static, axially symmetric, thin shells is elucidated. In particular, a clear distinction between global and local frames is made. An algorithm is given for obtaining the fields of disks. There are two significant gravitational potentials $\ensuremath{\lambda}$ and $\ensuremath{\varphi}$. The potential $\ensuremath{\lambda}$ is straightforwardly determined from the radial stresses by solving a two-dimensional potential problem. This potential is analytic everywhere except on the disk and, together with its stream function $\overline{z}$, can be used to generate a conformal transformation which brings the equation for $\ensuremath{\varphi}$ into the form of Laplace's equation. This potential can then be found by solving a Neumann boundary-value problem. However, the surface in the new coordinate system is not a disk since $\overline{z}$ is discontinuous across the disk. This is due to the fact that the Cauchy-Riemann equations imply that if the normal derivative of $\overline{\ensuremath{\rho}}$ is discontinuous, then the tangential derivative of $\overline{z}$ will be discontinuous.

Differentiable solutions of symmetrizable and singular symmetric first order systems

In this paper, we derive some differentiability theorems for boundary problems for first order systems of partial differential equations. The results stem from the observation that if the coefficient of normal differentiation near the boundary is independent of the tangential coordinates, then the commutator of the differential operator with any tangential derivative is independent of that coefficient; a similar observation was used by LAX & PHILLIPS [5] in introducing tangential mollifiers. With the help of this observation, we derive differentiability estimates for the symmetrizable systems of FRmDmCrlS & LAX [2] and for certain singular symmetric systems. In the latter case the boundary may be characteristic and the associated boundary form may be semi-definite. Our results contain, and provide an alternate derivation for, differentiability theorems of FRmDRICHS [1], G. PEYSER [71 and CrIAO-HAO GtJ [31. We have used LAX'S method of dual inequalities to derive a priori estimates for higher order derivatives; similar results could probably be obtained with PEYSER'S technique of using overdetermined systems. We also wish to observe that the techniques used in section 5 are imitative of procedures used by KOHN & NIRENBERG [41. I. Symmetrizable Systems

Polarographic maxima and dipole repulsion forces

Strong dipole repulsion forces at the mercury interface can give rise to polarographic maxima of the first kind even in the absence of net free surface charge. The dipole energy and forces yield an expression for the surface velocity of the mercury droplet. It is shown that movement of the mercury surface should cease when the dipole concentration is uniformly depleted by electrolytic action even though the potential increases. Maxima of the first kind may be interpreted in terms of dipole repulsion forces. The force is given by the product of the dipole moment and the tangential derivative of the electric intensity field. The force appears to be substantial because of the size of the tangential derivatives of the field. The field arises from the dipoles and quadrupoles created by “bond” formation between the electrode site and adsorbed species. Non-uniform surface concentration of the adsorbed species is responsible for the non-uniform fields and large tangential field derivatives. Non-uniform surface concentrations occur whenever the current distribution is asymmetric as in polarographic systems. Here the dipole repulsion forces bring about the migration of matter at the interface. An expression for the velocity of the migration has been derived. The velocity may be determined from reasonable assumptions about the hydrodynamic boundary layer thickness. The expression for the velocity indicates an association of polarographic maxima with electrocapillary maxima and it predicts the cessation of polarographic maxima with increasing potential because of dipole depletion.

The field of a dipole above an infinite corrugated plane

We investigate the excitation and propagation of the three dimensional electromagnetic field over an infinite corrugated plane which is approximated by an anisotropic impedance boundary condition. Emphasis is placed upon effects of surface anisotropy which are not evident in two dimensional treatments. In particular we consider the excitation by a magnetic point dipole in detail. It turns out that the fields are determined by a scalar wave function which satisfies a mixed boundary condition involving a linear combination of the wave function, its normal derivative and its second order tangential derivative. The exact formal solution is first derived, and then the radiated far field and the surface wave far field are evaluated separately. Both the phase and the amplitude of the excited surface wave are dependent upon the direction of observation. Numerical results are given. The physical significance of this solution is discussed. A comparison is made between this problem and the theory of ship waves.

Optical Properties and Electronic Structure of Solid Silicon

The existence of an empty (conduction) band 1.5 or 2.0 volts above the highest filled band in metallic-looking semi-conductors such as silicon and certain sulfides is inferred from the optical properties of these substances. Reflectivity data of silicon are examined with the aid of the expression for the complex index of refraction as given by classical electromagnetic theory. Through use of this expression, which is checked for internal consistency, an oscillator strength of 1.6 electrons is calculated for the absorption, and a dielectric constant at low frequencies of 12.5 is computed. The Wigner-Seitz-Slater method of computing electronic energy bands in crystals is used to determine the band structure of silicon. A combination of one $s$, three $p$, and three $d$ functions for each of the two atoms in the unit cell is made, and through use of boundary conditions of continuity of value, of normal, and of tangential derivatives, a solution is obtained for the plane $x=y$. The resulting band structure closely resembles that obtained by Kimball for the diamond, except that silicon is more nearly metallic than is the diamond. The $3s$ level of the silicon atom splits into two bands, the $3p$ level into six bands, and the $3d$ level into six bands; the two $3s$ bands and the lowest two $3p$ bands are completely filled by electrons. At the observed half-distance between nearest neighbors, the gap between the uppermost filled band and the lowest empty band is much greater than that expected from consideration of the optical data. The form of the optical absorption as expected from band structure considerations is proposed. The width of the filled bands as observed in soft x-ray emission spectra is about equal to the width of the computed bands.