Joseph B. Keller

Geometrical Theory of Diffraction

The geometrical theory of diffraction is an extension of geometrical optics which accounts for diffraction. It introduces diffracted rays in addition to the usual rays of geometrical optics. These rays are produced by incident rays which hit edges, corners, or vertices of boundary surfaces, or which graze such surfaces. Various laws of diffraction, analogous to the laws of reflection and refraction, are employed to characterize the diffracted rays. A modified form of Fermat’s principle, equivalent to these laws, can also be used. Diffracted wave fronts are defined, which can be found by a Huygens wavelet construction. There is an associated phase or eikonal function which satisfies the eikonal equation. In addition complex or imaginary rays are introduced. A field is associated with each ray and the total field at a point is the sum of the fields on all rays through the point. The phase of the field on a ray is proportional to the optical length of the ray from some reference point. The amplitude varies in accordance with the principle of conservation of energy in a narrow tube of rays. The initial value of the field on a diffracted ray is determined from the incident field with the aid of an appropriate diffraction coefficient. These diffraction coefficients are determined from certain canonical problems. They all vanish as the wavelength tends to zero. The theory is applied to diffraction by an aperture in a thin screen diffraction by a disk, etc., to illustrate it. Agreement is shown between the predictions of the theory and various other theoretical analyses of some of these problems. Experimental confirmation of the theory is also presented. The mathematical justification of the theory on the basis of electromagnetic theory is described. Finally, the applicability of this theory, or a modification of it, to other branches of physics is explained.

The transverse force on a spinning sphere moving in a viscous fluid

The flow about a spinning sphere moving in a viscous fluid is calculated for small values of the Reynolds number. With this solution the force and torque on the sphere are computed. It is found that in addition to the drag force determined by Stokes, the sphere experiences a force F L orthogonal to its direction of motion. This force is given by

Bubble oscillations of large amplitude

{\bf F}_L = \pi a^3 \rho \Omega \times {\bf V}[1 + O(R)]

Exact non-reflecting boundary conditions

. Here a is the radius of the sphere, Ω is its angular velocity, V is its velocity, ρ is the fluid density and R is the Reynolds number,

TRANSPORT EQUATIONS FOR ELASTIC AND OTHER WAVES IN RANDOM MEDIA

R = \rho \mu ^{-1} Va

A Theorem on the Conductivity of a Composite Medium

. For small values of R , the transverse force is independent of the viscosity μ. This force is in such a direction as to account for the curving of a pitched baseball, the long range of a spinning golf ball, etc. It is used as a basis for the discussion of the flow of a suspension of spheres through a tube. The calculation involves the Stokes and Oseen expansions. A representation of solutions of the Oseen equations in terms of two scalar functions is also presented.

Corrected bohr-sommerfeld quantum conditions for nonseparable systems☆

A new equation is derived for large amplitude forced radial oscillations of a bubble in an incident sound field. It includes the effects of acoustic radiation, as in Keller and Kolodner’s equation, and the effects of viscosity and surface tension, as in the modified Rayleigh equation due to Plesset, Noltingk and Neppiras, and Poritsky. The free and forced periodic solutions are computed numerically. For large bubbles, such as underwater explosion bubbles, the free oscillations agree with those obtained by Keller and Kolodner. For small bubbles, such as cavitation bubbles, with small or intermediate forcing amplitudes, the results agree with those calculated by Lauterborn from the modified Rayleigh equation of Plesset et al. For large forcing amplitudes that equation yielded unsatisfactory results whereas the new equation yields quite satisfactory ones.

Poroelasticity equations derived from microstructure

An exact non-reflecting boundary condition is devised for use in solving the reduced wave equation in an infinite domain. The domain is made finite by the introduction of an artificial boundary on which this exact condition is imposed. In the finite domain a finite element method is employed. Although the boundary condition is non-local, that does not affect the efficiency of the computational scheme. Numerical examples are presented which show that the use of the exact non-local boundary condition yields results which are much more accurate than those obtained with various approximate local conditions. The method can also be used to solve problems in large finite domains by reducing them to smaller domains, and it can be adapted to other differential equations.

Traveling wave solutions of a nerve conduction equation

Abstract We derive and analyze transport equations for the energy density of waves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization effects, coupling of different types of waves, etc. We also show that diffusive behavior occurs on long time and distance scales and we determine the diffusion coefficients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.

Elastic, Electromagnetic, and Other Waves in a Random Medium

A composite medium consisting of a rectangular lattice of identical parallel cylinders of arbitrary cross section is considered. The cylinders have conductivity σ2 and are imbedded in a medium of conductivity σ1. Simple properties of the conductivity tensor of the composite medium are deduced from the theory of harmonic functions.