occurs between the onset of the edipse and its midpoint. This agrees qualitatively with the work of Allais' with a paraconical pendulum, where the change of azimuth increased substantially in the 6rst half of the eclipse of 30 June 1954. Both these e6ects would seem to have a gI'RvltRt1onRl bRsls which cannot-bc explained by accepted classical theory.Both our experimental 6ndings and those of Allais cause one to question whether the classical laws of gravitation hold without modi6cation. CONCLUSIONQuantitative observations made with a precise torsion pendulum show, in agreement with many earlier, less precise recordings made at Harvard since 3.953, that ' Maurice F. C. Allais, Aerospace Kng. 18, 46 (1959). the times required to traverse a 6xed. fraction of its total angular path vary markedly during the hours before the eclipse and during its 6rst half, i.e. , up to its midpoint. Also the signihcant changes in these times do not coincide exactly with the astronomically determined. onset, midpoint, and endpoint of the eclipse.These variations are too great to be explained, on the basis of classical gravitational theory, by the relative change in position of the moon with respect to the earth and sun. This leads to the same conclusion arrived. at by Allaisthat classical gravitational theory needs to be modiaed to interpret his (and our) experimental results.Moreover, the 6ndings with the torsion pendulum, the sig~~~~~n™~~~~~~~~~~0~~~perpendicularly to the geogravitic vector, seem to indicate the possibility of a 6ne structure in these observations neither predicted nor recorded using the orthodox methods of quasistationary gravitational investigations.Upon de6ning vector spherical partial waves {Q ) as a basis, a matrix equation is derived describing scattering for general incidence on objects of arbitrary shape. With no losses present, the scattering matrix is then obtained in the symmetric, unitary form 8= -Q'~P, where (perfect conductor) Q is the Schmidt orthogonalisation of Q""= (k/v) J'de p(vXReg")Xg".g, integration extending over the object surface.For quadric (separable) surfaces, Q itself becomes symmetric, e fecting considerable simplihcation. A secular equation is given for constructing eigenfunctions of general objects. Finally, numerical results are presented and compared with experimental measurements.

Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward-amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square-root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.

No abstract

Upon introducing the outgoing spherical (or circular cylinder) partial waves {ψn} as a basis, the equation QT = − Re (Q) is obtained for the transition matrix T describing scattering for general incidence on a smooth object of arbitrary shape. Elements of Q involve integrals over the object surface, e.g. Qmn = ±(i2)δmn+(k8π)∫dσ⋅∇[Re(ψm)ψn]. where the −, + apply for Dirichlet and Neumann conditions, respectively. For quadric (separable) surfaces, Q is symmetric. Symmetry and unitarity lead to a secular equation defining eigenfunctions for general bodies. Some apparently new closed-form results are obtained in the low frequency limit, and the transition matrix is computed numerically for the infinite strip.

Scalar multiple scattering effects due to a random distribution of spheres are considered in detail. Transformation from a volume to a surface integral allows one to take account of the ``hole corrections'' involved in the equation of multiple scattering, and yields a secular equation for the propagation constant K of the composite medium. In the low-frequency limit a result is given which appears to be exact over the entire range 0 ≤ δ ≤ 1, where δ is the fractional volume occupied by scatterers. Also in this limit, the boundary conditions appropriate to the boundary of the composite medium are established from examination of the total transmitted and reflected fields.

Upon invoking Huygen's principle, matrix equations are obtained describing the scattering of waves by an obstacle of arbitrary shape immersed in an elastic medium. New relations are found connecting surface tractions with the divergence and curl of the displacement, and conservation laws are discussed. When mode conversion effects are arbitrarily suppressed by resetting appropriate matrix elements to zero, the equations reduce to a simultaneous description of acoustic and electromagnetic scattering by the obstacle at hand. Unification with acoustic/electromagnetics should provide useful guidelines in elasticity. Approximate numerical equality is shown to exist between certain of the scattering coefficients for hard and soft spheres. For penetrable spheres, explicit analytical results are found for the first time. Subject Classification: [43120.15, [43]20.30. When a purely transverse (solenoidal) or longitudinal (irrotational) wave is incident on an obstacle, in general both transverse and longitudinal waves are generated. This phenomenon is known as mode conversion, and is expressed in our theory by the presence of certain nonvanishing matrix elements. If mode conversion be artificially suppressed, by resetting the matrix elements in question to zero, then the present equations reduce to an independent superposition of the matrix equations for acoustic •a and electromagnetic •a scattering. We thus have a unified theory of acoustic, electromagnetic and elastic wave scattering by an obstacle of specified geometry. Such unification should prove invaluable, by providing the entire body of theoretical and experimenlal results from acoustics and electromagnetics to use as comparison standards in the elastic case. I. HUYGFNS' PRINCIPLE We seek the scattering from an object bounded by the closed surface c•, as shown in Fig. 1, upon illumination with a given incident wave having particle displacemeat •. The object is situated in a homogeneous, iso

Reflection behavior is examined for an interface having periodic height variation. Upon employing the Bloch theorem, in conjunction with the extended boundary condition, exact matrix equations are obtained for surface fields, as well as transmitted and reflected wave amplitudes. Tunneling considerations then lead to new energy constraints for problems of this type, in which propagating waves are coupled with evanescent modes. In the limit of surface corrugations shallow compared with impinging wavelength, analytic results are obtained confirming both the energy constraints and early computations by Rayleigh. Numerical results demonstrate the efficiency of the method. Subject Classification: 20.30.

Employing self-consistency, the multiple-scattering problem is formulated for periodic arrays of particles having constitutive parameters distinct from those of the embedding material. A T-matrix description of individual particle scattering is employed, so that particles need not be spherical. Explicit analytical and numerical results are obtained for the effective complex dielectric constant ε̄ and permeability μ̄ in the quasistatic and infinitesimal lattice limits for several lattice geometries, and shown to agree with existing static computations under appropriate conditions. Random arrays are also considered briefly, and the role of single-particle resonance effects is examined. Finally, longitudinal electric and magnetic waves are predicted to exist at certain discrete frequencies where ε̄ or μ̄ vanish.