William B. Gordon

Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields

The standard far-field approximation to the Kirchhoff formula for the field scattered by a flat metallic plate S of arbitrary shape is given by a certain surface (double) integral. This double integral can be reduced to a line integral evaluated around the boundary of S. Moreover, if S is a polygon, this line integral can be reduced to a closed form expression involving no integrations at all. The use of such line integral representations can easily reduce the costs of numerical calculation by orders of magnitude. If the integrands are to be sampled p times per wavelength to achieve an acceptable degree of precision, and if A is the area of S , then the numerical evaluation of the double integral requires p^{2}A/\lambda^{2} functional evaluations whereas the line integral only requires p\sqrt{A/\lambda^{2}} . If S is a polygon with N vertices, then only 2N functional evaluations are required to evaluate the closed form expression with no quadrature error at all.

High frequency approximations to the physical optics scattering integral

Discusses two high frequency (HF) approximations to the physical optics (PO) scattering integral for the far field radar backscatter from a general curved edged reflecting surface viewed at arbitary aspect. The PO scattering integral is first approximated as the sum of a specular effect and an edge effect, where the latter is represented explicitly as a certain line integral evaluated over the boundary edge of the reflector. A closed form result is then obtained by applying the method of stationary phase to the line integral. With the exception of singularities that can occur at caustics, or when the specular point falls on the boundary edge, these HF approximations are found to work reasonably well for smooth surfaces whose Gaussian curvatures have constant sign (positive or negative, but never zero). >

Vector potentials and physical optics

The general problem considered is to obtain solutions w to the vector equation v = curl(w), where v is a given divergence−free vector field with singularities. Two methods are discussed: A special method, which applies when v is of the type which occurs in the Kirchhoff theory of diffraction, and a general method, which applies to any divergence−free vector field whatever. As an example the general method is used to obtain the Maggi−Rubinowicz representations of the Kirchhoff−Helmholtz (double) integral as a line integral. The singularities of the solutions w are known to produce important optical effects, and the nature of these singularities is largely determined by topological properties of the domain on which v is regular.

Reduction of surface integrals to contour integrals

The closed-form reduction of surface integrals to contour integrals has potential applications to the numerical solution of scattering problems. The standard mathematical technique for the reduction of a general surface integral requires the solution of a partial differential equation of the Poisson type. In this paper, we show how the reduction can be accomplished by calculating an indefinite integral of type /spl int/ f (t) dt, where f = f (t) is a known function in a single variable t.

High-frequency approximations for edge scatter from surfaces of revolution

We consider the radar backscatter from a truncated surface of revolution viewed at axial incidence and derive a closed-form high-frequency (HF) approximation for the physical-optics (PO) scattering integral. The PO predicted edge effect is extracted from this result and used to obtain a closed-form HF representation for the edge effect predicted by the geometrical theory of diffraction. The predictions for total radar backscatter thus obtained are compared with the results of some method-of-moment calculations (for paraboloids) and also with the older theoretical results of Raybin [ IEEE Trans. Antennas Propag.AP-13, 754 ( 1965)] for spherical segments.

Contour Integral Representation for Near Field Backscatter From a Flat Plate

We consider the near field backscatter from a flat plate illuminated by a dipole source. The physical optics scattering integral is reduced to a contour integral evaluated around the boundary edge of the plate.

Reflections From Multiple Surfaces Without Edges

An algorithm is presented for calculating the positions of the specular points that appear when a collection of reflecting surfaces is illuminated by an external source. The set of specular points is represented as the fixed point of a certain mapping, and this fixed point is calculated by the method of successive approximations (MSA). The MSA is an iterative technique which is essentially different from a search or shooting and bouncing ray technique. The latter require much larger numbers of functional evaluations, especially when the number N of reflecting surfaces is greater than unity. A search technique requires a number of function evaluations that varies exponentially with N, whereas the number of function evaluations required by the MSA varies linearly with N.

An effect of receiver noise on the measurement of Doppler spectral parameters

We consider the Doppler power spectra calculated from radar precipitation echos. It is shown that receiver noise can produce a spurious linear realtion between the spectral width and radar range. This effect occurs at all elevation angles and disappears when the effects of receiver noise are eliminated by means of a thresholding technique. A simple model explains why this effect occurs when the signal spectrum is sufficiently narrowband and the variation of radar reflectivity isnot too great over the range interval of interest. Examples are given from data taken with a coherent X-band radar.

PERIODIC SOLUTIONS TO HAMILTONIAN SYSTEMS WITH INFINITELY DEEP POTENTIAL WELLS

Publisher Summary This chapter discusses periodic solutions to Hamiltonian systems within finitely deep potential wells. Variational techniques have recently been used to obtain information concerning the existence of periodic solutions to conservative Hamiltonian systems with convex potential wells. The chapter presents an account of similar results that have been obtained for Hamiltonian systems with infinitely deep potential wells. In order to extend the theorem to higher dimensional configuration spaces Rn, it is necessary that the set S of singularities satisfy a certain geometric condition.

Statistical moments of radar cross section

The radar cross section (RCS) for backscatter from a smooth closed convex surface reflector, as calculated from geometrical optics, is shown to have a mean value equal to A/4, where A is the total surface area of the reflector, and the mean is obtained by averaging over all aspects. Results are also given for the calculation of higher-order moments and probability density functions. >