Rough Sphere Research Articles

Field Theory of Depolarization of Radar Backscatter — With Application to a Distant, Slightly Rough Sphere

This paper sets out to develop a vector field theory of time‐dependent backscatter from a large target and shows how the cross‐polarization of the return signal follows as a natural consequence of certain spatial asymmetry of the target with respect to the illuminating source. It presents an analysis of radar backscattered power from a distant, slightly rough sphere for both pulsed and sinusoidally steady‐state sources. At first, an integral expression for the received power is developed for an arbitrarily rough sphere. Then the height deviation from the mean sphere is specialized to have a bivariate Gaussian distribution and a Gaussian correlation function. The average direct‐ and cross‐polarized received powers are derived for both the pulsed and the steady‐state cases. Numerical results are computed for the case of a very slightly rough sphere for which σ/δ is of the otder of 10‒4 and presented in contrast with experimental data from a target with a much larger scale roughness, namely the moon. (In this paper σ represents the standard deviation of heights and δ is the correlation distance.)

Radar echo variations of a large rough sphere

The effect of surface roughness on the radar backscattering cross section of a perfectly conducting nominally spherical target is examined by applying the Kirchhoff method. It is shown that, for the type of roughness and sphere size to which the Kirchhoff method is applicable, the standard deviation of the cross section increases with frequency according to the law 2\sqrt{2} \sigma_{0} k\zeta until the first Fresnel zone reduces in size to the scale length of the roughness. At this point a knee in the curve occurs and its further course is determined by a more detailed statistical description of the surface. Here \sigma_{0} is the nominal cross section, \zeta is the standard deviation of the surface height h and k=2\pi/\lambda , where \lambda is the wavelength. The average cross section is shown to be given by \sigma_{0}[1+0 \{(kh)^{3}\}] . Some experimental results are reported which support the theoretical conclusions.

Boundary-Layer Transition on a Cooled Rough Sphere in Hypersonic Flow

Measurements are given of the combined effects of two‐dimensional roughness and surface cooling on boundary layer transition at 45° from the stagnation point on a sphere in simulated hypersonic flow. With the roughness elements at 22.5° the combined effects of roughness and cooling are represented by a single functional relation between the transition length Reynolds number and the ratio between roughness height and the displacement thickness of the boundary layer at the roughness position. The measurements cover a range of ratios of surface‐to‐stagnation temperature of 0.5 to 1.0 and roughness heights of 1.8 × 10−4 to 4 × 10−3 in. Comparisons are made with measurements in incompressible flow and with one datum point in hypersonic flow.

Relation between Thermal Conductivity and Viscosity for Nonpolar Gases. II. Rotational Relaxation of Polyatomic Molecules

The dimensionless ratio f = λM/ηCv relating the thermal conductivity, molecular weight, viscosity, and constant volume molar heat capacity has been determined for several nonpolar polyatomic gases in the neighborhood of room temperature (270°–295° K). The experimental method, due to Eckert and Irvine, provides a direct determination of f by measurement of the subsonic temperature recovery factor. A recent theory of Mason and Monchick has been used to calculate collision numbers for rotational relaxation from the experimental data as follows: CH4, 9.4; CF4, 3.0; SF6, 2.5; C2H4, 2.4; C2H6, 4.0; O2, 12; N2, 7.3; CO2, 2.4; and C2H2, 1.8. Collision numbers for the near-spherical molecules were in close accord with a classical theory for rough sphere molecules with attractive forces; ethylene, which deviates appreciably from spherical symmetry, exhibited a smaller collision number. The data on linear molecules were in qualitative agreement with a quantum treatment. In general, collision numbers for rotational relaxation are determined by the following factors: (1) The molecular mass distribution, (2) the strength of the intermolecular attractive forces, and (3) the molecular asymmetry.

A Study of Surface Roughness and Its Effect on the Backscattering Cross Section of Spheres

The effect on the scattering cross section of a perfectly conducting object produced by surface roughaess whose scale is only a small fraction of a wavelength is discussed. The object itself is assumed large compared with the wavelength. The roughness is assumed to be statistically random in nature and it is shown that this can be treated by means of an impedance boundary condition. This permits the use of known results to determine the resulting modification to the cross section of the unperturbed object. Experimental data obtained by measuring the backscattering cross section of a large rough sphere at three frequencies, S, X and K band, are presented. It is found that even for a sphere whose depth of roughness is as large as 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> ,λ the measured change in cross section is no nore than about 0.1 db. This is in good agreement with the theoretical prediction.

LXVII.The motion of a particle on a rough sphere, including the case of a rotating sphere

LXVII.<i>The motion of a particle on a rough sphere, including the case of a rotating sphere</i>

IV. Impact with a liquid surface studied by the aid of instantaneous photography. Paper II

In a previous paper (‘Philosophical Transactions,’ A, 1897, vol. 189, p. 137) we have drawn attention to the fact that the disturbance set up in a liquid by the impact of a rough sphere falling into it, differs in a very remarkable manner from that which follows the entry of a smooth sphere. In the present paper we describe further experiments, made with the object of ascertaining the reason of this difference, and give the conclusions reached. It appeared desirable, in the first place, to take instantaneous photographs of the disturbed liquid below the water-line. These were easily obtained by letting the splash take place in an approximately parallel-sided thin glass vessel (an inverted clock-shade) illuminated from behind. The liquid surface when undisturbed was about level with the middle of the camera-lens, which was focussed for the sphere when under water. The general arrangement of the optical apparatus will be suffi­ciently understood from the accompanying cut (fig. 1). The method of timing the illumination was that already described ( loc. cit. ).