On the Bending of Electromagnetic Micro-Waves below the Horizon

An interesting phase in the development of the modern radio technique are the experiments conducted during the last few years with very short wave-lengths. Marchese Marconi [1] reported about an extensive series of successful radio connections over distances up to 260 km., in which waves of from 50 cm. to 60 cm. were used, concentrated with the help of a parabolic reflector. Clavier and Gallant [2] went even to still shorter waves of only 17.4 cm. which they sent over a distance of 61 km. also concentrating them with a reflector of 3.8 m. in diameter. The most remarkable feature of Marchese Marconi's results is that the distances covered by him exceed several times the range of rectilinear visibility from the sending station. The memory is still fresh of the great surprise which was caused among physicists by the unusually long range of long wave radio-reception. The explanation of these puzzling facts about long waves was traced, in the meantime [3], to the influence of the Kennelly-Heaviside layer of the upper atmosphere, and the question, naturally, arises to what extent atmospheric influences are responsible for the phenomena observed by Marchese Marconi with micro-waves. The first step in answering this question must be an investigation of how much bending is to be expected from the point of view of the wave theory completely neglecting the atmosphere. Such an investigation is the subject of this paper. The simple method which we propose is based on Huyghens' principle and treats the surface of the earth as a perfectly absorbing screen. As far as we know, it was not used heretofore and there are good reasons for this: In the case of long waves, the properties of the soil play an important part both in their production and their propagation. The height of the receiving station is only a fraction of the wave-length, so that only the so-called "surface wave" is of practical interest. On the other hand, the micro-waves are produced away from the soil and independently from it and, after they strike the earth, the surface wave is so thin as to be entirely unimportant. The transmission is, in this case, a matter of space propagation on which the physical properties of the earth surface have no material influence. It is therefore, perfectly permissible to replace it by a perfectly absorbing screen. The results of our calculations and their comparison with Marchese Marconi's observations are summarized in the last section

Aspects of the Hypermultiplet Moduli Space in String Duality

A type IIA string (or F-theory) compactified on a Calabi-Yau threefold is believed to be dual to a heterotic string on a K3 surface times a 2-torus (or on a K3 surface). We consider how the resulting moduli space of hypermultiplets is identified between these two pictures in the case of the E8xE8 heterotic string. As examples we discuss SU(2)-bundles and G2-bundles on the K3 surface and the case of point-like instantons. We are lead to a rather beautiful identification between the integral cohomology of the Calabi-Yau threefold and some integral structures on the heterotic side somewhat reminiscent of mirror symmetry. We discuss the consequences for probing nonperturbative effects in the both the type IIA string and the heterotic string.Comment: 27 Pages, LaTeX 2e, 2 separate figure