On the excitation of the waves of proper solutions

Abstract

An exact and explicit solution is obtained for an electro-magnetic field excited by a vertical electric dipole located over a flat plane of arbitrary surface impedance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</tex> . Though there are four asymptotic expansions, the expansion by Hadamard's method seems to have the most explicit physical meaning. According to this expansion only the proper solution of the ordinary surface wave can be present explicitly on the established condition of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arg(Z)&gt;\pi/4</tex> . The variation of the field on a spherical surface due to the change of surface impedance is investigated by the use of the ordinary B. Van der Pol and H. Bremmer formula. In this case, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\arg(Z) &gt;\pi/3</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z|</tex> is larger than some definite value <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z_{0}|</tex> , one of the terms of the formula is found to have the asymptotic form similar to the surface wave term in a flat surface case and, when the surface is highly inductive, it becomes the leading term at large distance from the dipole. The height-gain factor of this term first decreases rapidly with the height up to some point and then gradually increases. The growing up at large height is due to the radiation field. On the other hand, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|Z|\ll|Z_{0}|</tex> , the leading term can scarcely have the correspondence with that in a flat surface case. The propagation of surface wave over a spherical surface across several boundaries of discontinuity of the surface impedance is considered. The leading term is found to take the simple form as in the ordinary transmission line. The results in the case of flat plane are derived from these results as the asymptotic forms in the limit of infinite radius of curvature of spherical surface. The physical meaning of the proper solution such as ordinary surface wave is discussed. The physical condition is not the support condition but rather the condition on which, in some domain of space, the proper solution becomes the leading term of wave by an excitation whose dimension is of the order of magnitude of the wavelength or smaller. As in the case of spherical surface, the radiation field and the surface wave field are generally inseparable. The physical wave will be the leading term of the excited wave itself which may sometimes be a proper solution, such as ordinary surface wave, and sometimes be a mixture with the radiation field. The set of proper solutions is not unique. The Green function of the electromagnetic field constitutes the infinite sets of proper solutions of a continuous spectrum, by each set of which the observed field at a given point can be represented, and in each set of which some members of solutions have the properties of surface wave if there is a highly inductive surface. The field strength excited by a dipole is investigated as a function of the surface impedance in the case where the surface is a fiat plane (Section I) or a spherical surface (Section II). The results in the latter case are then applied to the surface wave propagating across several boundaries of discontinuity of surface impedance (Section III). The physical meaning of the waves of proper solutions, such as the surface wave, is finally discussed in Section IV.