Journal of Electromagnetic Analysis and Applications, 2011, 3, 199-207 doi:doi:10.4236/jemaa.2011.36033 Published Online June 2011 (http://www.SciRP.org/journal/jemaa) Copyright © 2011 SciRes. JEMAA 199 Lateral Waves near the Surface of Sea Osama M. Abo-Seida 1 , Samira T. Bishay 2 , Khaled M. El-Morabie 3 1 Department of Mathematics, Faculty of Science, Kafer El-Sheikh University, Kafer El-Sheikh, Egypt; 2 Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt; 3 Department of Mathematics, Faculty of Science, Tanta Univer- sity, Tanta, Egypt. Email: {aboseida, stbishay, km_morabie}yahoo.com Received February 25 th , 2011; revised April 14 th , 2011; accepted May 8 th , 2011. ABSTRACT In this research, we investigate the propagation of lateral electromagnetic wave near the surface of sea. Interference patterns generated by the superposition of the lateral and direct waves along the sea surface (flat and rough) are shown. The field generated by a vertical magnetic dipole embedded below the sea surface (having a flat and perturbed upper surface) is shown to consist of a lateral-wave and a reflected-wave. Closed-form expressions for the lateral waves near the surface of the sea are obtained and compared with those mentioned for the reflected waves numerically for the con- sidered model. Keywords: Stratified Media, Rough Surface, Radiation in Sea, Lateral Waves 1. Introduction Lateral electromagnetic waves generated by a vertical electric or magnetic dipole near the plane boundary be- tween two different media like air and earth or air and sea have been the subject of investigation for many years beginning with the work of Sommerfeld. King [1] de- rived simple formulas for the transient field generated by a vertical electric dipole on the boundary between two dielectric half-space when the permittivity of one of these is much greater than that of the other. The rough- ness of the upper surface of the sea is considered by Bishay [2,3] to indicate the effect of the rough surface on the electromagnetic fields. Recently, Abo-Seida et al. [4] calculated the far-field radiated from a vertical magnetic dipole in sea with a rough upper surface. Besides, in pre- vious studies [4], the Hankel transformations are esti- mated by using new technique developed by Long et al. [5] and Chew [6]. The present study is a further contribution to [4], so the Hankel transformations which were estimated by Abo-Seida et al. [4] are employed here. The previous studies [2,3] have obtained the formulas of the reflected waves in the region of the seawater, due to a vertical magnetic dipole in a three-layered conducting media by resolving the problem using the residue and saddle-point methods. However, these methods, involve lengthy algebra and several transformations, which are very tedious and complicated. The new technique utilized in [4] was used in this study in order to obtain closed-form expressions of the lateral waves. Firstly the form solutions of the far-field, due to a ver- tical magnetic dipole in a sea (three-layered conducting media) with variable interface are expanded as an infinite series. Then with the aid of the complex image theory [7], closed-form expression of the lateral waves near the sea surface due to the dipole are obtained. Besides, the physical meaning of the results is presented. 2. Geometrical Structure We shall adopt the following model as illustrated in Fig- ure 1. A small loop antenna, whose magnetic moment is 0 IS , is located in the middle layer (i.e. in the sea) at depth 1 , horizontally. An observing point is also located in the sea at depth 2 . We suppose that the thickness of the sea is a, and that air is infinite upward and the ground downward along the z-axis. The ground- sea interface is taken to be planar, while the air-sea in- terface varies slightly from its mean value, as shown in the figure. r is the distance between the source and the observing point P, and 1 is that between P and the im- age of the source in the air. d P d r The material constants are assumed to be as follows. The dielectric constant in the air, sea, and ground are 0 2 ,   and 3  , respectively. The magnetic permeability is taken equal to that of the free space in every layer. The