Although in recent years, the surface impedance boundary condition (SIBC) concept has taken increasing importance both at low and at high frequencies, its origins are rather modest and are deeply rooted in the concept of skin depth and skin effect. Schelkunoff [1] is credited with coining the term and introducing the concept in the early 1930s, by analogy with circuit theory, as the ratio of the electric and magnetic field intensities. With the rapid advance of the theory of radio waves and propagation, the concept has been further developed for the analysis of wave propagation over the earth’s surface. The need for solutions for propagation of waves generated by antennas in the presence of layered, curved surfaces has led to what we refer here as the classical SIBCs. The troubled times before and during World War II seem to have been a catalyst in this development, as the need for these important solutions became more acute. The first systematic attempt to treat SIBCs was undertaken by Rytov [2,3] in 1939. His perturbation method approach allowed calculation of fields inside and outside conductors based on power series expansions, and encompassed the Leontovich condition as the first-order term in the expansion. Inclusion of higher-order terms allows treatment of curved boundaries as well as variations of the field on the surface of conductors. The method is not as well known as others, primarily because Rytov’s work was published in Russian and, with the exception of one paper [4] which was translated and published in French [3], none of his work has ever been translated. The work of Leontovich and the SIBC bearing his name are much better known and have found wide acceptance in the past as well as in contemporary work. The first-order SIBC developed by Leontovich [5,6] was published in 1940 but was in existence at least since 1938 [2]. Later, a modified form, which applies to curved surfaces, has been developed [6] and has been discussed (and corrected) by Mitzner [7]. Mitzner developed a second-order SIBC using an integral equations approach solving for the scattered field due to a general scatterer. Although Mitzner’s SIBC is equivalent to Rytov’s second-order SIBC and can in fact be developed based on the expansion method of Rytov, Mitzner’s contribution includes discussion of errors for his own and

Leontovich’s conditions as well as limits of applicability and modifications for large skin depths. The SIBCs due to Leontovich, Mitzner, and Rytov, together with the limi-

ting case of the perfect electric conductor (PEC), form what we call the classical SIBCs and are discussed in this chapter. The purpose is to show how these SIBCs were developed, discuss the limits on their applicability and in so doing to pave the way for additional, new developments to be taken up in subsequent chapters. We start with a short discussion of the skin effect and the skin effect approximation, followed by the Leontovich SIBC. The Mitzner SIBC is introduced as an example of higher-order SIBCs and also to point out to the method by which he arrives at the SIBC. The general approach of Rytov is discussed and used to analyze the conditions of applicability for low-and high-order SIBCs. In the following chapter, we will also show how Rytov’s approach can be used to obtain the Leontovich and Mitzner’s SIBCs as special cases of the more general higher-order SIBCs.