Effective boundary conditions for reflection and transmission by an absorbing shell of arbitrary shape

The reflection and transmission properties of an absorbing shell of arbitrary shape can be described by two algebraic boundary conditions relating the four tangential fields <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E₊, H₊, E₋, H₋</tex> at corresponding points on the outer surface <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S₊</tex> and inner surface <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S₋</tex> , respectively. The boundary conditions contain first-order curvature correction terms, but it is still necessary that the shell curvature be reasonably small compared to the attenuation per unit distance in the shell material. Especially simple conditions are obtained for shells thin compared to the attenuation length. The shell boundary conditions can also be used to determine an effective surface impedance for a conducting body coated by multiple layers of absorbing material.