A: Derivation of Geometric Mean Beam Length Relations


The Geometric Mean Beam Lengths depend on both geometry and wavelength through the definitions






The double integral in (A-1) must be evaluated for various orientations of surfaces Aj and Ak; the result will depend on kλ.  Derivations for some specific geometries are now considered.


A.1 Hemisphere to Differential Area at Center of its Base

Let Aj be the surface of a hemisphere of radius R, and dAk be a differential area at the center of the base (Fig. A-1). Then (A-1) becomes, since S=R and = 0,



Hemisphere filled with isothermal medium.


The convenient dAj is a ring element dAj = 2πR2 sin k d k, and the factors involving R can be taken out of the integral. This gives



With  and , this reduces to




This especially simple relation is used later in the concept of mean beam length where radiation from an actual volume of a medium is replaced by that from an equivalent hemisphere.