The Geometric Mean Beam Lengths depend
on both geometry and wavelength through the definitions
(A-1)
(A-2) The double integral in (A-1) must be
evaluated for various orientations of surfaces A
the result will depend on _{k};k. Derivations
for some specific geometries are now considered._{λ}
Let R, and dA be a differential
area at the center of the base (Fig. A-1). Then (A-1) becomes, since _{k}S=R
and
= 0,
The convenient dA = 2π_{j}R^{2} sin
, and the factors involving _{k }d
_{k}R can be
taken out of the integral. This gives
With
and
, this reduces to
(A-3) 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. |