L. B Felsen. # Relation between a class of two-dimensional and three-dimensional diffraction problems online

. **(page 2 of 3)**

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function t = V-e + V-E - V- 6. , with e given in (l9)-(2l), is a scalar wave

â€” â€” â€” mc â€”

function which is outgoing at infinity. This follows from the fact that the

Cartesian components of the vector (Â£ + E - ^. ), and therefore their deriva-

tives as well, possess these properties. Furthermore, we know that i|f vanishes

on the screen. [in fact VÂ»e = on the screen by (15)^ V-E vanishes there be-

cause of (^), while V. â‚¬. =0 at the screen since the sources do not extend

' â€” mc

to the screen.^ These properties of f ensure that it will vanish identically

provided that it vanishes as the edge is approached radially. We shall there-

fore examine the behavior of ^ near the edge; as we shall see, the divergence

\ri.ll vanish there provided the functions F^ and Fp are suitably chosen.

First, as mentioned above, V'Â£. =0 near the edge. Next we ob-

' ' â€” mc

serve from (12), that the singular part of the expansion of V-E is given by

the form\ila (note: Ty^WP sin /2) = - -SyWP cos

â€” â€” â€” mc â€”

function which is outgoing at infinity. This follows from the fact that the

Cartesian components of the vector (Â£ + E - ^. ), and therefore their deriva-

tives as well, possess these properties. Furthermore, we know that i|f vanishes

on the screen. [in fact VÂ»e = on the screen by (15)^ V-E vanishes there be-

cause of (^), while V. â‚¬. =0 at the screen since the sources do not extend

' â€” mc

to the screen.^ These properties of f ensure that it will vanish identically

provided that it vanishes as the edge is approached radially. We shall there-

fore examine the behavior of ^ near the edge; as we shall see, the divergence

\ri.ll vanish there provided the functions F^ and Fp are suitably chosen.

First, as mentioned above, V'Â£. =0 near the edge. Next we ob-

' ' â€” mc

serve from (12), that the singular part of the expansion of V-E is given by

the form\ila (note: Ty^WP sin /2) = - -SyWP cos

Online Library → L. B Felsen → Relation between a class of two-dimensional and three-dimensional diffraction problems → online text (page 2 of 3)