ON THE PROPAGATION OF WAVES
and so on. In these equations the c's and the fi's are real, and also 1 a's, unless there is " total reflection " ; the s's are pure imaginaries, with 1 same reservation.
When there are three media, we are to suppose in the problem of reflect] that J2"3 = - 1, Ka = 1. Thus from (21), (22),
so that
A,
'
If there be no " total reflection," the relative intensity of the reflecl waves is
.(25;
where d2 = cos2 az (ocz 0$, s-f = sin2 a2 (scz ^)..........(26;
The reflection will vanish independently of the values of c3 and s1} ' whatever may be the thickness of the middle layer, provided
since these quantities are all positive. Reference to (9) shows that th are the conditions of vanishing reflection at the two surfaces of transit considered separately.
If these conditions be not satisfied, the evanescence of (25) requires t' either CT or sl be zero. The latter case is realized if the intermediate la be abolished, and the remaining condition is equivalent to 0-3/0^ = as/a,: was to be expected from (9). We learn now that, if there would be reflection in the absence of an intermediate layer, its introduction will hi no effect provided a2(#2 Ğa) be a multiple of TT. An obvious example when the first and third media are similar, as in the usual theory "thin plates."
On the other hand, if d, or cos az (xz ac-^), vanish, the remaining requi ment for the evanescence of (25) is that /32/a2 = A/fti-
In this case ~------ = ~------;
so that by (9) the reflections at the two faces are equal in all respects. In general, if the third and first media are similar, (25) reduces to
sn2
4 cos2 a2 (x.2 flTj) 4-
which may readily be identified with the expression usually given in tei of (9).
It remains to consider the cases of so-called total reflection. If 1 occurs only at the second surface of transition, a1} C62 are real, while as iB2 = 0, we get