1912] THROUGH A STRATIFIED MEDIUM, ETC. 73
the same, and ^ sin2 61 = c-2 sin2 82. The general formula (9) then identifies itself with Fresnel's expression
tan (#! #2)
On the other hand, if cr2 = o-j , the change being one of compressibility only, we find
sin (*«'- ft)
Fresnel's other expression.
In the above it is supposed that a2 (and $2) are real. If the wave be incident in the more refractive medium and the angle of incidence be too great, a2 becomes imaginary, say ia2'. In this case, of course, the reflection is total, the modulus of (9) becoming unity. The change of phase incurred is given by (9). In accordance with what has been said these results are at once available for the corresponding optical problems.
If there are more than two media, the boundary conditions at x = oc2 are
a2 {B2 eia* (x>~*i] ~ A«e-ia* ^x>] } = as(Bs-AB), ............ (12)
o-a [Bzeia^-^] + Aze~ia* <*-"'> } = a-3 (Bs + A,), ............ (13)
and so on. For extended calculations it is desirable to write these equations in an abbreviated shape. We set
Ba-Aa=*Ha, Bz + Az = Kv, etc., ............... (14)
i sin a2 (sc2 ac^) = s^ etc., ......... (15)
cr3/cr2 = /32) etc.; .................. (16)
and the series of equations then takes the form
JT^aA, -ffi-A*, ............. (17)
(18)
(19)
and so on. In the reflection problem the special condition is the numerical equality of H and K of highest suffix. We may make
H=-l, K = + l ......................... (20)
As we have to work backwards from the terms of highest suffix, it is convenient to solve algebraically each pair of simple equations. In this way, remembering that ca s- = 1, we get
#!= a.H,, K,= 0&, ......... (21)
......... (22)
......... (23)case of the juxtaposition of two media both of infinite depth. Supposing B2 = 0, we get