1917] FROM A REGULARLY STRATIFIED MEDIUM 499
If /91 = 0, sinh j3 = i sin ftz, which can occur only when sin2 p < cos2 #, or, which is the same, sin2 6 < cos2 p. Again, if sin /32 = 0, sinh2 /3 = sinh2 0lt occurring when sin2 6 > cos2 p.
It thus appears that, of the four cases at first apparently possible, «i = & = 0, sin aa = sin /92 = 0, are excluded. There are two remaining alternatives:
(i) sinh2 a = —; sin2 0 > cos2 p; «j — 0, sin /32 = 0; (ii) sinh2 a = +; sin2 Q < cos2 p ; /9a = 0, sin a2 = 0.
Between these there is an important distinction in respect of what happens when m is increased. For
<j>m = sinh ra/3/sinh (a + tnj3). In case (i) this becomes
l/0m = cos a2 + i coth m/3i sin a2, ..................(48)
and 1/| 0m|2 = 1 + sin2a2/sinh2w/31................(48 it's)
If/3j be finite, sinh2m/31 tends to oo w,hen m increases, so that | <^>m|2 tends to unity, that is, the reflection tends to become complete. We see also that, whatever m may be, <j)m cannot vanish, unless & = 0, when also r = 0.
In case (ii)
+ l/<£m = cosh a.1 — i cot mft., sinh OLI, ...............(49)
and 1/j <j>m 2 = 1 + sinh2 ofi/sin2 m/32, ...............(49 bis)
so that <^>m continues to fluctuate, however great wi may be. Here ^>m may vanish, since there is nothing to forbid m/3a = STT. Of this behaviour we have already seen an example, where cos2 p = 1.
In order to discriminate the two cases more clearly, we may calculate the value of sinh2 a from (43), writing temporarily for brevity
ei^ = A, ^«'=A'.........................(50)
Thus by (9) and (10)
__ ^(A2-l) (1-7?2)A
sothat r±tss^^.ft or -^^7; ...............(52)
whence
. ,, {(A-77)2A'2-(l-7;A)2}'{(A + 77)2A/2-(l
sinh2 a = —------~---------—-.— ' *• —^~--------^~-
4?;2A 2(A- — I)2
The two factors in the numerator of the fraction differ only by the sign of r), so that the fraction itself is an even function of ?j. The first factor may be written
= - (1 + A A' - 7i (A + A')} (1 - AA' + T? (A' - A)};
32—2..(47)