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Full text of "Scientific Papers - Vi"

/^ i *
1919]                     .           FROM A PERFORATED   WALL                                          ° >'>
making 5 = 0 when, for example, a> = 0, cos 0 = 1.     The reflexion inuy also vanish when the obliquity of incidence is such as to compensate-, (or a limt.o, cr . In examining the formula for the general case we almll writo for brevity
cos 6 (a- + cr'}jo- = S, ......................... :-(-1)
and drop I, so that k1} Jc2, k stand respectively for ktl, /M, kl.    r^» m;lkl'H 11() difference to the first of equations (9), while the second b
rp,                                 „                  — ik' sin k'                                        ( .>.>\
Thus                            B=r7< - r, — r^~r — T'~ ......................... l~"";
kS cos & + ^k sin /c
Separating real and imaginary parts, we find for the immtM'ni.or ol II in (L-cos /Cj cos i'7c2   7c$ — - — -- -•— — 7c2 tan. 7<H
L                *
/-'a t-au •//•,)
+ i \ kS tan \ —.—2 — 7«?i tan 7ca -h
The denominator of (22) is obtained (with altoriul wi^n) by writing for S in (23).
In what follows we are concerned with the modal UN of II    Leaving oii!< factgrs common to the numerator and denominator, we* may Lake
Mod2 Numerator = \kS-----—:—2 — lcn tan ,
,   f/70tani7ca     , \ ,       ,     , fa* I.a.!iv7r,]y         ,.
+ •{ [JcS — . — — MI ) tan /?i -1-         .       - .  ...(2- !•)
(\         i            /                       i       1         •
The evanescence of B requires that of both the NqiwroH in (24), or that
7 cv    "'i tan i/Cn     ,         .       ,.       .   -i        -i       ,»                        i \ * >.
kb = - r ----- h /c2 tan 7<?: = t7^ cot t/^ — A?g cob ^ ,   ,.,,,,.. ,(2.'>)
"i
or again with elimination of 8,
ifa (tan ilcz 4- cot ik2) = Jcz (tan /^ -f- cot ^,), whence                       •    &j sin 2^ + 1'&2 sin 2t'7ca == 0,      ........... . ......... (20)
or in the notation of the hyperbolic sine
fa sin 2kt = 7<?2 sinh 2/<?a ......... , ..... . ......... (27)
If this equation, independent of a, <r', and cos 0, can bcs HiU.iHfu'd, it allowH us to find ki from an assumed k2, or conversely, and thoneo k by motuiH of (0).
The next step is to calculate S by means of one of e.quatioriH (25). If $, BO found, > cos 6, we may choose o-'/o- so that .B shall vanish ; but if 8< COB B, no ratio o-'/a- will serve to annul the reflexion. If the incidence bo perpendicular, 8 must exceed unity. . If S were negative, the reflexion would bo finifce, whatever may be the angle of incidence and the ratio cr'l
a-.rds, the linear period at any place is proportional, cceteris paribus, to the distance from the original boundary. In this argument the thickness of the film— another linear quantity—is omitted, as is probably for the most part legitimate. In imagination we may suppose the film to be infinitely thin or, if it be of finite thickness, that the diffusion takes place strictly in one dimension.