176 ON THE PASSAGE OP WAVES THROUGH
When A and B are found, we have in (16)
[37,
f+6
/_•
From (65) we get
IT A = -2
— qr
ps— qr
__ q + r — s —p ps — qr
.(66) .(67)
= -1-0007+ T4447*, = -0-2217 + 11198 i,
so that
Thus for kb = 1 we have ^ = -0-65528+1
r =-0-63141+ T0798 • whence
irA = + 0-60008 + 0-51828 i, 2bB = - 0-2652 + 0-1073 i,
and (67) - + 0'3349 + 0'6256 i.
The above values of irA and 2&JB are derived according to (17) from th values at the centre and edges of the aperture. The success of the metho< may be judged by substitution of the values for ^2/i2 = \. Using these i; (17) we get — 0'9801 — 0'0082 i, for what should be — 1, a very fair approxi mation.
In like manner, for kb — 2
(67) =+ 0-259 + 1'2415»; and for kb = $ (67) = + 0'3378 + 0-3526 i.
As appears from (16), when k is given, the modulus of (67) may b taken to represent the amplitude of disturbance at a distant point imme diately in front, and it is this wibh which we are mainly concerned. Th following table gives the values of Mod. and Mod.2 for several values of fa The first three have been calculated from the simple formula, see (20).
TABLE IV.
kb Mod.2 Mod.
o-oi 0-0174 0-1320
0-05 0-0590 0-2429
0-25 0-1372 0-3704
0-50 0-2384 0-4883
1-00 0-5035 0-7096
2-00 1-608 1-268
The results are applicable to the problem of aerial waves, or shallow watt waves, transmitted through a slit in a thin fixed wall, and to electriand the second s, the equations of •condition from (17) are