1913]
FINE SLITS IN THIN OPAQUE SCREENS
165
the direction of original propagation, equality of phase obtains, and (16) remains applicable even in the case of a wide slit. It only remains to determine Wm as a function of y, so that for all points upon the aperture
6
(17)
where, since kr is supposed moderate throughout, the second form in (13) may be employed.
Before proceeding further it may be well to exhibit the solution, as formerly given, for the case of a very narrow slit. Interpreting $ as the velocity-potential of aerial vibrations and having regard to the known solution for the flow of incompressible fluid through a slit in an infinite plane wall, we may infer that tym will be of the form A (b2 y2)"*, where A is some constant. Thus (17) becomes
In this equation the first part is obviously independent of the position of the point chosen, and if the form of tym has been rightly taken the second integral must also be independent of it. If its coordinate be 77, lying between + b,
_ " log Qy - y} dy b log_(y -ifidy
, . ...... ^ }
must be independent of 17. To this we shall presently return ; but merely to determine A in (18) it suffices to consider the particular case of 17 = 0. Here
Thus A (7 + log^ikb)TT = - 1, and I ^fmdy = vA ;
J b
j,..__Łą__(-Z-V
so that (16) becomes
v '
.(20)
From this, -\jrp is derived by simply prefixing a negative sign.
The realised solution is obtained from (20) by omitting the imaginary part after introduction of the suppressed factor eint. If the imaginary part of \og(^ikb) be neglected, the result is
7
corresponding to %w = 2 cos nt 003 kx ......................... (22)
Perhaps the most remarkable feature of the solution is the very limited dependence of the transmitted vibration on the width (26) of the aperture. slit be not very narrow, the partial waves arising at different parts of the width will arrive in various phases, of which due account must !>< takem The disturbance is no longer circularly symmetrical as in (16). But if, as is usual in observations with the microscope, we restrict ourselves to