1^4 ON THE PASSAGE OF WAVES THROUGH
In the original paper these results were applied to an aperture, especially of elliptical form, whose dimensions are small in comparison with X, For our present purpose we may pass this over and proceed at once to consider the case where the aperture is an infinitely long slit with parallel edges, whose width is small, or at the most comparable with A,
The velocity-potential of a point-source, viz., er^jr, is now to be replaeod by that of a linear source, and this, in general, is much more complicated. If we denote it by D(kr), r being the distance from the line of the point where the potential is required, the expressions are*
/ \i ( I2 I2 32
D (kr) = - (j^J e~ikr |1 - jTg^; + f
. 1KT\ j - 1C V 1C T
' + iog-2J|l-^- + 2r^-
~ST $t ~ 02 At $2 +
23.4r.62 S '"'............
where 7 is Euler's constant (O577215), and
+ l/m......................(14)
Of these the first is "semi-convergent" and is applicable when kr is the second is fully convergent and gives the form of the function whon kr is moderate. The function D may be regarded as being derived front e~lkr/r by integration over an infinitely long and infinitely narrow strip of the surface &
As the present problem is only a particular case, equations (()) and ( JO) remain valid, while (9) may be written in the form
fm = /^mD (kr)dy, ^p = -jymD(kr)dy, .........(1/3)
the integrations extending over the width of the slit from y = - 6 t,o y = + 6. It remains to determine ¥m) so that on the aperture ^wt « — 1, ^ = + 1-
At a suflficient distance from the slit, supposed for the moment to bo very narrow, D (kr) may be removed from under the integral sign and also In-replaced by its limiting form given in (13). Thus
C'«)
If the 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
See Theory of Sound, § 341.case (i).