160 Passage of Sound through Narrow Slits. 



It will be seen that at any rate the sound is much less 

 sensitive to an alteration of width than to one of length. 

 If we apply the formula (6) applicable to an ellipse, identi- 

 fying the length of the slit with 2a and its width with 2b y 

 we get using logarithms to base 10 



(i.) M' = 



(ii.) M' = 



•25 



log 10 (4x"25-r--001) 

 •14 



= •083, 



•068 



j 10 (4x-14^--005) 



so that the formula in question gives results not very wide of 

 the mark. Other observations also were in fair accordance. 



Appendix. 



Mean potential over the circumference of a circle whose 

 plane is parallel to the direction of propagation of plane waves. 



Fur. 3. 



P is any point on the circle. OP = r, OM = #. Potential 

 at P = e ikx = e ikr cos 6 . Hence for the mean we have 



if 



e ikrco * e d6 = J Q (kr), 



J (kr) vanishes when Jcr = 27rr/\- 



2-404, or 



'2r=-77\. 



If the plane of the circle be inclined at an angle a to the 

 direction of propagation, J (kr) is replaced by J (kr cos a). 

 If oL = \ir, so that the plane is parallel to the waves, we find 

 unity for the mean value, as was to be expected. 



When the potential to be averaged varies in two dimensions 

 only, even though the waves may not be plane, we may pro- 

 ceed by the method of 'Theory of Sound/ § 339. Using 



