SIMPLE -HARMONIC WAVES. DIFFRACTION 227 



Sr = cos &c, where 6 denotes the inclination of OP to Ox. 

 Hence d/dx = cos 6 9/9r, and 



(21) 



Performing the differentiation, we find 



(22) 



For small values of kr, i.e. within distances from which are 

 small compared with X/2w, this becomes 



...................... (23) 



On the other hand, for large values of kr, 



,,-ikr 



tfc- - cos0, (24) 



r 



so that along any one radius vector the condensation (s = 

 varies ultimately as 1/r. The radial and transverse components 

 of the velocity are to be found by the formula (6) of 69 ; 

 viz. they are 9$/9r and 9</>/r90, respectively. It appears 

 that near the origin these are of the same order of magnitude, 

 whilst at a great distance the lateral velocity is less than the 

 radial in the ratio 1/Ar. 



Introducing the factor Ce int in (24), and taking the real 

 part, we find that the velocity-potential due to a double source, 

 of strength G cos nt y at a great distance, is 



bC! / r\ 



-- -sin n (*-- cos ............. (25) 



4?rr \ cj 



The waves sent out in any direction are therefore ultimately 

 plane, of the type (7), provided A=kCcos6/4<7rr, the mere 

 difference of phase being disregarded ; and the flux of energy 

 (across unit area) will therefore be pk*cC 2 cos* ^/32?r 2 r 2 . Multi- 

 plying by 27rrsin 6. rSO, which is the area of a zone of a 

 spherical surface of radius r bounded by the circles whose 

 angular radii are and 6 -f 0, and integrating from 6 to 

 6 = TT, we find that the total emission of energy by the double 

 source C cos nt is 



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152 



