372 Dr. J. W. Nicholson on the Scattering 



Comparing this relation with that in (24) between v n and 

 te^, it appears that for a large argument 



»»(") 



; ^g\/f- J — tW> 



when a, and therefore 0, is large. 

 Accordingly, under this condition, 



«U«) = r~" cos (*-?> • ••■ • (29) 



corresponds to the functions r? w near the origin, with the 

 same limits. 



The portion of the velocity potential representing the dis- 

 turbance due to the presence of the obstacle must, at a great 



e~ lkr . e~ % ® 



distance, be proportional to , or, ultimately, to -3— . 



v ' a 



Thus the appropriate function of a for such a diverging 

 w T ave, associated with the harmonic defined by ??, is 



v n U)-uo(-yw H (a), ..... (30) 



which, at a great distance, has the value — e~ l ®, 



or secha.e-^ ccosha (31) 



Expansion of the Divergent Wave. 

 The divergent wave, corresponding to the incident wave 

 <j) = e lkz , 

 whose expansion was given in (21, 23), is of the form 



^=2 * «>„-^(-)^)H». • • (32) 

 At the spheroidal obstacle defined by a = f, if perfectly rigid, 



since there can be no normal velocity at the surface. 



Tnus 3 / / \n \ dA w /00 . 



a »5|- < f >n-* w (-)X)=---^jr, • • • ( 33 ) 



where the values of A re appear in (23). 



The values of a n may now be calculated. L will denote 



logcoth- . 



