of Sound hy Spheroids and Disks. 369 



Expansion of an Incident Wave. 



A plane wave travelling parallel to the axis (z negative) 

 of a spheroid may be represented by a velocity potential 



£=«**=**, (20) 



where 6 = kc cosh a. 



Since 

 and 



Therefore C l .„ TT , s , ~ TT / d \ sin 





" 

 Thus if 0= S™ A„H», 



iA„£H^,=H„(^)^. • • (21) 



The function on the right will be called ic n {ci). It must 

 obviously be a typical associated function satisfying the 

 differential equation (3). A knowledge of these functions 

 at once determines the coefficients A of (21). 



Writing D= ,^, and expanding sin 6 as a series, 



, N ( A arB 2 *> 4 D 2 co 4 D 4 r \ sin 

 Wo («)«(l + ^— ___ + ___ + . ..j__ 



, 2 or/ 1 12(9 2 \co 4 « 4 



= l-^l(l + 3cosh 2 a) + ^cosh 4 a + 2cosh 2 a+^+... 



+ 2cosh 2 «-|>)...} 

 ^ 2 (a) = — (1 — 3 cosh 2 a) 



^ 3 (a)= — ^(cosh 3 a— f cosha), ....... (22) 



these being the values when « is not great, that is, in the 

 neighbourhood of the origin, at which the obstacle is placed. 



