87 



How let us specify that Six) be different from zero only 

 In a very narrow range about x ; 0, and that / S(.x)dx. z !• 

 As we make ^(x) narrower and narrower the last two terms on 

 the left of (11) will eventually vanish, while the rest of 

 the equation approaches 



^ -^ t«0 -^ oo 



rsr, 



Intecratlng (12) on r^ gives now, for t s 



t)dt (12) 



oo 



~^^\^^- dr, = caa-(£)-^ J /f (r^ -ct,t) dtdr, (13) 

 a ^-^^ 



The double integral on the right represents the correction 

 to (6) necessitated by the fact that the amplitude of motion 

 is large. It can be approximately evaluated by inserting 

 for F a function obtained from the non-compresslve approxi- 

 mation. Je have from (S) and (9) 



where 



Nov/ 



P z rV^$ 5 ?''(rf ). (14) 



J- B r^ 



df r-7„ „ 1 



_ —* 



-^ V. V V 



so we have f = .2v ^ - ^^ J_7_ ^5) 



