Sound in narrow Tubes of variable section. 91 



Our equation thus becomes 



(£«)<m-3HfSS 



= (l + py)g'.F(V> 3 • • O) 



in which on the right the first approximation (7) suffices. 

 Thus 



yF(x) = ^-[e ik 4Y(B + Ae~ 2ik *)dx 



-e- ik *{Y(A + Be 2ik *)dx\, . . (10) 

 where 



Y= i+MYA (11) 



y ax 2, v 



In (10) the lower limit of the integrals is undetermined ; 

 if we introduce arbitrary constants, we may take the inte- 

 gration from -co to x. 



In order to attack a more definite problem, let us suppose 

 that d 2 y/dx 2 , and therefore Y, vanishes everywhere except 

 over the finite range from x = to x = b, b being positive. 

 When x is negative the integrals disappear, only the 

 arbitrary constants remaining ; and when x is positive 

 the integrals may be taken from to x. As regards the 

 values of the constants of integration, (10) may be supposed 

 to identify itself with (7) on the negative side. Thus 



y¥(x) = e ~ ikx ( A -^ ( *Y(A + B^)d*| 



+ 6 ^|b + ~ p Y(B 4- Ke~ 2ikx )dx\ . (12) 



The integrals disappear when x is negative, and when 

 x exceeds b they assume constant values. 



Let us now further suppose that when x exceeds b there 

 is no negative wave, i. e. no wave travelling in the negative 

 direction. The negative wave on the negative side may 

 then be regarded as the reflexion of the there travelling- 

 positive wave. The condition is 



B { i+ ^So Ydx } + £S Ye - 2ikidx=o > 



(13) 



giving the reflected wave (B) in terms of the incident 



H2 



