92 Lord Rayleigh on the Propagation of 



wave (A). There is no reflexion if 



| Ye- 2ikx dsc = ; (14) 



and then the transmitted wave (a?>b) is given by 



v w-,\ 1 -m Y *'}- ■ ■ ■ (15) 



Even when there is reflexion, it is at most of the second 

 order of smallness, since Y is of that order. For the 

 transmitted wave our equations give (a? > b) 



w N Ae~ ik * j n 1 C\ T , 



F(f)a — V"2Sj. 



£ Jo Jo 1 f . 



41 >+4J> }1 ' ( ' 



but if we stop at the second order of smallness the last part 

 is to be omitted, and (16) reduces to (12). It appears that 

 to this order of approximation the intensity of the trans- 

 mitted sound is equal to that of the incident sound, at least 

 if the tube recovers its original diameter. If the final value 

 of y differs from the initial value, the intensity is changed 

 so as to secure an equal propagation of energy. 



The effect of Y in (15) is upon the phase of the trans- 

 mitted wave. It appears, rather unexpectedly, that there is 

 a linear acceleration amounting to 



w I, 



b 



Ydoc, (17) 



or, since the ends of the disturbed region at and b are 

 cylindrical, 



ir?©' (i -**v)*. • • • <«> 



from which the term in k 2 y 2 may be dropped. 



That the reflected wave should be very small when the 

 changes are sufficiently gradual is what might have been 

 expected. We may take (13) in the form 



