Sound in narrow Tubes of variable section. 93 



As an example let ns suppose that from x = to x = b 



y = y + rj(l—coama:), .... (20) 



where y is the constant value of y outside the region of 

 disturbance, and m = 2ir/b. If we suppose farther that 

 7} is small, we may remove 1/y from under the sign of 

 integration, so that 



im 2 v P _ 2ikx 7 



-zn — 1 cos mxe Zlhx dx 



B 

 A 



m 2 7) 



{l — cos2kb + is'm2kb}. 



(21) 



Independently of the last factor (which may vanish in 

 certain cases) B is very small in virtue of the factors m 2 /k 2 

 and v /y . 



In the second problem proposed we consider the passage 

 of waves proceeding in the positive direction through a 

 tube (not necessarily of revolution) of uniform section a^ 

 and impinging on a region of irregularity, whose length is 

 small compared with the wave-length (X). Beyond this 

 region the tube again becomes regular of section cr 2 (fig. 1). 



Fiz. l. 



V 



v 



It is convenient to imagine the axes of the initial and final 

 portions to be coincident, but our principal results will 

 remain valid even when the irregularity includes a bend. 

 We seek to determine the transmitted and reflected waves as 

 proportional to the given incident wave. 



The velocity-potentials of the incident and reflected waves 

 on the left of the irregularity and of the transmitted wave 

 on the' right are represented respectively by 



</>! = Ae~ ikc -r- Be ikx , </> 2 = Ce~ ikx ; . . 



so that at x 1 and x 2 we have 

 d(f)i/dx = ih{ — Ae 



cj) 2 = Ce" 



ikxr 



ikr \ + Be ,tc ')> dfc/d* = —ikC <r" 



(22) 



(23) 

 '■ (24) 



