2k 



For x < 0, complete the contour C wi,-.h a large seai-circle in 

 R . There is no contribution from the large semi-circle, and applica- 

 tion of Cauchy's theorem now gives 



y(x < 0) - ^r (+ 2tt1) (Residue of Y (s) at s - + k ) exp(- ik :) 



(2ik ) 



- + i 



■ R exp(- ik x) 

 o 



with a reflection coefficient 



(+ k - k,) exp(- ik x) 



o i o 



\ k o + V 



(4.2) 



It is a trivial matter to check that these solutions for the reflscr.ed 

 and transmitted waves satisfy the conditions (2.23) and (2.24). 



In the kirds of problems encountered in acoustics, the funcr.ion 

 I (b) usually has a branch point singularity in R as well as simple 

 poles, and correspondingly Y (s) may have one or more branch pcJjits 

 and simple poles in R . The pole contributions are unaffected (ex.-.ept 

 in certain critical circumstances) and give rise to n atural modes of 

 the syste a, analogous to the reflected and transmitted waves found here. 

 When the dissipation factors are small, some poles will lie close to 

 the real axis and give rise to propagating modes , as here. Others 

 may lie close to the imaginary axis and give non-propagating modes, of 

 the kind found in yave-guide problems and in the motion of plates and 

 beatas. In ether problems poles may be present in the complex plane 



21 



swtes W MMMJ m 



