WlHffiyiTIITW^lltljIWWrtW^ffllWWMil'^WiWHI^iiiipiiiwin imm ,~.~~- - - 



whole phenomenon depends oc strong coupling between the ends of the 

 system. 



We shall try here to upe our one-dimensional examples to illustrate 

 the possibility of generalizing the W-H technique to deal with three- 

 part problems. For such one-dimensional problems, however, there is 

 not generally any "weak interaction" approximation that one can make, 

 because waves on strings do not decay in the way that two and three 

 dimensional acoustic fields do, so that iu a sense one is always con- 

 fronted with the strong coupling situation. Nonetheless, a number 

 of interesting points arise in the string problems which have direct 

 comparisons in more serious three-part problems. 



Consider a uniform 3tring of line density p in - <*> < x < 



o 



and in t < x < + », thsse two semi- infinite strings being joined by a 



string of line density p .in < x < I. A wave with displacement 



exp ik x is incident from x • - », and we want to find the displacements 



everywhere, subject to conditions at r » and at x ■ £ which we will 



specify later. 



Write the total displacement in x < as y + exp ik x, so that y 



o 



represents an outgoing field as x ■*-<". Define 



r 



*_(s) - j y(x) exp isx dx (7.1) 



— flO 



so that Y (s) is analytic in R (Iin s < k .) and of algebraic decrease 

 - — oi 



everywhere at infinity in B (because y(x) is finite at x - 0). From 

 the differential equation 



&~Z- + k 2 y «■ (- «<x < 0) 

 dx 2 ° 



34 



