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The system (2.18), (2.20) is completed i-hen the boundary condition 

 at x * Is specified. In the present simplest case, in which there can 

 ba no difference of transverse force from x ■ 0- to x ■ 0+„ the boundary 

 condition is that 



9A Z «_ T f Cy + « ik ° X> - *?««. T ? . (2.21) 



x ■* 0- dx ' x ■+ 0+ dx 



so that 



y*(0-) + ik - y'(0+) (2.22) 



while the total displacement of the string must also be continuous, so 

 that 



Aim . , ik_x x Aim ,, ,,* 



x .o_ (y + eN>) -x*o+y (2 - 23) 



or 



y(O-) + 1 - y(0+) (2.24) 



Eliminate say y(O-) and y' (0-) from (2.20) by using (2.22) and (2. 24), 

 giving 



(s 2 - k 2 ) Y (s) - y'(0+) - isy(0+) + i(s - k ) (2.25) 



o — o 



and if we now add this is to (2.18), the unknowns y'(0+) and y(0+) 

 disappear and we have 



(s 2 - k 2 ) Y + (s) + (s 2 - k* ) YJp) » i(s - k o ) . 



It is convenient to divide through by (s 2 - k 2 ), wliich ws may do because 

 the equation itself is only meaningful in the strip D and the zeros +k 

 lie outside thet strip. Then we get a standard form of Wiecer-Hopf 

 functional equation 



12 



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