■■. \'~' r *';-?^.«r.-:z.-,^.-: iv..-.A^^r»«w>it-«.w 



6. WAVES ON BEAMS 



We now replace the strings of <j52 by beams of specific mass p , p 

 and bending stiffness B , B in x < 0, x > 0, respectively. The free 

 wavenumbers will be denoted by k , k. as before, for time dependence 

 exp(-itut), wheie 



k :-<p M ' /B o> • k . •■ <p»»*/v- 



(6.1) 



For the moment we leave conditions at the junction x ■ unspecified 



and take a wave with displacement exp(ik x) incident from x » -», 



o 



denoting the tota l displacements again by y + exp(ik x) in x < 

 o 



and by y in x > 0. This ensures exponential decay of y(x) as 



x -»• + °° and as x + - ■» provided In k , Im k are gi**en small positive 



values, fcr we anticipate that in x > 



and 



y(x) ■ A exp(ikjX) + B expC-kjX) 



y(x) - C exp(-ik x) + D exp(k x) . 



(6.2) 



The near-field terms here, B exp(-kx) and D exp(k x) , decay 



» o 



as x •*■ + °» and as x ■* - °°, respectively, because k and k, havt oositive 



o » 



real parts; they arise because the equations of motion are 



dx" 



k' y-0 



in x < 



(6.3) 



^-f - k* y - inx>0 

 dx* l 



Taking half -range Fourier transforms of (6.3) gives 



26 



