i. < x < "+ « in three different forms. There are, however, a number of 



problems which have been successfully tackled — though always only in 



approximate form — in this category, using modified W-H methods. We quote 



as examples the problems of diffraction by an infinite rigid strip of 



finite width in the high frequency limit, cf the resonances of a circular 



tube of finite length open at both ends and of wave motion in an elastic 



plate set in an infinite rigid surrounding baffle. 



Now in many of these problems the approximate solution is achieved 



on the basis of a "weak interaction" simplification. For example, in 



diffraction of waves of wavenumber k by a rigid strip of width f- in 



the high frequency limit, (k I) » 1, one can suppose that to first 



o 



order each edge is unaware of the presence of the other, so that one 

 can start the approximation with two semi-iufinite problems of standard 

 W-H type. Then if the incident wave amplitude is 0(1), the first inter- 

 action of one edge with the other will te through a cylindrical wave 



emanating from one edge, due to the incident wave, and of amplitude 



-X 

 0(k i) 2 ) near the other edge. For the second approximation we there- 

 fore solve another two semi- infinite problems, but now with more 

 complicated forcing terns arising from the mutual interaction between 

 the edges. While one can see how to continue the process in that 

 simple case, it is advantageous to derive a modified W-H equation whose 

 approximate solution throws up these successive interaction problems in 

 a natural way. The advantage is that one can see from the generalized 

 W-H equation what to try in more complicated problems where the 

 physical situation is less clear. For example, near resonance the 

 "weak interaction" sort of approximation is quite inappropriate, as the 



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