:':' '*yyjAXifc^ P &&oBt**r**W4t**f**, 



7. GENERALIZED W-H EQUATIONS: THSEE-PART BOUNDARY VALUE PROBLEMS 



The standard W-H equation, (2.26), arises in many, though by no 

 means all, boundary value problems in which different boundary data are 

 prescribed on, say, x < and x > 0. Many problems of interest in 

 acoustics fall into this category; for instance, the problem of energy 

 conversion from the elastic to the acoustic mode when a surface wave 

 in an elastic plate encounters a junction in the plate across which 

 the plate properties change abruptly can be modeled in terms of two 

 semi-infinite plates x < and x > for many purposes. The boundaries 

 concerned do not always have to coincide with just the x-axis. For 

 example, the standard W-ri equation arises In the diffraction of waves 

 by an open-ended parallel plate waveguide, or by an open ended circular 

 duct, provided these are both semi-infinite. On the other hand, dif- 

 fraction by three parallel equi-spaced semi-infinite plates is a com- 

 pletely open problem, though diffraction by an infinite cascade of 

 semi-infinite staggered plates is a relatively simple standard W-H 

 problem. If the waveguides referred to above have closed ends (diffrac- 

 tion by a semi-infinite thick rigid plate, or by a semi-infinite solid 

 rod) the standard W-H method does not lead to a closed form solution, 

 but to an infinite set of coupled linear equations, whose solution can 

 only be approximated by the solution of a finite subset of the equations 

 in the low frequency limit. 



Thus it is clearly difficult to give any general guidelines as to 

 when the standard W-H method would work except to say that it will not 

 work, without modification, in the case of three-part boundary value 

 problems where data is given on, S3y, -»< x<0; < x < £; 



32 



....:.■., ^...a^. .,«..n .— ..■.i-i.-iw,-.-..,^' i aiiigaaa aia ^sMaaaisgfe i iMa feaaa 



