rise to various Unda of standard and generalized forms of W-K equations. 

 Hot only is tha derivasion of the W-H functional equation much simpler 

 than usual for these problems, but its solution is also much easier F and 

 the final inversion of a Fourier transform integral can be readily per- 

 formed and the results seen to correspond with familiar undergraduate 

 ideas. 



The W-H technique is a method of solving certain types of boundary 

 value problem in which, typically, we have information about the 

 pressure, any, on the half-plane x < and about the velocity on the half- 

 plane x > 0, and we cannot solve for the radiated acoustic field until 

 we know the pressure all over the whole boundary, -«° <x< +«°, or 

 about the velocity there. In the W-H method, a single equation is 

 derived, relating the Fourier transforms of the unknown pressure 

 (for x > 0), of the unknown velocity (for x < 0) and of the given 

 forci.ig field arising from some prescribed f oi .«* or source or incident 

 field. The transformsof the two unknown distributions are known (partly 

 because of information supplied by the anticipated physical behavior of 

 the system under study) to have certcin analyticity properties as 

 functions of the wavenumber regarded as a complex variable , and a 

 certain crucial step (the W-H method) and the use of some fundamental 

 theorems of the calculus of functions of a complex variable together 

 enable one equation to be solved for two unknown functions. Then the 

 field everywhere can be found in terms of an inverse Fourier integral 

 which in many instances can bs estimated by stationary phase or steepest 

 descent/saddle point techniques [e.g., 9,10] or by generalized function 

 methods [11,12], In the problems to be discussed here such elaborate 

 methods are no; needed, and an exact inversion of the Fourier Integrals 



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