8. TOO-DIMENSIONAL HALF-PLANE PROBLEMS 



We go on now to look at a simple problem involving a half-plane 

 embedded in an acoustic fluid. In the first place the fluid will have 

 no bulk motion, whereas later we shall allow the fluid to flow over the 

 half-plane with uniform subsonic velocity, leaving a wake behind the plate 

 if the edge is a trailing edge. The issue we want to examine is the 

 following one. Suppose that the plate were infinite in both the positive 

 and the negative x-directions and a wave were forced to propagate along 

 the plate with some prescribed frequency u and wavenumber q, the wave- 

 number q being real and greater than the acoustic wavenumber k ■ w/c 



o o. 



Thus the velocity in the positive y-direction is prescribed in the form 



v(x,t) = v exp(iqx -:Luit) . (8.1) 



Then the solution for the potential 0(x,y) in the fluid in y > is 

 (dropping the factor exp - iut) 



4>(x»y>=-— exp(iqx - Y n y) (8.2) 



q 



where 



9 ° 



for this makes 



\3x 2 dy 2 °) 



, 



makes <{> -<■ as y + + » 



and makes 7p (x,0) ■ v exp(iqx) . 



The field described by (8.2) is that of a subsonic trapped surface wave . 

 No radiation takes place across any plane y ■ const, because the pressure, 



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