p ■ piu4>, and the velocity 3<{>/3y are 90 degrees out of phase. The 

 energy is locked in a thin layer oi thickness 0(Y -1 ) adjacent to the 

 surface, and none escapes as sound. 



Suppose now that the surface is semi-infinite, occupying 

 (_ oo < x < o, y = 0), all dependence on the z-coordinate parallel to 

 the surface edge being excluded. Let the surface - °° < x < still be 

 forced to move with the prescribed velocity 



v(x) - v exp(iqx) . 



Then (8.2) cannot be the solution, because it is easy to see that since 



3<J)/3y is prescribed on y = C, for some range of x at least, ty must be an 



odd function of y and since <{> must be continuous across the extension 



(y"0, < x < °°) of the surface, $ must be zero there — whereas (8.2) 



is not zero. Clearly, no single mode like (8.2), nor even any discrete 



set of modes of this kind, is capable of making 3<p/3y have the value 



v expiqx on y ■= 0, x < and of making 4> = 0ony = 0, x>0. The 



solution for <J) must therefore contain a continuous spectrum of modes 



like (8.2) with all values of the wavenumber. In particular, it must 



contain modes with wavenumbers a, say, with a < k , and for these 



1 ° 

 modes the exponential decay exp {- (q 2 - k 2 ) 7 y} must be replaced by 



l ° ± 



oscillatory behavior exp {i(k 2 - a 2 ) y}, the choice cf + i(k 2 - a 2 ) 2 y 



rather than - i(k 2 - ot 2 ) 2 y being dictated by the radiation condition , that 



the phase factor exp {i(k 2 - a 2 ) y - iwt)} be that of an outgoing wave 



i 



as y ■+ +°°(when y <. we take - i(k 2 - i 2 ) y) . Energy is radiated 



across a plane y ■ const, by such a mode — and we say that the energy 



which was trapped in the subsonic mode (8.2) on an infinite plate has 



51 



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