Response of a Vibrating Plate in a Fluid 



where Po Is the density of the fluid above the plate, and where partial 

 integration has been utilized. Had Eq. (11) been used instead, 

 Eq. (12) would read 



2 C° C^ 2 



p_(x,y,z) =i^ \ \ dx» dy' W(x,y)|L| G(x,y , z |x' ,y' ,0) (13) 



which is reducible to Eq. (12) by partial integration. Thus, super- 

 sonic flow does not present any especial difficulty aside from the 

 fact that G is singular all along the Mach cone, and this is an 

 integrable singularity. 



Inserting the expressions for pg and P| into Eq. (1) results 

 in a single partial integro- differential equation to solve, viz. , 



a b 



BA^W - PpCo^W = F + 5p+^^r r G(x,y,|x',y')|L|V(x',y')dx'dy' 

 ^ *^ 4Tr' Jo Jo 



^If-'II g(xe.yc|xi,y^)W(x,y) dx dy (14) 



plate 

 where the subscript c refers to the cavity, thus 



Equation (14) presents a formidable computational problem. The 

 Green's function g is known as an infinite series which is slow to 

 converge (l/n) thus compounding the difficulty by an increasing num- 

 ber of necessary operations to maintain a given accuracy. 



An alternative to solving Eq. (14) is to convert it to an integral 

 equation for its Fourier amplitudes and to solve the resulting equation. 

 The advantage is that this equation is simpler (though it is a singular 

 integral equation). The following notation shall be employed: 



f(K) = -i^ J "^""^ ^^""^ 



supp {f} 



f(r) = -i^ J ^^^ ^<^) 



K- space 



489 



