In order to obtain, then, some information about finite closed cavities, which are known 

 to occur physically, it has become necessary to investigate not the problem set above, but 

 approximations to that problem involving the so-called finite cavity models. The prominent 

 models are those of Riabouchinsky and Wagner,^ their justification being that the nonideal- 

 ness of the flow at the near of the cavity may be approximated quite roughly while reasonable 

 results may still be obtained for the drag of the body and the shape of the forepart of the cav- 

 ity. The fact that the two different models lead to almost identical results for body drag-^ lends 

 considerable force to the justification argument. 



At the present time, the use of neither cavitation model allows even an approximate 

 solution to be obtained for an arbitrary body; solutions for even those simple bodies treated 

 are obtained at the expense of (relatively speaking) considerable labor. Because of these 

 reasons it seems appropriate that a suitable linearized approximation to the exact problem be 

 discussed. The discussion must thus be limited to the case of slender bodies, but these are 

 bodies of great practical interest. 



THE LINEARIZED THEORY 



The history of attempts to solve hydrodynamic problems by means of linearizing assump- 

 tions is about as old as the history of mathematical hydrodynamics. The linearization of both 

 equations of motion and boundary conditions is, for instance, essential in a great part of the 

 classical theory of waves. Probably the first discussion of a linearized theory of flow past 

 a practical configuration was given by J.H. Michell in 1898 in his now famous analysis of 

 "The Wave-Resistance of a Ship."^ Michell made, as a matter of fact, linearizing assump- 

 tions very similar to those which are made in the present paper on cavitation flows. Despite 

 certain similarities between the problems, the present method of solution does not resemble 

 that of Michell who used a Fourier series development. Here the boundary conditions are sat- 

 isfied by making use of a suitable singularity distribution. The integral equation that results 

 and which must be solved for the determination of the suitable singularity distribution is iden- 

 tical with that which occurs in the linearized theory of lifting airfoils. This theory was ap- 

 parently first suggested by L. Prandtl about 1918. * Certain results of the thin airfoil theory 

 and in particular the inversion formula of A. Betz (1919)^ for the important integral equation 

 will be utilized. 



Consider the flow schematically illustrated in Figure 1 and let 



V = Velocity of fluid at any point in the flow field = U^ + v 



V = Perturbation velocity 



u,v = X and y components, respectively, of v 

 U = Uniform velocity at oo, parallel to the a!-axis 

 f/ = Component in ^-direction of velocity on cavity wall = U^ + u{x, y^) 



