In water tunnels, we observe that attached cavities grow and shrink in re- 

 sponse to system pressure changes. To the eye, at least for slow changes, 

 the cavity appears to be a closed body of changing shape and length. It 

 is actually a material surface. In the linear theory the coordinate, Y, 

 of the end of the cavity, is 



J o \ o o / 



(38) 



where 1(t) is the fluctuating end of the cavity, v is the perturbation 

 velocity component normal to the cavity, and U is the reference speed. 

 For steady flows, Y = is Tulin's closed cavity model; Y ^ corresponds 

 to an open wake model. For unsteady flows, Equation (38) is actually quite 

 a difficult integral to evaluate, even in simple flows. Most of these 

 problems were foreshadowed by Parkin's work on unsteady hydrofoils 

 (Parkin 1957) . As it has turned out, most work in the field since then has 

 avoided direct use of Equation (38), and, because of these formal diffi- 

 culties, other approaches have been adopted. Among these, two are well- 

 known;* Guerst permitted no change in cavity volume, whereas Leehey assumed 

 that the end of the cavity remained at its steady value (thereby circum- 

 venting Equation (38)), but allowed the cavity volume to change. This ap- 

 proach was also used later by Kim and Acosta (1975) to evaluate the terms 

 of the transfer matrix, z. Still, this seemed not entirely a satisfactory 

 resolution because it was clear that very slow oscillations should approach 

 known "quasi-steady" values, wherein, the actual changes in cavity length 

 that are seen, are properly modeled. It has been shown more recently 

 (Acosta and Furuya) that application of Equation (38) , with Y = for small 

 frequency, does, in fact, lead to the quasi-steady limit of Tulin's closed 

 cavity model. It was also shown that the fixed- terminus cavity model for 

 unsteady flow had a rather unnatural singular behavior of one of the 

 coefficients of z. To appreciate this we now need to comment on each of 

 the terms of the cascade transfer function z. 



*Refer to the review by Wu for details, 



78 



