Wang 



displacement of the free surface and the solid body from their cor- 

 responding locations in the basic flow is denoted by n = h(s ,t). If 

 the wetted side of the solid body and the free surface of the basic 

 flow are taken to be n = 0, h(s ,t) is assumed to be a very small 

 quantity. In W we have found that the linearized kinennatic and 

 dynamic boundary conditions on the free surface for the unsteady 

 perturbation potential <|>, are 



8<f)i 8h , 8h f. ... 



and 



^''^'^-^^^ °" ■> = «• (^> 



and that the boundary condition on the solid body is 



In the above equations R and q^ are respectively the radius of curva- 

 ture and the constant speed on the free surface of the basic flow, and 

 q^j is the speed of the basic flow. The + (or -) sign on the right- 

 hand side of (E) holds for the upper (or lower) branch of the cavity 

 wall; these signs are necessary to make R a positive quantity. We 

 should mention here that in obtaining (2) we have assumed that the 

 cavity pressure remains unchanged during the unsteady perturbation. 



If we regard qc/R as an equivcilent gravitational acceleration 

 and the s-coordinate rectilinear, then (1) and (2) are in the same 

 form as the linearized free surface boundary conditions in water wave 

 problems. Thus we expect that the centrifugal acceleration qg/R 

 due to the curvature of the basic flow streamiline should play the role 

 of a restoring force in producing and propagating the surface waves 

 along the curved cavity wall. 



The purpose of the present paper is to use this perturbation 

 theory to study some unsteady behavior of the Klrchhoff flow when the 

 solid plate is in small harmonic oscillations. 



II. THE BASIC FLOW 



In this paper we consider the basic flow to be a flat plate held 

 normal to an Incoming uniform stream of infinite breadth, with a 

 cavity formation of Infinite length as shown in Fig. 2. This is the 

 so-called Kirchhoff flow. Both the speed of the incoming stream and 

 the length of the plate AB are taken to be unity. A set of Cartesian 

 coordinates (x,y) with its origin at the stagnation point C Is chosen 

 as Indicated In Fig. 2, where the point I denotes the point at Infinity, 



190 



