504 



lar to 33, and forming the normal cross-section+'f', 

 a^. The surface swept out by the closed boundary 

 curve, 3an, of a^ is called the lateral surface. 

 Such a three-dimensional body is called jet-like if 

 the dimensions in the plane of the normal cross- 

 section are small compared to some characteristic 

 dimension, L(c), of c (see Figure 1), e.g., its 

 local radius of curvature 1/k , or the length of c 

 in the case of a straight curve. A jet-like body 

 is said to be slender if the largest dimension of 

 Un is much smaller than L(c). If a^ is independent 

 of C, the body is said to be of uniform cross- 

 section, otherwise of variable cross-section. Since 

 a material curve in the three-dimensional body, g, 

 can be defined by the equations, 9*^ = 6°'(£;) , it 

 follows that the equation resulting from (1) with 

 goi = 801(5) represents the parametric form of the 

 material curve in the present configuration and de- 

 fines a curve, c, in space at time, t, which we as- 

 sume to be sufficiently smooth and nonintersecting. 

 Every point of this curve has a position vector 

 specified by (10) . Let the (three-dimensional) jet- 

 like body in some neighborhood of c be bounded by 

 material surfaces, 5 = Clf 5 = S2 ' (indicated in 

 Figure 1) and a material surface of the form 



F(ei,e2,c) 







(17) 



FIGURE 1. A jet- like body in the present configuration 

 showing the line of centroids with position vector r 

 and the end normal cross-sections f| = ?i, 5 = Kz- Also 

 shown are the unit principal normal a] , the unit binor- 

 mal a2 and the tangent vector 33 to the curve with po- 

 sition vector r. 



which is chosen such that 5 = constant are curved 

 sections of the body bounded by closed curves on 

 this surface with c lying on or within (17) . In 

 the development of a general theory, it is preferable 

 to leave unspecified the choice of the relation of 

 the curve, c, to one on the boundary surface (17) . 

 In special cases or in specific applications, how- 

 ever, it is necessary to fix the relation of c to 

 the surface (17) . 



Suppose now that r* in (1) is a continuous func- 

 tion of 6^,t and has continuous space derivatives 

 of order 1 and continuous time derivatives of order 

 2 in the boimded region lying inside the surface (17) 

 and between E, = E,i , E, = C2- Hence, to any required 

 degree of approximation f* may be represented as a 

 polynomial in 9 , 6 with coefficients which are con- 

 tinuously dif ferentiable functions of £, < t. Instead 

 of considering a general representation of this kind, 

 we restrict attention here to the approximation. 



a sheet if the dimension of the body along the nor- 

 mals, called the height and denoted by h, is small. 

 A sheet is said to be thin if its thickness is much 

 smaller than a certain characteristic length, L(s), 

 of the surface, s, for example, the local minimum 

 radius of curvature of the surface, or the smallest 

 dimension of s in the case of a plane sheet. If h 

 is constant, the sheet is said to be of uniform 

 thickness, otherwise of variable thickness. Since 

 a material surface in the three-dimensional body can 

 be defined by the equation, E, = 5(6"), it follows 

 that the equations resulting from (1) and (2) with 

 C = 5(9") represent the parametric forms of the 

 material surface in the present and the reference 

 configurations, respectively. In particular, the 

 equation, 5=0, defines a surface in space at time. 



= r + 



d 

 ~a 



(18) 



where r is defined by (10) and d = d (5,t). 



~a -a 



Definition of a Sheet-like Body. A Representation 

 for the Motion of a Thin Sheet. 



Consider a two-dimensional surface, s, defined by 

 the parametric equation, E, = , over a finite co- 

 ordinate patch, a' = 9I = a", B' = 6^ = g".- Let r 

 and 33 denote, respectively, the position vector and 

 the unit normal to s . At each point of s, imagine 

 material filaments projecting normally above and 

 below the surface, s. The surface formed by the 

 material filaments constructed at the points of the 

 closed boundary curve of s is called the iateral 

 surface. Such a three-dimensional body is called 



The normal cross-section of a jet is a portion of the 

 normal plane to the curve, c, i.e., the intersection of the 

 body and the normal plane. 



FIGURE 2. Sketch of the cross-section (y = const.) of 

 a sheet of vertical thickness 4' showing a wave motion 

 propagating over a bottom of variable depth. Also shown 

 is the surface 9^ = (with position vector r and height 

 f) chosen such that the center mass of the (three- 

 dimensional) fluid region lies on this surface. The top 

 and bottom surfaces of height S and a are specified by 

 9^ = 1/2 and 9^ = -1/2, respectively. 



