FISHERY BULLETIN; VOL. 72, NO. 4 



integrals above; however, later in the discussion 

 s and his) will be converted to appropriate func- 

 tions of X and t so that the integrations may be 

 performed. 



The accelerative or virtual mass element of 

 work can be calculated using the fact that the 

 virtual mass of a flat plate accelerating parallel 

 to its normal vector is equal to the mass of the 

 fluid enclosed in a circumscribing cylinder hav- 

 ing the plate chord as diameter (Fung, 1969). 

 Hence from Figure 2 the virtual mass is given by 



dM = p TTh^(s) ds 

 and the magnitude of the acceleration by 

 a sin ©1= a sinie - 02^ 



where a = a 



-1 c/v -1 V'y 



Now ^ = tan -f- and 89 = tan ^jf^ 



ax y X 



^j, dVy J ,r, dVx . . 

 where V y = —jf- and V x = —j^, giving 



Figure 2. — Diagram illustrating body element undergoing ac- 

 celeration d and relationships of orientation of element to vector 

 a in terms of the angles 0, 6 i , 82- 



dx 



and 



COSiFy H] 



dx) 



iVy + Vx ) ''- dt 



-1 V'y -1 Vy 



cos(tan pTT^ - tan -y^ 



dt 



dt 



we get finally 



a s\niQ-Q.,) 

 Since 



a sinie - e. 



. / _i dy ^ -1 V'y\ 

 as.n(^tan ^ ^ tan ^j 



^ iv'y + V'i Y'^ , we get 



sin 



/f -1 dy 

 ('"" Tx 



tan 



-1 K 



V 



^) 



Thus the magnitude of force F^•^; on an element 

 of body ribbon due to induced mass dM is 



Fy,j I = adM = pTTh'\s){V'y + V'x )'''' 



. ,, ., dy ^ _, V'y , 

 •sinltan -^ -tan tti- '• 

 dx V X 



Since the element of work is given by 



dW= \Fy!^^ \-\dx\  cos(Fv'M | dx) 



where 

 888 



dW = pirh^ is) iVy + Vl ) '''' {Vy + V'h ''^ 



5in ( 



sin I tan -f — tan 



-1 vy\ 



V'x j 



■cos 



(, -^V'y 



i'"" vi 



- tan" 



Vy_ 

 Yx 



dsdt. 



Using the identities for cos (a - 6 ), sin (a - b), 

 and ds gives finally, 



dW^ = 



where c?W^ is part I of the element of work per- 

 formed by the acceleration of the surrounding 

 fluid. 



In addition to c?W/^, above we also want the 

 work done in accelerating the body itself Calling 

 Pg (s) the linear mass density of the body we get, 

 using Figure 2, 



