VLYMEN: SWIMMING ENERGETICS OF THE LARVAL ANCHOVY 



ROSSER, J. B. 



1948. Theory and application of /Qe"''^dx and Jga'P^y^'dy 

 J -^ e ""^ dx. Part I. Methods of computation. Mapleton 

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SCHLICTING, H. 



1960. Boundary layer theory. 2d ed. McGraw-Hill Book Co., 

 N.Y., 122 p. 

 Stroud, A. H. 



1971. Approximate calculation of multiple integrals. 

 Prentice-Hall Inc., Englewood Cliffs, N.J. 

 Taylor, G. 



1951. Analysis of the swimming of microscopic organisms. 



Proc. R. Soc. Lond. Ser. A, 209:447-461. 

 1952a. The action of waving cylindrical tails in propelling 



microscopic organisms. Proc. R. Soc. Lond., Ser. A, 

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1952b. Analysis of the swimming of long and narrow ani- 

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 U.S. Navy Hydrographic Office. 



1956. Tables for rapid computation of density and electrical 

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Vlymen, W. J. 



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Webb, P. W. 



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APPENDIX 



The integration of the iterated integrals was 

 accomplished via a two-dimensional extension of 

 the standard Gauss-Legendre guadrature. The 

 one-dimensional fixed limit integration formula 

 was used by Holwell and Miles (1971) for similar 

 classes of functions with good results. The type of 

 integrals requiring evaluation were of the general 

 form 



fh rgit) 

 J" Jm 



F(x, t) dxdt 



a, b, fixed. 



Defining 



rgit) 

 Jf^t) 



G(t) = |_^^ Fix, t) dx, 



we get 



/ y ' 



■git) fb 



Fix, t)dxdt = L G(t)dt. 



i: 



By n- point Gauss-Legendre quadrature Ab- 

 ramovitz and Stegun (1966) this is given approxi- 

 mately by, 



/' 



Git)dt = ^-—^ X f^,G(^) 



where?, = ^-^ v, 



i = 1 



b + a 



y, = ith zero of P„(x), the n- order 

 Legendre polynomial and 



u-,= 2/(1-^,2) [P'„ (?,)]' . 



Using Gauss-Legendre quadrature on Gi^^) 

 yields, 



/■ 



Git)dt 



b - a 



1: 



(^,) 



I = 1 •' 'y^ii 



b^± a;i^zf^t u,;nn,i,), 



I = 1 



J = 1 



where t?^ = 2 _ '- y^ + 



w* = 2/il - y^) [p;,iVj)y and 

 y^ = Jth root ofP^fjr). 

 We have finally the result. 



a J fit I 



Fix, t) dxdt ^ ^ " f 



I = 1 



i w,wr S(^^ f^''^ Fin,,^,), 



J = 1 

 where the above definitions hold. 



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