Ship Maneuvering in Deep and Confined Waters 



For steady horizontal drift at moderate forward speeds one 

 finds a similar condition 



a$" 2 Z 8^^" 



dz nu g^^„ 



which shall govern the local accelerations of the flow in the trans- 

 verse plane penetrated by the moving body, [ 18]. 



As is seen from the two equations above the vertical velocities 

 at the water surface are zero i n the limit of zero frequency or zero 

 drift, and negligible for w « Vg/L or P^F^l « 1. The water sur- 

 face may therefore be treated as a rigid wall, in which the underwater 

 hull and streamlines are mirrored, i.e. the image moves in phase 

 with the hull. 



For high frequencies , v^ere co » Vg/l-» the condition at the 

 free surface is $ = 0. The water particles move up and down normal 

 to the surface, but no progressive waves are radiated. At the juncture 

 of the horizontally oscillating submerged section contour and the free 

 surface this condition may be realized by the added effect of an image 

 contour, which moves in opposite phase, (Cf. Weinblum [ 19] .) The 

 value of added mass in this case, "neglecting gravity, " is smaller 

 than the deeply submerged value by an aimount equal to twice the 

 image effect. 



Added masses A^ for two-dimensional forms oscillating 

 laterally with very low frequencies in a free surface have been cal- 

 culated by Grim [ 20] and by Landweber and Macagno [ 21] , using 

 a LAURENT series with odd terms to transform the exterior of a 

 symmetric contour into the exterior of a circle (TEODORSEN map- 

 ping). By retaining the first three terms this transformation yields 

 the well-known two-parameter LEWIS forms [ 22] ; other combina- 

 tions of three terms have been studied by Prohaska in connection 

 with the vertical vibrations of ships [ 23] . Two terms (and one single 

 selectable parameter for the excentricity) define the semi-elllptlc 

 contour as that special case with given draught, for which the added 

 mass is a minimum. Landweber and Macagno also made calculations 

 of the added masses A^ In the high-frequency case. For the seml- 

 elllptlc contour A^Ja!. = 4:/tt , which result was first found by 

 Lockwood-Taylor , [24] , 



A basic theory for the dependence of the hydrodynamlc forces 

 on finite frequencies was developed for the semi-submerged circular 

 cylinder by Ursell, [ 25] . By use of a special set of non- orthogonal 

 harmonic polynomials he found the velocity potential and stream 

 function that satisfied the boundary conditions and represented a 

 diverging wave train at Infinity. Based upon similar principles 

 Tasal extended the calculations of added masses (and damping 

 forces) for two-dimensional LEWIS forms to Include the total 



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