Norrbin 



be Illustrated by a pulsating source) additional waves will form, 

 which move with speed g/co. At low frequencies these waves move 

 faster than the body, so that the diverging wave front folds forward, 

 and at a certain forward speed there is now a new requirement on 

 basin width to avoid wall interference. For connbinations of co and 

 V (or u) , in which y = w"u" = 0,272, the opening angle equals 

 P = 90°, and with a further reduction in speed it rapidly reduces 

 again to 55° as y approaches 1/4. This latter condition is associ- 

 ated with a special phenomenon of critical wave damping, as has 

 been shown from theory as well as experiments by Brard, [ 33] . 



In model tests with a ship form in latered oscillations a narrow 

 range of critical frequencies may be identified by a change of the 

 distribution of the hydrodynamic forces, which was clearly demon- 

 strated by van Leeuwen's analysis. 



Whereas there is a discrepancy in the absolute values of 

 added masses compared in Fig, 8 this discrepancy could be reduced 

 by the application of a three-dimensional corrector; more elaborate 

 theories of forward speed effects for slender bodies at low fre- 

 quencies may further improve the comparison. In the main, there- 

 fore, it may be stated that the variation of added mass with frequency 

 is well documented. 



Added Masses in Maneuvering Applications 



The performance problems set up in maneuvering studies 

 usually involve a short- time prediction of a transient response to a 

 control action, and it is therefore convenient to be in the position 

 to use ordinary non-linear differential equations with constant coef- 

 ficients. This, of course, is in contrast to the linearized spectrum 

 approach to the statistical seakeeping problem, which will more 

 readily accept frequency-dependent coefficients. (Frequency- or 

 time-dependence as a result of viscous phenomena will be touched 

 upon below.) Which values of added mass are now to be used in the 

 equations for the manoeuvring ship? It shall be noted that it is hard 

 to judge from the behaviour of a free-sailing ship or ship model which 

 is the correct answer unless special motions are carefully examined. 



It was early suggested by Weinblum that the low added mass 

 values of the high-frequency approximation should be adequate for use 

 in dealing with problems of directional stability, where starting con- 

 ditions should simulate impulsive motion, [ 19] , Weinblum also drew 

 attention to Ref, [ 34] , in which Havelock proved that the high- 

 frequency values appeared in horizontal translations with uniform 

 acceleration, regardless of the initial velocity. 



The impulsive pressures experienced on the tapered bow and 

 stern portions of a slender body In oblique translation may be calcu- 



836 



