Brard 



J [^ (St ' ^ (u 



t '-t') dr', 



2') The non-linearity is also to be taken into account when the mo- 

 tions are not really quasi-rectilinear. From a practical point of view, this is 

 very serious since, in many cases, the trajectory of the origin of the moving 

 axis is not a straight line. In such a case, the nuclei depend upon r' and t ' - r' , 

 and not upon t ' - r' only. Consequently, we encounter here a new problem, 

 which consists in the empirical determination of the new function 0(t' , t '- t') 

 which have to be substituted for <;6(t ' - t') . 



A similar circumstance happens when the heel becomes great even if the 

 trajectory of is nearly a straight line, for, in this case, the wake cannot be 

 considered as a plane surface but is an helicoidal surface. The first phase in a 

 change of heading would be different and the wake due to a gyration in the verti- 

 cal plane could have a severe effect on the trim. 



3*) The non-linearity may affect also the scale effect since the coef- 

 ficients a, b, . . . , depend upon the Reynolds' number. This cause of error 

 exists also in the quasi- static theory, but it has no effect on the functions 



20.2. Other Causes of Errors 



They are the effects of the free surface and those of the walls and of the 

 bottom of the tank. 



In order to get accurate measurements of the sets of forces, it is necessary 

 to operate at a sufficiently high speed. But U becomes great and the range of 

 values of wL/U which is accessible becomes narrow. In order to increase this 

 range, one may be obliged to operate sometimes beyond the critical speed U/x/gH, 

 where H is the depth of the tank, and sometimes below. On the other hand, the 

 coefficient V/x/gT, where i is the depth of may be great and consequently, the 

 waves generated by the model may be not negligible at all. Lastly because the 

 range of values of oj is not very wide (from 1.1 to 3.27) it may be necessary to 

 work at various values of wL/Ux u^/gL in order to keep constant the values of 

 aL/U and, consequently, the changes of the wave patterns which results from that, 

 may lead to errors about the true effect of the reduced frequency. 



That means that experiments conceived in order to determine the functions 

 ^j. . . , 0^ require a very caution approach. 



21. The Solving of the True Equations 



Generally, one admits that, when the forces acting on the model are known, 

 the equations of the maneuvering ship may be solved by analog computers. 



Such computers are most often fitted with curve-plotters, and it is possible 

 to get the curves which give the motion of the body following a given maneuver 

 as a function of time. 



902 



