Brard 



Then we have to add the effects of a motion with three degrees of freedom 

 u U, w u, Lq U, the total motion being still parallel to the (z,x) -plane. In order 

 to do that, we admit that these three parameters have negligible squares and 

 products, and, consequently, that the wake surface is practically the same as in 

 the previous case. 



22.3. In par. 5, we introduce the hydrodynamic forces due to the total 

 velocity potential of the absolute motion of the fluid. For the sake of simplifi- 

 cation, we assume here that the body is not fitted with planes or fins. We con- 

 sider firstly the distribution of the pressure on the hull. It is shown that it is 

 the sum of three terms. One is due to the velocity potential when the fluid is 

 quite perfect, that is, when the wake is not taken into account. The two other 

 terms are due to the wake. The first of them is generated by a local Kutta- 

 Joukowsky or gyroscopic effect; the second is due to the partial derivative with 

 respect to the time. A second method, more rapid, gives the total force and 

 moment starting from the absolute momentum of the fluid and from its moment 

 with respect to the fixed axis. In order to do that, we substitute to the vortices 

 a distribution of doublets normal to the hull and to the wake. We show that it is 

 possible to express the difference between the set of forces yielded by the quasi- 

 steady motion theory and the real set of forces in terms of convolution functions: 



for the lift. 



for the drag. 



1/ 



k 



k 



2/ 



k 



fokC^') 0(t'-T')dT' 



fokC-^') ^(t'-T')dr' 



iol^(T')^'(t'-T')dT' 



for the moment; f^^(t'), k = o,l,2, are the arbitrarily given functions 



(u).,'(u).,. (i? 



We call "deficiencies" these differences. In fact, each deficiency is made of two 

 terms, one of them is really a deficiency, because it is due to the fact that the 

 circulation is unable to take instantaneously the value relating to the quasi- 

 steady motion; but the second one which is due to the partial derivative 9/3t, 

 acts in the opposite sense. The three functions 0,0,0' are probably the same 

 whatever k may be; but we did not prove that rigorously. Moreover we cannot 

 prove that they are proportional to one another (which is the case for an airfoil 

 of infinite aspect ratio). From a theoretical point of view, something is lacking 

 there but, from a practical point. of view, that is without great importance, be- 

 cause these functions may be obtained through experiments. 



904 



