Brard 



irregular waves are integro-differential equations of the Volterra's type [1]. 

 That is already true even on regular seas, because the waves generated by the 

 ship have to be added to the incident waves. 



The problem which the present paper is devoted to is that of the maneu- 

 vering ship. 



For this problem, a "classical" theory already exists. That is, the quasi- 

 steady motion theory. It is admitted that, with the exception of the effects of the 

 so-called "added masses," the hydrodynamic forces exerted on the maneuvering 

 ship are identical to those found for a steady motion with the same angles of 

 attack and the same linear and angular velocities. 



That leads to a set of differential equations of the second order. 



This set is rather complicated in the case of a submerged body in an infinite 

 fluid because the number of the degrees of freedom is high. Moreover, the lin- 

 ear approximation is most often insufficient. Consequently, the equations con- 

 tain many, many terms. As the theory is unable to yield them, it is necessary 

 to resort to an empirical determination of their numerical values. When the 

 equations are written, it is necessary to solve them by using analog computers. 

 And the work is not finished by this time. The empirical determination of the 

 coefficients of the equations would have been practically impossible if the mo- 

 tion had not been split in its components; then the results so obtained must be 

 gathered. That is not so easy since the equations are not linear. Consequently 

 a comparison between the calculated motion and the real motion of the model or 

 of the full scale ship must be undertaken. 



Finally, the precise study of the maneuvering qualities of a ship, especially 

 of a submarine, requires a great deal of work. 



Therefore, the idea that the quasi -steady motion theory might be too simple 

 is attractive to very few. That is, however, the question about which the author 

 of this paper has tried to make up his mind. 



The starting point of the present investigation is that the hydrodynamic set 

 of forces exerted on a maneuvering ship is partly due to some circulation around 

 the ship. If so, this circulation around the body generates a vortex wake since 

 the circulation along a closed fluid circuit is null. And the vortex wake is what 

 prevents the equations to be purely differential. As in the Karman-Sears theory 

 of the unsteady motion on an airfoil of infinite aspect ratio [2], we shall expect 

 to deal with Volterra's integro-differential equations. Consequently the forces 

 in the real motion and those calculated by using the quasi -steady motion theory 

 must differ from one another, no circulating being able to take instantaneously 

 the value relating to the steady motion. This starting point needs some comments. 



For the quasi-steady motion theory does not preclude some circulation. In- 

 deed, this circulation cannot come from the set of forces deduced by Lagrange's 

 method from the kinetic energy of the absolute motion of the fluid surrounding 

 the body: it is assumed that this motion depends upon a velocity potential reg- 

 ular at the infinity. But, if some circulation exists in the steady motion, we 

 shall find it in the equations expressing the quasi-steady motion theory. 



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