SOME TOPICS IN THE THEORY OF 

 COUPLED SHIP MOTIONS 



J. Kotik and J. Lurye 



TRG Incorporated 



Melville, New York 



1. INTRODUCTION 



In this paper we present several different results in the theory of ship mo- 

 tions. Some of the results express certain physical quantities in terms of other 

 such quantities, while the remaining results are in the direction of computing 

 physical quantities by solving boundary value problems. The following of our 

 results are of the first type: 



Kramers-Kronig relations with forward speed and cross-coupling. 



Impulse response in terms of force coefficient for simple harmonic motion. 



As work of the second type we present a numerical approach which seeks sim- 

 plicity by avoiding integrations over curved surfaces and approximations to or 

 representations of curved surfaces. These results already obtained are only a 

 beginning, since they assume zero forward speed, but they are sufficiently 

 promising to encourage us to extend them to include forward speed. 



2. KRAMERS-KRONIG RELATIONS FOR HYDRODYNAMIC 



CROSS- COUPLING COEFFICIENTS AT FORWARD 

 SPEED 



In this section we sketch the proof that the real and imaginary parts of the 

 complex hydrodynamic cross- coupling coefficients are connected by the Kramers- 

 Kronig relations* in the case of a submerged body having forward speed. We 

 begin by defining these coefficients. 



Let the aforementioned body at first be at rest in a steady flow which (1) 

 satisfies the usual normal velocity condition on the body surface, (2) satisfies 

 the linearized free surface condition, and (3) becomes uniform with velocity -ex 

 as x^+oo. (Here x is a unit vector in the direction of the positive x axis.) This 

 flow evidently represents forward motion of the body at speed c in the positive 

 x direction. (The x and y axes are horizontal, the z axis is positive upwards, 

 and the origin of the x,y, z coordinate system is at the center of gravity of the 



'■'See footnote after Eq. (2.2). 



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