Murthy 



III. 3 Transformation of Co-ordinates 



It is easy to derive the following equations for the transforma- 

 tion of co-ordinates between the moving system (x, y, z) and the body- 

 fixed system (x 1 , y' , z 1 ). 



x = x + x 1 cos 9 + (z' - z ) sinG x 1 = (x - x) cos + (z + z_) sin 



G G 



z = z + z + (z 1 -z ) cos - x 1 sin0 z'= (x - x) sin - (z+z -z)cos0 +z„, 

 G G G G 



(2-1) 



where x, z and are the surge, heave and pitch displacements 



III. 4 Perturbation Expansions 



We will now expend some of the physical variables represent- 

 ing the motion of the ACV and that of the water in powers of four 

 perturbation parameters 8 , (3 , a and e defined below. 



. semi -width 



d = — ratio of the side hulls. 



length 



_ draught ,. . ,, . , . , . cushion pressure,. 



p = — sz r— ratio of the air cushion ( l. e. , ) 



length p g L 



a = amplitude of small motion (and oscillation) 



amplitude 

 length 



The first two parameters have been selected from the require- 

 ment of the linearized theory that the amplitude /length ratio of the 

 waves induced by the side hulls and the air cushion due to the motion 

 of the ACV should be small. The amplitudes of the induced waves may 

 be assumed to be proportional to the beam of the side hulls and the 

 cushion pressure (in head of water) and the length is proportional to 

 F n . It is therefore clear that the speed of the ACV should be suffi- 

 ciently large. 



The linearized theory is therefore inapplicable to very low 

 speeds on account of the unacceptable steepness of the induced wa- 

 ves. 



120 



