Murthy 



turbed free surface of water and the Z-axis taken vertically down- 

 wards. The second is a moving co-ordinate system (x, y, z) with the 

 x,y-plane coinciding with the X, Y-plane, i. e. also lying in the un- 

 disturbed free surface and the x-axis having the instantaneous direc- 

 tion of the horizontal component of the velocity of the C. G. The x-axis 

 is therefore along the tangent to the course of the craft. The z-axis 

 is taken positive downwards and contains the C. G. (on the negative 

 side, in the case of ACVs ) . As we have assumed that the ACV 

 has an angular velocity in the horizontal plane, the x-axis will be 

 continually rotating away from the X-axis (see figure2). 



A third co-ordinate system fixed in the ACV and moving with 

 it will be introduced in the next section. 



The following equations for the transformation of co-ordinates 

 from the (X, Y, Z) system to the (x, y, z) system and vice versa are 

 easily derived : 



X = X + xcos a - ysina x = (X - X ) cos a + (Y - Y ) sina 



G G G 



Y = Y + xsina + ycos a y = (Y - Y ) cosa - (X - X ) sina 



(j G G 



Z = z z = Z (1-1) 



where X, and Yr are the co-ordinates of the C. G. in the fixed 

 system and a is the angle by which the x-axis has rotated from the 

 X-axis at any instant. 



I* 



I »( r ) dr 



i. e. a = J o>( r ) dr (1 -2) 



II. 2 Laplace's Equation 



The water is assumed to be inviscid and incompressible. It 

 is also assumed to be incapable of sustaining surface tension so that 

 the pressure of the water particles on the free surface may be equa- 

 ted directly to the air pressure thereon. There exists therefore a 

 velocity potential for the motion of the water $(X, Y, Z:t) satisfying 

 Laplace's equation 



110 



