Minimum Wave Resistance of Surface Pressure Distribution 

 If the auxiliary function m(x,y) is defined by the partial differential equation 



p(x.y) 



3" 2 /^^ 3' 

 ^x' W <3y' 



m(x,y) 



(11) 



then it will be determined uniquely except for some arbitrary boundary condi- 

 tions, say, 



ni(±l,y) = — m(il,y) = 0, m(x,±b) = 0, (12) 



dx 



where S is assumed as a rectangle with length 2 and breadth 2b. 



Putting Eqs. (11) and (12) into Eq. (5), and integrating partially, yields (10) 



dy' , (13) 



(14) 



Since the first term has no trailing wave, this formula shows that the po- 

 tential consists of two parts; one is, say, the wave-free potential, and the other 

 is the part having the trailing wave, which is a sum of singularity distributions 

 along its periphery. 



If m(x,y) satisfies also the conditions 



_L^ m(±l,y) = ^m(±l,y) = , — m(x,±h) = , (1^) 



3x2 Bx^ By 



along with Eq. (12), then the potential is wave -free. 



In this case, integrating Eq. (11) and imposing the conditions of Eqs. (12) 

 and (15), it is found that the total pressure is zero: 



IJ p(x,y) dxdy = . (16) 



s 



This is similar to the conclusion in thin ship theory (10). 



777 



