Eggers 



,( 3 ) C \ I ,<')/( 2 ) ^ r° [,(!),( 



2) ^ ,( 1) ,(2) (1) (2) 



^ + 4^2 ^1^ - ^X '/'l^ 



dC dT? 



and 



(26) 



.(3) 



S^'^C^.^) 4*^ d^d77 



^0 It Jx^ 



,( 1) ,(2) ( 1) (2) ( 1) ( 2) 



^X ^^2^ + ^Z "^'22 - ^X ^2^ 



da d77 . 



(27) 



Resistance Due to Additional Singularities Within the Hull 



Let us now, only to save labor in writing down formulas, assume that the 

 depth of the tank is large enough that we may put H= ». If 0( i) can at X= X^ be 

 represented by a system of free waves (we omit terms nonsymmetric in Y for 

 reasons of simplicity) as 



^5c'^^"'= ^ [a^'^ ^°^(W.7oX) + Bi'^sin(W^7oX) 6^'^''°^ cos (U^ y^ Y) . (28) 



where AU = 77/(7oT) and U^ = vAU = sec ^^^ sin 0^, M^ = l + 4U^^ K^= (l + M^)/2= sec 2^^, 

 *v = Vk7 = sec^^, A^^^ = Ail\ and B^^) - B^;^) (where e^ stands for the angle of 

 wave propagation against the X axis), then (6) we can evaluate the integrals in 

 closed form as 



( 2) 



R = T 



CD 



2 - COS' 



A„ + B 



•Xn • 



(29) 



(This formula reflects the fact that the resistance is essentially equal to the av- 

 erage energy in the wave components times the difference between the ship's 

 speed c and the x component of the group-velocity, divided by c.) 



,( 1) 



K (//j has a corresponding far -field representation 



( 2)free 

 *x 



.(3) 



7] aJ'^ cos (W^7oX) + b['^ sin (W^y^X) 



K„rnZ 



COS (U^7oY) . 



(30) 



then R*^ as interference between both systems can be written down directly as 



R<,'>=Tf'(2-cos^fi,)(A',;'A: 



( 2) .( 1) _(2) „( 1)' 



+ B, B. 



(31) 



660 



