Ship Waves and Wave Resistance 

 APPENDIX 



A FOURIER SERIES REPRESENTATION OF THE SOURCE 

 POTENTIAL IN A TANK OF FINITE WIDTH 



The expression to be presented here has essentially been derived in Ref. 6. 

 We shall confine ourselves to show certain properties which are needed in the 

 foregoing applications. 



The expression describes a wave potential of a source of output +4tt , i.e., 

 a singularity like negative inverse distance, in coordinates made dimensionless 

 by ship's half length as introduced under the first subheading of the text. The 

 expression is 



G = 27 



(X,^,z,o + gi°"'^ (X,^,z,o 



/2 L^'' 



1/ = — CO 



+ sin (U^7oY) sin (U^7o^) [1 " (" 1 )"] 



AU 



jcos (U^7oY) cos (U,7o^) [1 + (-1)"] 



(Al) 



with 



AU = 77/(7oT) , 



U = z^AU = sec 2 sin 0„ , 



f ree _ free 



'1/ 6- V 



[sign(X-0 -1] ^ e^^^°^^'^^ sin W^7o(X- ^) , 



K = Vl + 4U^2 ^ 

 W^ = yii; = sec 6^ 



and 



local _ local 



•U7ni X-^l 



v=o 

 - U^ sin (V7oO|| - 5J'(U) V2 muCU'' + V^)] dV 



Vcos(V7oZ) - U^ sin (V7oZ) 



Vcos (V7o0 

 (A2) 



withu = +n/v^ + U^2 and S^^^(u) = l for i^ = and 8^° = elsewhere or only the 

 Hadamard "part fini" should be taken for i/ = o. As shown in Ref. 6 it is easy to 

 find out by investigation of single terms of the series that the function G is sub- 

 ject to the following conditions: 



669 



