More recently, one-fold integral representations (involving the exponential 



integral in the integrand) have been obtained and used by Inglis and Price and 



3 

 Guevel and Bougis, These single- integral representations are modifications of 



the double- integral Fourier representation in terms of the polar coordinates (X,e), 

 Single- integral representations associated with the Cartesian Fourier representation 

 (3.5), in the manner shown in Reference 12 for the particular problem of ship wave 

 resistance (f=0, F^O), have not been obtained to the author's knowledge. However, 

 such single- integral representations are considerably more complex than the corre- 

 sponding integral representations for the ship wave resistance problem and the series 

 representations obtained in Reference 13 for the particular case of wave radiation 

 and diffraction at zero forward speed (F=0, f?^0) . For the practical purpose of 

 numerically evaluating the velocity potential defined by a surface (or line) distri- 

 bution of singularities (sources or dipoles) with known strength, it may actually be 

 preferable to use a double- integral Fourier representation, such as that given by 

 Equation (3.5), together with an interchange in the order of integration between the 

 Fourier variables (y,v) and the space variables (x,y,z), as is shown in Section 6. 



11 



