2, THE BASIC POTENTIAL- FLOW PROBLEM 



The basic potential-flow problem of the linearized theory of ship motions in a 

 regular sea is briefly formulated in this section. The sea is assumed to be of 

 infinite depth and horizontal extent. Water is regarded as homogeneous and incom- 

 pressible, with density p. Viscosity effects are ignored, and irrotational flow is 

 assumed. Surface tension, wavebreaking, spray formation at the ship bow, and non- 

 linearities in the sea-surface boundary condition are neglected. A moving system of 

 coordinates (X,Y,Z) in steady translation with the mean forward velocity U of the 

 ship is defined. Specifically, the mean (undisturbed) sea surface and the center- 

 plane of the ship in its mean position are taken as the planes Z = and Y = 0, 

 respectively; the Z axis is directed vertically upwards, and the X axis is directed 

 toward the ship bow. 



In the above-defined translating system of coordinates, the linearized sea- 

 surface boundary condition takes the form 



[g32+(U8^-8^)^]$^ = (U8^-9^)pVp - gQ' on Z = (2.1) 



where g is the acceleration of gravity, T is the time, $' = $'(X,T) is the velocity 

 potential, ?" = P'(X,Y,T) and Q' = Q'(X,Y,T) correspond to distributions of pressure 

 and flux, respectively, at the sea surface (we have Q' = for all practical applica- 

 tions, and P' E except for surface-effect ships), and the notation 9^, 9^, 9^ is 

 meant for the differential operators 9/3Z, 9/9X, 9/9T, respectively. 



The present study is concerned with flows simple- harmonic in time, with radiant 

 frequency to where OJ is the frequency of encounter. However, such free-surface flows 

 are not completely (or uniquely) determined unless an appropriate "radiation condi- 

 tion" is imposed, as is well known and is discussed by Stoker, for instance. A 



.15 

 convenient alternative approach, employed previously in Lighthill and 



Noblesse,"""^' "'■■^ to the use of such a "radiation condition" consists in defining a 

 time-harmonic flow as the limit, as the small positive auxiliary parameter o 

 vanishes, of a flow defined by a velocity potential of the form 



$'(X,T) = Re<l'(X)exp[(a-ioj)T] (2.2) 



