3. THE GREEN FUNCTION 

 The Green function, G(^,x), associated with the sea-surface boundary condition 

 (2.5) satisfies the equations 



V^ G = 6(?-x)6(n-y)6(?-z) in C < (3.1a) 



[8 -(f-iF8^+ie)^]G = on ? = (3.1b) 



where 6 ( ) is the usual Dirac "delta function," and V^ represents the differential 



^ ->- -> 

 operator (9^,3 ,9^^). Physically, the Green function G(C,x) is the "spatial compo- 



-^ -»- 

 nent" of the velocity potential ReG(C,x)exp [ (e/f-i)t] of the flow created at the 



field point c,iK,T],^^0) by a moving source of pulsating strength Re exp [(e/f-i)t] 



located at point x(x,y,z<0). In the limiting case, z = 0, the source at point x 



evidently is no longer fully submerged, so that this physical interpretation of the 



Green function becomes ambiguous. A complementary physical interpretation for this 



limiting case is that the pulsating flow created at point x(x,y,z=0) stems from a 



flux across the plane z = of the mean sea surface. In the limit z = 0, the Green 



function G(C,x) must then satisfy the equations 



V^G = in C < (3.2a) 



[9 -(f-iF9 +ie)^]G = -6(?-x)6(n-y) on C = (3.2b) 



as may be seen from the sea-surface boundary condition (2.5). Equations (3.2a and 



b), justified above on physical grounds, can be justified mathematically in the man- 



12 13 

 ner shown in Noblesse ' for the particular problems of wave radiation and diffrac- 

 tion at zero mean forward speed (F=0) and of steady flow about a ship advancing at 

 constant speed in calm water (f=0) . 



The Green function G(E,,x) actually is a function of the four variables C-x, 

 ri-y, C+z, and (^-z) , and thus is invariant under the substitutions ^ -<-> - x, 

 n 4-+ _ y^ <;-!->- z. By performing these changes of variables in Equations (3.1a and b) 



