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Extra info for Deep Water Gravity Waves: On the Simpler Aspects of Nonlinear Fluctuating (Weak Interaction Theory)

Example text

Qo and replaces the coordinate transformation (8) for the linear oscillators. The generating function (13) is easily shown to satisfy the equations [see Corben and Stehle, Sect. 61, for a complete discussion] : VES(9, ~) ; ~ = V S(£, E) so that any function satisfying (14) (14) gives rise to a canonical transformation, provided one can express (~,E) in terms of (~,E)The time independent Hamiltonian H(£,~) is expressed using (13) and (14) a H(~,V£S) and is related to the Hamiltonian in the (Q,E) system by H(£,V£S) : H(E) (15) The condition (15) is satisfied for any canonical transformation cyclic in ~, however, the generating function for the transformation (13) is generally multi-valued.

V27 = O, the solution of which generates a family of equip o t e n t i a l surfaces, i . e . , surface of constant ~ in the f l u i d . is constant along the free surface. In p a r t i c u l a r The v e l o c i t y f i e l d is therefore i r r o - t a t i o n a l on these surfaces of constant ~ and we assume v ( r , t ) : V#(r,t) (4) throughout t h i s work, i . e . , the water wave f i e l d is assumed to be i r r o t a t i o n a l in both two and three dimensions. We note that hydrodynamic turbulence is related to v o r t i c i t y 97'98 and since we are r e s t r i c t i n g our analysis to i r r o t a t i o n a l flow, we w i l l not discuss turbulence here.

S--'~I d2x - - . (x-x ' ) ZO ~ . . - - : I k = k' - 0 k ~ k' I (ll) a(_x - x') where 8k_ k, is the Kronecker and 6(x - x ' ) the Dirac delta function. Using ( I I ) we again apply Parsavel's theorem and obtain from (I0) H3 : ~ ~_,m,~ {6&+m+~V~m~B~BmB~+ . . . . . 8 JL+m-~ V p~mB ~BmB* . . *** + -_~-m-_p a Vm--PR R'R* + ~+m+~V-~--PBjLBmBp ~ -_E-m-~ . . . } (12) 45 where we note the r e s t r i c t i o n on wave vectors in each of the three-wave interactions. 34) H4 ~ ½ ~ d2x { ~ s ( X , t ) < ~ ( x , t ) ~ ( x , t ) K ~ s ( X , t ) _ 1 - ~s(X,t)~(x,t)K2~(x,t)~¢s(X,t) Cs(~,t)~2(~,t)m3@s(~,t) + @s(X,t)[K@s(X,t)]Vx~(X,t)'Vx@s(X,t) (13) Again applying Parsavel's theorem to (13) we obtain H4 = ~ ~,~,p,g + ~ {~+m+p+qV~m~qB~BmBpBq+ 6~+m+o_qV~mpB~BmBpB~ .

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