By Bruce J. West (auth.)

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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~ .