571 



"'% 



"■%. 



FIGURE 1. Coordinate system, looking along the direc- 

 tion of wave propagation, ox, and along the labeling 

 coordinate direction, oq. 



Displacement Field 



Using labeling variables, q, r, s, to identify fluid 

 particles, define the field of particle positions in 

 terms of 1, r, s and time, t, as follows: 



X = q + Ut - a (exp mr) sin (kq - at) (1) 



y = r cos B - s sin B 



+ a (exp mr) cos (kq - at) (2) 



z = r sin B + s cos B (3) 



for r < R < 



0) is the wave encounter frequency, and differs from 

 the particle oscillation frequency by the Doppler 

 shift, Uk. 



Mass Conservation 



The displacement field defined by Eqs . 1, 2, and 3 

 can be made to satisfy the requirement that the 

 density of a fluid particle is independent of time 

 by requiring that the Jacobian: 



3 (x,y,z)/3(q,r,s) 



= 1 - a^km (exp2mr) cos 6 



+ (m cos B - k) a (exp mr) cos (kq - at) (7) 



is independent of time. This requires 



k = m cos B (8) 



Now proceed to apply the momentum equations to cal- 

 culate the pressure, which in turn will be set con- 

 stant at the free surface. 



3. PRESSURE FLUCTUATIONS 



The momentum equation in Lagrangian variables gives, 

 for the derivative of pressure with respect to the 

 labeling variable q: 



-p„/p = (x + z f sina - y f cos a)x 



+ (y + X f cos a)y + (z - x f sin a)z 



q q 



+ g z 



(9) 



The equations for the r and s-derivatives are 

 similar, f = 2fJ is the angular velocity of rotation 

 of the coordinate system, the angular velocity being 

 vertical as mentioned before. Substituting for x, 

 y, and z from Eqs. 1, 2, and 3 into Eq. 9, one ob- 

 tains : 



U is a constant mean particle velocity in the x- 

 direction, a is an oscillation amplitude parameter, 

 m is an inverse decay distance measure, K is wave- 

 number and a is the frequency of particle motion. 



First consider the kinematics of wave motion, 

 next find the condition for incompressibility before 

 proceeding to apply dynamics to give the dispersion 

 relation. A surface defined by letting r be a func- 

 tion of s will have waves that proceed in the x- 

 direction. For example, a string (line) of particles 

 defined by fixed values of r and s will have maxima 

 in y-displacement at 



kq - at = 2mr 



(4) 



From Eq. 1, substituting for q from Eq. 4 gives the 

 x-positions of crests to be at 



X = [2mr + (a + Uk)t]/k 

 crest 



The crests move at a speed of 



c = (a + Uk)k = co/k 



(5) 



(6) 



-Pq/P = [a' 



f cos a(a + Uk) 



-Pr/P = - to" 



-gk sin a] a (exp mr) sin 



f o cos a] a exp2mr 



(10) 



+ [-a^ cos B + f a cos (a + B) 



+ fUm cos a + gm sin a] a (exp mr) cos 



+ fu cos (a + B) + g sin (a + B) 



(11) 



-Ps/P = [°^ ^i" B - f a sin (a + B)]a(exp mr) cos 9 



- fU sin (a + B) +9 cos (a + B) (12) 



where 6 = kg - at is the phase of particle oscilla- 

 tion. 



At the free surface, which consists of particles 

 with a specified relation between r and s, and with 

 values of labeling variable, q, from - ■= to + ", the 

 pressure must be independent of q and t. This is 

 satisfied, as can be seen from Eqs. 10, 11, and 12, 



