572 



if the pressure is independent of phase 6, and 



, and p are independent of 9 . 

 From Eq. 12, p is independent of 



when 



cot 



f sin a 



cot a 



Pq' 



(13) 



Since a is given by the slope of the boundary, Eq. 

 13 gives g for a given a and a. Equation 10 shows 

 Pq to be independent of 9 when 



f cos a(a + Uk) 



g k sin a = 



(14) 



For a given value of a, Eq. 14 yields k, and m is 

 then found from Eqs. 8 and 13. 



This leaves Eq. 11 unused, but it can be shown 

 that the requirement that p^. be independent of 6 is 

 not independent of Eqs. 13 and 14. Equation 11 also 

 shows that there will be a mean pressure gradient 

 across the wave propagation direction, proportional 

 to a . This is a nonlinear effect of the presence 

 of waves. 



4. DISCUSSION 



The equivalent to a linear dispersion relation con- 

 sists of Eqs. 8, 13, and 14, relating particle fre- 

 quency, a, decay direction angle, 6, horizontal 

 wavenumber, k, and decay parameter, m, with f, a, 

 and U as parameters . 



Note that the introduction of a mean drift veloc- 

 ity, U, has a now-trivial effect on dispersion, as 

 can be seen from Eq. 14, where the effect is not a 

 simple Doppler shift in frequency. The equations 

 of rotating fluids are not invariant to Galilean 

 transformations. Also note that the dispersion is 

 independent of the amplitude parameter, a; this is 

 an unexpected result for non-linear waves. But the 

 amplitude of particle motion parallel to oy is really 

 a exp[2mR(s)], where R is the value of r at the sur- 

 face. Since m is found from the equations involved 

 in determining dispersion, one cannot really claim 

 that dispersion is independent of amplitude. 



With the dependence on phase, 9, eliminated in 

 Eqs. 10, 11, and 12 by satisfying the dispersion 

 relations, one can see that the mean surface slope 

 across the wave propagation direction will vary with 

 wave amplitude and with y- position. 



As pointed out by Dubreil-Jacotin (1932) , and 

 later by Yih (1966) the results are valid for a 

 fluid of arbitrary stable density stratification. 



The solutions given here can be further extended 

 to replace the free surface by an interface between 

 the given flow field and a homogeneous wave trapped 

 fluid, giving the gravitational billows described 

 elsewhere [Mollo-Christensen (1978)]. This will re- 

 place the acceleration of gravity, g, by g' = 

 g(Ap/p), where Apis the density difference between 

 the two fluids and p the density of the lower fluid 

 at the interface. 



Similarly, the flow field at the off-shore or 

 inside end may be bounded by a field of geostrophic 

 billows or a combination of gravitational and geo- 

 strophic billows [see Mollo-Christensen (1978)]. 



I 



FIGURE 2. High-passed and contrast enhanced satellite 

 infrared images from January 27, 1975, at 1600, 1700, 

 and 1800 hrs. , GMT. Florida on the right side. Gulf 

 Coast on top. 



