600 ROSSITER [chap. 16 



for variations of many kinds, yet its distribution and magnitude cannot yet be 

 said to have been determined analytically. The development of the tide- 

 generating potential indicates the uncorrected equilibrium form to be given by 



18.5 (sin2 A-i) cos N mm, (2) 



where A denotes latitude and N is the mean longitude of the moon's ascending 

 node. The above expression includes the factor 1+k — h, taken as 0.7. Antici- 

 pating the analytical results presented in section 4 of this Chapter, it is necessary 

 to demonstrate that perturbations of the nodal tide can exist under certain 

 circumstances. 



The non-linear terms occurring in the hydrodynamical equations governing 

 tidal motion give rise to higher species of tides ; especially is this the case in 

 shallow water. The particular case of a closed rectangular basin has been 

 examined by Proudman (1952, §145), and in general it may be stated that a 

 single constituent (such as M2) will generate its first harmonic (M4) and a 

 "constant" term. It can be shown from the following greatly simplified con- 

 siderations that the "constant" term in fact contains a truly constant term 

 plus a slowly varying term with a period of 18.6 years. 



In the Admiralty Manual of Tides (Doodson and Warburg, 1941) it is demon- 

 strated that the quarter-diurnal tide is approximately proportional to the 

 square of the semidiurnal tide. Now, 



(M2)2 = (/of M2)2^2cos2(F + W-g) 



= (/of M2)2iy2|i + cos2(F + w-g)}, (3) 



where/ and u are the nodal terms, H and g are the harmonic constants of M2. 

 If M4* be the amplitude of that portion of M4 given by the second term in the 

 bracket in equation (3), then the first term is given by (/ of M4)M4*, since/ of 

 M4 = (/of M2)2. Now, the/of M4= 1 —0.074 cos iV, and hence the above suggests 

 a truly constant contribution to mean level equal to M4* and a perturbation of 

 the nodal tide amounting to O.O74M4*. This would affect the amplitude but 

 not the phase of the equilibrium nodal tide. Similar perturbations would be 

 generated from many of the other primary tidal constituents. M4* cannot be 

 determined from an analysis of observed tides, but for the special case of a 

 standing oscillation in a narrow gulf Doodson (1956) has obtained results 

 which suggest that the truly constant contribution equal to M4* can be ap- 

 preciable. It must therefore be admitted that the accompanying nodal per- 

 turbation could be of the same order of magnitude as the equilibrium tide. 



The generation of nodal perturbations is not exhausted by the foregoing 

 remarks. Certain tidal constituents can interact by virtue of particular proper- 

 ties of their angular speeds and nodal factors. Such a group is M2, Ki and Oi, 

 but it must be admitted that no estimate can be made of the magnitude of such 

 perturbations. 



It must, therefore, be concluded that observational data cannot be expected, 

 in general, to provide confirmation of an equilibrium nodal tide, even if it is 

 corrected for the presence of land-masses and a yielding earth. 



