HAEMONIC A]SrALYSIS AISTD PREDICTION OF TIDES. 59 



18. OVERTIDES. 



In the development of the equilibrium theory the absence of 

 friction and sufficiency of depth were assumed. Under these condi- 

 tions each term of the result represented a simple harmonic Vvave. 

 When a wave runs into shallow water, the trough is retarded more 

 than its crest, so that the duration of rise of the tide becomes some- 

 what less than the duration of fall and the wave loses its simple 

 harmonic form. We may, however, represent this modified form of 

 the wave by the introduction of a series of components whose speeds 

 are simple multiples of the speed of the fundamental astronomical 

 tides. These are called overtides because of their analogy to over- 

 tones in musical sounds. The only overtides usually considered in 

 the analysis are those for the principal lunar and solar components 

 M2 and S2 [Terms (A)^ and (-B)i of formulas (100) and (215), and are 

 designated by M^, Mg, Mg, and S^ and Sg, the subscript indicating the 

 number of periods in a component day. 



The arguments of the overtides are taken as exact multiples of the 

 argument of the fundamental tide. There is no theoretical expres- 

 sion for the coefficients of these tides, but it is probable that the 

 amplitudes as determined from observations will be subject to 

 variations due to changes in the longitude of the moon's node analogous 

 to the variations in the fundamental tide. It is assumed that the 

 variability of the overtides may be represented by the square, cube, 

 fourth power, etc., of the fundamental tides, and the factors of 

 reduction are taken accordingly. 



Thus, 



FoiM.,= {FoiM,y (242) 



FoiM,= {FoiM,r (243) 



FoiM,= {FoiU^y (244) 



The F of S4 and F oi Sg are taken as unity. 



19. COMPOUND TIDES. 



Compound tides are components whose speeds are the sums or 

 differences of the speeds of the elementary components. They were 

 suggested by Helmholtz's theory of sound waves, and, like the over- 

 tides, are due to shallow water. 



The arguments of the compound tides are taken as the sums and 

 differences of the elementary tides. 



Thus, 



Arg. (MS) 4 =Arg. Ma + Arg. S, (245) 



Arg. (MN)4 =ATg. Mg+Arg. N2 (246) 



Arg. (MK)3 =Arg. M^ + Arg. K, (247) 



Arg. <2MK)3 = Arg. M,-Arg. Kj (248) 



Arg. .(2SM)2 =Arg. S^-Arg. M2 (249) 



Also, 



Arg. (2MS) 2 = Arg. M4 - Arg. S2 = Equilibrium Arg. fi^ + (26 -2u) (250) 



Arg. MSf = Arg. S2 - Arg. M^ ^ Equilibrium Arg. MSf -{2$-2u) (251) 

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