113 



tinuous expression for the friction term may be derived when the 

 velocity has the simple harmonic variation: 



v=B sin {at-\-^), 



(115) 



in which B is the maximum numerical value of v during the tidal cycle. 

 Designating the friction term as F, then: 



F=±(ByC'r) sin^ {at+^). 



(116) 



The positive sign is to be applied when (ai+jS) has values between 

 and IT, 2ir and Stt, 47r and Stt, etc. ; and the negative sign when {at-\-0) 

 has values between tt and 2ir, Stt and 47r, etc. 



The graph of such a function is shown by the solid line in figure 36. 



-fl 



-1 



Figure 36.— Graph of friction term of an harmonic current. 



225. Placing, for convenience, ai^+j8=cc, the function isin^ x is by 

 definition such that sin^ (— a;) = — sin- x. By Fourier's theorem it 

 should therefore be expressed by the series: 



Ax sin a;+^2 sin 2x-^Az sin 3x+ . . . yl^ sin ?ia; . . . (117) 



In which the coefficients, for values of x between and tt, are : 



Ax = (2/7r) I sin^ x sin x dx, ^2= (S/tt) I sin^ x sin 2x dx, 

 Jo Jo 



. . . An— (2 /it) I sin^ X sin nx dx. 

 Jo 



And, for values of x between tt and 27r, the coefficients are: 

 Ax=(2/Tr) I sin^ X sin x dx, Ao— {21 tt) I sin' x sin 2x dx, 



J -K J V 



J2-K 

 sin^ X sin nx dx 



