SECKEL and YONG : HARMONIC FUNCTIONS 



F{t) is the observed temperature or salinity 

 at the time t. Furthermore, F{t) is known only 

 at finite intervals of time so that the above 

 Fourier integrals must be obtained by numer- 

 ical integration. This integration, approxi- 

 mating the area under the curves F{t) cos (nU) 

 and F{t) sin (?i<.>f), is performed by summing 

 areas of rectangles with height G(t) cos (wuO 

 or G{t) sin (no>t), and with width Af, the 

 sampling interval. 



The finite difference form of the Fourier in- 

 tegrals is 



A„ = 



and 



X G{l)iCos{nut)Ati. n = 0,1,2, ...k. 



B„ = — X G(l)iSm(nut)M,. n = 1,2,3, ...k. 



• i = \ 



The number of samples in the interval ( 

 io t = T \s m + 1, 







and G(t)i = ViiFU,) 4- F(/,. i)], / = 1,2,3, ...m. 



The time used to evaluate the geometric factor 

 is 1/2 (^' + i'-\)- Other schemes of obtaining 

 the best estimate of G(t) cos {riud) during the 

 interval Ai can be used but would not signifi- 

 cantly affect the results in our application (see 

 Kaplan, 1953: p. 168-172). 



Library programs for the evaluation of Four- 

 ier coefficients by computer usually require that 

 the sampling interval, A?, be constant. Since 

 this condition is not necessarily met in our ap- 

 plication, a more flexible computer program was 

 written to evaluate the coefficients. In this 

 program the sampling interval may vary, and 

 the number of samples for the basic period of 

 analysis need not be the same in each application. 



The Fourier coefficients evaluated in the 

 above manner enable us to describe anal.vtically 

 the temperature or salinity as a function of time. 

 If we wish to go further and gain insight into 

 the properties of the temperature or salinity 

 distribution, it is more useful to express the 



Fourier series as a sum of cosines: 



S„ (!) = — + X Cicos u(ni — a„), 



2 " n = 1,2,3 k. 



The transformation is accomplished by the use 

 of the trigonometric indentities 



A„ = C„ cos wa„, 

 B,, = C, sin uia.i. 



and 



C„=±{Al+ B^y\ 

 B„ 



ua„ — arctan - 



A, 



In the application described in this report 

 the fundamental period in the Fourier series is 

 the sampling duration or any portion of this 

 duration that may be arbitrarily chosen; the 

 amplitudes and phase angles do not necessarily 

 coincide with natural variations in temperature 

 or salinity; and the harmonic functions have no 

 predictive value. 



In some cases, such as the Koko Head tem- 

 peratures with a well-defined annual cycle, the 

 fundamental period of the Fourier series de- 

 rived for each year approximates the annual 

 cycle. At Christmas Island, however, an annual 

 temperature cycle is not always clearly apparent. 

 Despite the fact that choice of the fundamental 

 period may be arbitrary and may not coincide 

 with a naturally occurring period, the spectrum 

 is resolved beyond the first few harmonics. For 

 example, if the fundamental period, n — 1, is 12 

 months then the period of the first harmonic, 

 n = 2, is 6 months. A naturally occurring 9 

 months cycle in the observations would in this 

 case not be resolved. As n increases, however, 

 resolution improves to 4, 3, 2.4, 2, etc., months. 



The highest harmonic, or ?i-value, to which 

 harmonic analysis can be carried, is limited by 

 the number of observations. In the ideal case 

 and when samples are equally spaced in time, 

 there must be at least 2n observations, i.e., at 

 least two samples per cycle. In nature, where 

 we are dealing with noncyclical variations and 

 unequal spacing of samples a sinusoidal curve 

 cannot be resolved with only two samples, and 



183 



