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54 > = , sin (0)1 + 6.) + 2 sin ( >2 + 62) Each set of characteristic variles, ,, ft,, and . <U2, can be different, but both waves art measured at the same point at the same time. Consider an example that lets the phase constants bj and 62 be zero: y, = 5 sin 4 Figure S6 shoTra the individual regular waves yi and jj. and the compound wave over the interval 0° to 180° Note that », andjj began the normal upward cycle at O"; but by 180°, they are perfectly outofphase. During the next 180". the two waves come back into phase. When combining periodic waves, it is useful to know the maximum and miiumum amplitude of the resulting wave. Because the peaks of the two elementacy waves do not necessarily 1 at the same point, the mazdmum amplitude of either wave may not be reached. A mathematical technique called differenlialKyn. is used to find the maximum and minimum amplitudes. The first derivative, with respect to angle , is written dy/d or y, where >> is the firmula to he differentiated The rules are: FIGURE 8t Compound line w
 (sin ) = cos ;  (cos ) = sin di)  (Sin (Ou) = to cos 0) d<9  (sin (< + fc)) = to COS (0) + fc)  ( , sin (tu,(t) + 61) + 2 sin (0) + &2)) = , , cos («1), + + aibh cos ( + Applying this method to the previous example, > = 381 4 + 51 5 dv ~ =y = 12 cos 4 + 30 cos 5 The points of maximum and/or minimum value occur wheny  0 For;" = 12 cos 4 . the maxima and minima occur wben 4 = 90° and 270= ( = 22/;" and 67!") (Figure 86). ForJ = 30 cos 5 . the maximum and minimum values occur at 5 = 90" and 270° ( = 18 and 54°). It must be pointed out that the first derivative identifies the location of the extreme highs and lows, but does not tell which one is the maximum and which is the minimum. The second derivative, , calculated by taking the derivative ofy. is used for this purpose as follows: (») 0 andy(*) > 0, thentx) te a minimum. Ify(*) = 0 andy(*) < 0, then ( :) is a majdmum. Tben,>>, = ZZA" and>2 = 18° are maxima and, = 67!;" and = 54" are minima. Anyone interested in pursuing the analysis of extrema will find more complete discussions in a text on calculus. Rather than concentrating on these theoretical aspects of curves, consider a practical example of finding a cycle in the price of scrap copper, shown in Table 81 and charted in Figure fl?. The price peaks seem evenly spaced, cKcurring at mid1966, January 1970, and January 1974, about 4 years apart. The soluttons to these problems are tedious; therefore, calculations will be performed using the computer programs in Appendix 4. The results obtained by using actual copper ptices will not be as dear as using fictitious dau. It is important to be able to understand the significance of practical results and apply them eefectively Because trigonometnc curves fluctuate above and below a honzonial line of value zero, the first step Is to detrend the dau usig the leastsquares method. This results in the equation for a straight line representing the upward bias of the data. The value of the detrending line Is then subtracted from the original dau to produce copper prices that vary equally above and below the line from positive to negative values. The straight linea + bx, which best represents the trend, can be found by solving the leastsquares equations: •n of mgonometnc curve fining can be found in Claude Ckcion, Tbe Art oflndepen
TABLE 81 Dealers Bujing Price, No. 2 Copper Scrap at NewYork* Average Quantily Price ( ) Year  l«     1963  2112  22.46  22,17  22,00  1964  23.18  24.56  25.57  3059  1965  28.23  33.77  35,90  4005  1966  4622  51.48  4076  4016  1967  36.51  29J0  3036  3642  1968  39.7S  30.07  29.08  32.13  1969  38.94  42.95  43.36  46.23  1970  47.70  46.98  3578  2735  1971  2540   27.15  28.48  1972  32.74  33.53  30 1  29.25  1973  36.82  45.07  55.13  65.51  1974  66.56  70,06  47JO  35,62  1975  32.06  31.46  35.75  36.46  1976  3822  43.24  45.46  38.96  1977  37j08  38.72  34 1   1978  35.07  40.23  41.63  44.95  1979  51.12  63.71  59.56  63.38  4iiedo  n from Amtncan Metal Mart<et    
To do this, let be the date and be the price on that date. For convenience, instead of lettingjc=1967,1967!S. 19 7 ,.. ., let := 1, 2,3, The solution, using a computer program in Appendix 2 (also an integral part of the programs in Appendix 4) or the handcciilation method, is j= 28.89+ .267x Figure 8>7 displays the original copper pnces and the regression line. The onginal prices can now be detrended using the equation above, subtracting the line values firom the corresponding prices. Complete stepbystep results for this example can be found in Appendix 4. The detrended data is now used in the general trigonometric singlefrequency = e cos cof +1> sin va The variable t replaces to express the angle in integer units rather than in degrees. This will be more convenient to lisualize and to chart. To find the frequency (O. It will be necessary to first solve the equation: cos(oa=o using the system of equations, FIGURE 87 Copper prices 19631979.
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