. Graphical and mechanical computation . esent this curve approximately by a series ofthe above form, containing a finite number of terms, and to find theapproximate values of the coefficients ak and bk. The following sectionswill give some of the methods employed to determine these coefficients. 86. The fundamental and the harmonics of a trigonometric series. —In Fig. 86a we have drawn the curves y = ax cos x, y = bx sin x, andy = fli cos x -\- b\ sin x. The maximum height or amplitude of y = #i cos x is ax and theperiod is 2 t. The amplitude of y = b\ sin x is bi and the period is 2 we
. Graphical and mechanical computation . esent this curve approximately by a series ofthe above form, containing a finite number of terms, and to find theapproximate values of the coefficients ak and bk. The following sectionswill give some of the methods employed to determine these coefficients. 86. The fundamental and the harmonics of a trigonometric series. —In Fig. 86a we have drawn the curves y = ax cos x, y = bx sin x, andy = fli cos x -\- b\ sin x. The maximum height or amplitude of y = #i cos x is ax and theperiod is 2 t. The amplitude of y = b\ sin x is bi and the period is 2 we may write y = ai cos x + bi sin x = Vai2 + be . l = sin x H .. cosx , 170 Art. 86 and letting Vaf + bi2 = cuwe may write TRIGONOMETRIC SERIES \V + bx\ = cos fa, a>\ y/af+W 171 = sin fa, y = d sin (x + <£i), where Ci= Vax2 + &r, fa = tan ! , Here Ci is the amplitude and fa is called the phase. The wave rep-resented by y = C\ sin (x + fa) is called the fundamental wave andy = a\ cos x, y = bi sin x are called its Fig. 86a. Similarly, we may represent y = ak cos &x, y = bk sin &x,and y = fl/t cos kx -f- && sin &x = c& sin (&x + fa), where a = Vak2 + 6fc2 and <fo = tan-1 ak/bk. The wave represented by y = c* sin (&x + $*.) is called the &th har-monic, its amplitude is ck, its phase is fa, its period is 2 ir/k, since sin K*+¥ + = sin [fcx + 2 7r + 0A] = sin (&x + fa), and its frequency, or the number of complete waves in the interval 2 t,is k. The trigonometric series is often written in the form y = Co-f-Cisin (x+fa) +c2sin (2x + fa) + ? ? ? + cn sin (rax + fa) + ? • • ,showing explicitly the expressions for the fundamental wave and thesuccessive harmonics. The more complex wave represented by thisexpression may be built up by a combination of the waves represented bythe various harmonics. Fig. 866 shows how the wave for the equation y = 2 sin (x + ^J + sin Ux- ^j + h sin (3 x + ^J , or ^S , ^2 y- . . V2 . y = cos x cos 2 x i cos 3 x-f- V
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