Analysis of alternating-current waves by the method of Fourier with special reference to methods of facilitating the computations . ng use of the lemma ^ sin X + sin 2jc + sin 3^ + . . sm-sm(^ + i)- + sin (^ —i)rt; 4-sin^jc=i ^^^ . X sm - 2 the factor by which B^ is multiplied is also seen to be equal to zero. 7 Byerlys Fouriers Series and Spherical Harmonics, p. 32. 576 Bulletin of the Bureau of Stmidards Woi. There consequently remains in the second member only oneterm 2Afe^m=i sm-^^ = AJ (2n-i)-2^=, cos-^J which by (4) reduces to 2nAji, and we have finally 27^Afc = 22^=, y^ sm ——or (6) To o
Analysis of alternating-current waves by the method of Fourier with special reference to methods of facilitating the computations . ng use of the lemma ^ sin X + sin 2jc + sin 3^ + . . sm-sm(^ + i)- + sin (^ —i)rt; 4-sin^jc=i ^^^ . X sm - 2 the factor by which B^ is multiplied is also seen to be equal to zero. 7 Byerlys Fouriers Series and Spherical Harmonics, p. 32. 576 Bulletin of the Bureau of Stmidards Woi. There consequently remains in the second member only oneterm 2Afe^m=i sm-^^ = AJ (2n-i)-2^=, cos-^J which by (4) reduces to 2nAji, and we have finally 27^Afc = 22^=, y^ sm ——or (6) To obtain Bj, we multiply both sides of the equation for yo by2, the equation for y^ by 2 cos —, the equation for y^ by 2 cos —-etc., and add these equations. The first member of the resultingequation is 22,^=^0 ym cos . In the second member, the coefficient of ^^ is2 S^-! sm cos , ^-^ 2n 271 which we have already shown reduces to zero. The general coefficient Br, where r does not have the value k,is affected by the factor vw=2«-i mkTz mrn 2S^=o cos cos ° 2n 2n = 2+2 >W = 2» — cos—^^ ^+cos—^^ — L 2n 2^ J. which by use of the lemma (4) can be shown to be equal to remains therefore in the second member, one term only, 2B]X!L=T cos^ = 2nBT^ and the value of Bj, follows at once. 271/ It has been shown, therefore, that the alternating wave can berepresented by the finite series (2) at 271 points of the half wave,and that the coefficients in (2) are capable of calculation from the2n ordinates of these points, by the simple relations Grover] Aualysis of Alternating-CuYreut Waves 577 The coefficients of the sine terms in (2) are therefore foimd bytaking the averages of the measured ordinates of the cm-ve, eachordinate having been multipUed by the sine of an appropriatemultiple of that angle which indicates the position of the ordinatein question. Similarly, the coefficients Bj^ of the cosine terms arefound by averaging the products of the measured ordinat
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectphysics, bookyear1913
