. Graphical and mechanical computation . posite to that with which yroccurs, so that, an + CLZn + • * * = — (yo - yi + ? • • • ± yr • • • - yo+* + yi+* - 1 / = — (2 y0 - 2fl 2* + • • ±2yr ? = -(yo- yi n + • • • ± yr • • • )• =F yr+, ...) Hence we need merely divide the half-period into n equal intervals andaverage n ordinates. We may therefore restate our rules for determiningthe coefficients if the wave contains odd harmonics only. If, starting at x = o, we measure n ordinates at intervals of r/n, theaverage of these ordinates taken alternately plus and minus is equal to thesum of the am
. Graphical and mechanical computation . posite to that with which yroccurs, so that, an + CLZn + • * * = — (yo - yi + ? • • • ± yr • • • - yo+* + yi+* - 1 / = — (2 y0 - 2fl 2* + • • ±2yr ? = -(yo- yi n + • • • ± yr • • • )• =F yr+, ...) Hence we need merely divide the half-period into n equal intervals andaverage n ordinates. We may therefore restate our rules for determiningthe coefficients if the wave contains odd harmonics only. If, starting at x = o, we measure n ordinates at intervals of r/n, theaverage of these ordinates taken alternately plus and minus is equal to thesum of the amplitudes of the nth, 3 nth, 5 nth, . . cosine components. If, starting at x = 71-/2 n, we measure n ordinates at intervals of r/n, theaverage of these ordinates taken alternately plus and minus is equal to thesum of the amplitudes, taken alternately plus and minus, of the nth, 3 nth,5 nth, . . sine components. Furthermore, a0 = o since the sum of the ordinates over the entire periodis Example. Assuming that the symmetric wave of Fig. 92 contains nohigher harmonics than the 5th, we are to determine the 1st, 3d, and 5thharmonics. Applying the above rules we have 200 EMPIRICAL FORMULAS — PERIODIC CURVES Chap. VII 05 = i (jo — ^36 + yn — ym + ym) = Ho — + — + 19-0) = — fa = I (yi8-ya+yw-ym+yi62)= b (++) = = 3 (Jo - yeo + ym) = Mo - 2-8 + ) = = % (y30 - y™ + ywo) = \ ( - + ) = + a3 + a5 = j (y0) = o, .. ax = 0i - fa + 65 = T (yn) = , /. fa = + Result: y = — cos x + cos 3 x — cos 5 #+ sin x + sin 3 x + sin 5 x. We may compare this result with that obtained for the same curveby the use of the computing form on p. 187. If only the 1st and 3d harmonics had been present in the above wave,we should have a3 = \ (y0 - ym + ym); fa = i (yso — J90 + yuo); 0i + a3 — yo — o; fa — fa = y90. If all t
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