THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

4.8 The Effect of Initial Conditions 79In order that Equation (4.14) represents the string at all times, it also must describethe displacement at t = 0 and therefore is written asy 0 (x) = y(x, 0) =∞∑An sin k n x (4.15)The derivative of y with respect to time must also represent the velocity at t = 0,v 0 (x) = v(x, 0) =n=1∞∑ω n B n sin k n x (4.16)We apply Fourier’s theorem 1 to Equation (4.14) in order to obtain∫ LAn = 2 y 0 (x) sin k n xdx (4. 17a)L 0and then apply the theorem to Equation (4.16) to getn=1Bn = 2ω n L∫ L0v 0 (x) sin k n xdx(4.17b)Example 2: String Pulled and Suddenly Released. Consider a string that isplucked by pulling it at its center a distance d and then is suddenly released atinstant t = 0. In such a case, v 0 (x) = 0 and all the coefficients B n will be zero.The coefficients A n are given byA n = 2 L[ ∫ L202dxL= 8dn 2 π 2 sin nπ 2∫ Lsin k nxdx+ 2 dL L (L − x) sin kn dx2](4.18)1 The theorem states that a complex vibration of period T can be represented by a displacement x = f (t)written in terms of a harmonic seriesx = f (t) = A 0 + A 1 cos ωt + A 2 cos 2ωt +···A n cos nωt +···+ B 1 sin ωt+ B 2 sin 2ωt +···B n sin nωt + ...where ω = 2π/T and the constants are given byA 0 = 1 ∫ Tf (t) dtT 0A n = 2 ∫ Tf (t) cos nω dtT 0B n = 2 ∫ Tf (t) sin nω dtT 0