But setting t 0 in 427 yields a0 an cosn bn sinn

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Unformatted text preview: x/L) +[a2 cos(2cπt/L) + b2 sin(2cπt/L)] sin(2πx/L) + . . . . (4.21) (4.22) The vibration of the string is a superposition of a fundamental mode which has frequency T /ρ cπ 1 c = = , L 2π 2L 2L and higher modes which have frequencies which are exact integer multiples of this frequency. Step II consists of determining the constants an and bn in (4.22) so that the initial conditions u(x, 0) = h1 (x) and ∂u (x, 0) = h2 (x) ∂t are satisfied. Setting t = 0 in (4.22) yields h1 (x) = u(x, 0) = a1 sin(πx/L) + a2 sin(2πx/L) + . . . , so we see that the an ’s are the coefficients in the Fourier sine series of h1 . If we differentiate equation(4.22) with respect to t, we find that ∂u −cπ cπ (x, t) = [ a1 sin(cπt/L) + b1 cos(cπt/L)] sin(πx/L) ∂t L L 100 −2cπ cπ [a2 sin(2cπt/L) + b2 sin(2cπt/L)] cos(2πx/L) + . . . , L L and setting t = 0 yields + h2 (x) = ∂u 2cπ cπ (x, 0) = b1 sin(πx/L) + b2 sin(2πx/L) + . . . . ∂t L L We conclude that ncπ bn = the n-th coefficient in the Fourier s...
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