The periodicity condition f 2 f implies that n

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Unformatted text preview: s once again we need to find the nontrivial solutions to the boundary value problem, d2 f (x) = 2 (f (x)) = λf (x), f (0) = 0 = f (L), dx 99 and just as before, we find that the the only nontrivial solutions are constant multiples of f (x) = sin(nπx/L), with λ = −(nπ/L)2 , n = 1, 2, 3, . . . . For each of these solutions, we need to find a corresponding g (t) solving equation (4.19), g (t) = −(nπ/L)2 c2 g (t), or g (t) + (nπ/L)2 c2 g (t) = 0. This is just the equation of simple harmonic motion, and has the general solution g (t) = a cos(ncπt/L) + b sin(ncπt/L), where a and b are constants of integration. Thus we find that the nontrivial product solutions to the wave equation together with the homogeneous boundary condition u(0, t) = 0 = u(L, t) are constant multiples of un (x, t) = [an cos(ncπt/L) + bn sin(ncπt/L)] sin(nπx/L). The general solution to the wave equation together with this boundary condition is an arbitrary superposition of these product solutions: u(x, t) = [a1 cos(cπt/L) + b1 sin(cπt/L)] sin(...
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