To see what boundary conditions are natural to impose

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Unformatted text preview: and (4.35) is ∞ u(x, t) = fn (x) an cos( −λn t) + bn sin( −λn t) , n=0 where the an ’s and bn ’s are arbitrary constants. Each term in this sum represents one of the modes of oscillation of the vibrating string. In constrast to the case of constant density, it is usually not possible to find simple explicit eigenfunctions when the density varies. It is therefore usually necessary to use numerical methods. The simplest numerical method is the one outlined in §4.3. For 0 ≤ i ≤ n, we let xi = i/n and ui (t) = u(xi , t) = the displacement at xi at time t. Since u0 (t) = 0 = un (t) by the boundary conditions, the displacement at time t is approximated by u1 (t) u (t) , u(t) = 2 · un−1 (t) a vector-valued function of one variable. The partial derivative ∂2u ∂t2 is approximated by 112 d2 u , dt2 20 40 60 80 100 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 Figure 4.1: Shape of the lowest mode when ρ = 1/(x + .1). and as we saw in §4.3, the partial derivative ∂2u ∂x2 where is appro...
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