Differential Equations for Engineers · Linear higher-order equations and oscillatory response
A taut string of length 6 occupies 0 ≤ x ≤ 6
Problem
A taut string of length 6 occupies \(0 \leq x \leq 6\). The transverse displacement \(w(x,t)\) satisfies \(w_{tt} = 16 w_{xx}\) for \(t > 0\), with fixed ends \(w(0,t) = 0\) and \(w(6,t) = 0\). The initial displacement is \(w(x,0) = 2\sin(\pi x/6)\) and the initial velocity is \(w_t(x,0) = 8\sin(2\pi x/6)\). Using separation of variables, find \(w(x,t)\) for \(t > 0\).
Hint
Fixed ends on a string give sine eigenfunctions; each mode is a time oscillator at frequency \(c n\pi/L\).
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