Methods | Part IB, 2003

The transverse displacement y(x,t)y(x, t) of a stretched string clamped at its ends x=0,lx=0, l satisfies the equation

2yt2=c22yx22kyt,y(x,0)=0,yt(x,0)=δ(xa)\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}-2 k \frac{\partial y}{\partial t}, \quad y(x, 0)=0, \frac{\partial y}{\partial t}(x, 0)=\delta(x-a)

where c>0c>0 is the wave velocity, and k>0k>0 is the damping coefficient. The initial conditions correspond to a sharp blow at x=ax=a at time t=0t=0.

(a) Show that the subsequent motion of the string is given by

y(x,t)=1αn2k2n2ektsinαnacsinαnxcsin/(αn2k2t)y(x, t)=\frac{1}{\sqrt{\alpha_{n}^{2}-k^{2}}} \sum_{n} 2 e^{-k t} \sin \frac{\alpha_{n} a}{c} \sin \frac{\alpha_{n} x}{c} \sin /\left(\sqrt{\alpha_{n}^{2}-k^{2}} t\right)

where αn=πcn/l\alpha_{n}=\pi c n / l.

(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?

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