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Separated solution of the damped string equation

For initial sine coefficient

a_{n}

and zero initial velocity, put

ω_{n}^{2} = n^{2} π^{2} - 1/4

. The modal amplitude is

T_{n} (t) = a_{n} e^{- t /2} (cos (ω_{n} t) + \frac{s i n ( ω _{n} t )}{2 ω _{n}}) .

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