The middle$-$C string $($f$=256$Hz$)$ of a guitar is plucked, and the sound becomes

inaudible after approximately $2$ seconds. Assume that the transfer function of the

string can be modeled as a standard second order system without zeros. The settling

time is approximately equal to the time for the plucked string to become inaudible.

Note that the angular frequency $\backslash$omega in radian per second $($ra$($d$)/($s$))$ is equal to $2\backslash$pi times

the frequency f in hertz $($Hz$)$:$\backslash$omega ra$($d$)/($s$)=2\backslash$pi fHz$.$

$($a$)$ Find the approximate value of the damping ratio $\backslash$zeta of the string.

$($b$)$ Find the approximate value of the natural frequency $\backslash$omega $\_($n$)$ of the string.

$($c$)$ Write the transfer function of the string.