$2.$ One end of a string is attached to a device that vibrates at the fundamental frequency of the string $\backslash ($ f$\_\{$a$\}=220\backslash$mathrm$\{$~Hz$\}\backslash ).$ This produces transverse waves in the string at the same frequency. The other end of the string passes over a pulley, and the tension can be varied by attaching weights to this end. The string has approximate nodes next to both the vibrating device and the pulley.

$($a$)$ If the string has a linear mass density of $\backslash (\backslash$rho$=1\mathrm{.}4\backslash$mathrm$\{$~g$\}/\backslash$mathrm$\{$m$\}\backslash )$ and is $\backslash ($ L$=6\mathrm{.}5\backslash$mathrm$\{$~m$\}\backslash )$ long from the vibrating device to the pulley, what must be the tension in the string to vibrate in its fundamental mode?

$($b$)$ Find the tension necessary for the string to vibrate in its second, third and fourth harmonics?

$($c$)$ Draw the first three harmonics.