B$.$ A cable of length $\backslash ($ l $\backslash )$ and mass $\backslash ($ r $\backslash )$ per unit length is stretched under a tension $\backslash ($ P $\backslash ).$ One end of the cable is connected to a mass $\backslash ($ m $\backslash ),$ which can move in a frictionless slot, and the other end is fastened to a spring of stiffness $\backslash ($ k $\backslash ),$ as shown in figure.

$-$ Derive the frequency equation for the transverse vibration of the cable;

$-$ Discuss the effects of the mass $\backslash ($ m $\backslash )$ and stiffness $\backslash ($ k $\backslash )$ on the first natural frequency.