Problem $1.(15$ points total$)$

Use the equation for a perfectly elastic string

${}_0\frac{de{l}^{2}u}{del{t}^2}=T_0\frac{de{l}^{2}u}{del{x}^2}+Q(x,t){}_0,00$

with fixed ends $u(0,t)=0,u(L,t)=0$ to

$($a$)(10$ points$)$ Compute the sagged equilibrium displacement $u_E(x)$ as a solution of a steady$-$

state problem. Sagged means that the string is only affected by gravity $Q(x)=-g$;

$($b$)(5$ points$)$ Show that if $u(x,t)$ satisfies $(1)$ then $v(x,t)=u(x,t)-u_E(x)$ is a solution of

the homogeneous wave equation

$\frac{de{l}^{2}v}{del{t}^2}=c^2\frac{de{l}^{2}v}{del{x}^2},00$

with sound speed $c=\sqrt[\phantom{2}]{\frac{T_0}{{}_0}}.$