$10$B$-1($a$)$ Find the normal modes of the wave equation on $0$ x $$ $/2,$ t $0$ given by $2$t$2$ u $=$ c $22$x$2$ u$,$ u$(0,$ t$)=$ u$($$/2,$ t$)=0,$ t $>0($b$)$ If the solution in part $($a$)$ represents a vibrating string, then what frequencies will you hear if it is plucked? $($c$)$ If the length of the string is longer$/$shorter what happens to the sound? $($d$)$ When you tighten the string of a musical instrument such as a guitar, piano, or cello, the note gets higher. What has changed in the differential equation?