A string has endpoints which are fixed at \(x = 0\) and \(x = L\) . At \(t = 0\) , the string is hit at \(x = a\) so it starts vibrating:
\[ y ( x , 0 ) = 0 \qquad \partial _ { t } y ( x , 0 ) = L v _ { 0 } \delta ( x - a )\tag{42} \]
where y is the amplitude of the oscillations (assume the amplitude to be small and the wave velocity to be v). Find \(y ( x , t )\)