Consider the following problem of vibration of a string. A tightly stretched string with fixed end points $x=0$ and $x=L$ is initially in a position given by $y(x,0) = f(x)$. It is released from this rest position and allowed to vibrate. The mathematical representation of this problem which depicts the displacement of the string $y(x,t)$ at different times for different values of $x, 0 \le x \le L$ is