Show that the time derivatives, $v= \frac {\partial u} { \partial t}$, of any solution to the wave equation is also a solution. If you know the initial conditions of $u$, what initial conditions does $v$ satisfy?
I am having a hard time with this question because I do not get what are they are asking me to do. I am confused with the question itself.
You are expected to state the wave equation and assume you have a solution $u(x,t)$ (or maybe in more dimensions, but do the 1D case first). Now define $v(x,t)=\frac {\partial u(x,t)}{\partial t}$. Take the partial derivative of the wave equation with respect to $t$. You should find some terms that you can argue are $v(x,t)$ and note that the equation they satisfy is the wave equation.