Does the wave equation require an initial function for one of its derivative?

Question

Is it possible to find an explicit solution to the wave equation: $$ \partial_t^2u-c^2 \partial_x^2 u=0 \\ u(x,0)=f(x), \ u(cx,x)=g(x) $$ or do we need information about a derivative of $u$ as well?

Answer 1

The method of solving is shown below. The solution is : $$u(x,t)=f(x-ct)+g\left(\frac{x+ct}{2c}\right)-g\left(\frac{x-ct}{2c}\right)$$ So, the solution doesn't need more information to be determined.

Note : An obvious mistake in my first edition has been corrected.