D'Alembert's solution for the one-dimensional wave equation for a semi-infinite string.

Find the solution to:

PDE: u*xx-c(to the power of negative k)u*tt=0 , 0
ICs: u(x,0)=f(x) and u*t(x,0)=g(x), x>=0

BC: u*x(0,t)=0 , t>=0

This BC corresponds to a string with its end point free to move in a vertical direction.

I will use * to symbolize a partial derivative. For example, u*x denotes the partial derivative of u with repsect to x. Here is the problem: (also note that the PDE is the one-dimensional wave equation)