D'alembert for semi-infinite string (on R-)

[Lpv]

D'Alembert problem for semi infinite string on R-

u_tt=c² u_xx ( -$$\infty$$<x<0)

Initial condition:
u(x,0)=f(x)
u_t(x,0)=g(x)

Boundary condition:
u(0,t)=0

please help me to solve it

 

[Dr.D]

Start by assuming a separable form for the solution.