I have problem solving below wave equation using Fourier transform.
$U_{tt} = U_{xx} , 0<x<1, t>0$
$U(0,t) = U(1,t) = 0$
$U(x,0) = x , U_t(x,0) = 0$
Firstly (as I learned) Fourier sine transform formula is:
$F_s[u(x,t)]=\int_0^{\infty}u(x,t)sin(\lambda x)dx$
while solving the above problem, I had to transform initial data using F_sine transform as well. but i don't know what should my integral boundaries be? $\int_0^{1} or \int_0^{\infty}$? what about reverse Fourier boundaries? which is another integral like $u(x,y) = \int_0^\infty F[u(x,t)]sin(\lambda x)d\lambda$
Usually I see the wave problem which needed to solve with FT to be Unlimited on one side like $0<x$ but here i don't know what happens to formulas of $F[u_{xx}]$ and my initial data as well.