I want to verify that $$w(x,t)=\frac{1}{2c}\iint_Df(y,\tau)dyd\tau$$ is the solution of the problem $$w_{tt}=c^2w_{xx}+f(x,t) , \ \ x>0, t>0 \\ w(x,0)=w_t(x,0)=0, \ \ x>0 \\ w(0,t)=0 , \ \ t\geq 0$$
where $D$ is the following set: 
$$$$
Do we have the following bounds at the limits? $$w(x,t)=\frac{1}{2c}\int_0^t\int_{c(t-\tau)-x}^{x+c(t-\tau)}f(y,\tau)dyd\tau$$ Or how is $D$ defined?