D'Alembert formula with initial velocity
= D'Alembert formula with initial velocity
{c}
For $u_{tt}-u_{xx}=0$ with $u(0,x)=0$ and $u_t(0,x)=f(x)$,
$$
u(t,x)=\frac12\int_{x-t}^{x+t}f(y)\,dy.
$$
= D'Alembert formula with initial velocity
{c}
For $u_{tt}-u_{xx}=0$ with $u(0,x)=0$ and $u_t(0,x)=f(x)$,
$$
u(t,x)=\frac12\int_{x-t}^{x+t}f(y)\,dy.
$$