$frac{partial^2u}{partial t^2}=cfrac{partial^2u}{partial x^2}$ $~~~~~~xin(0,L)$

$c=1$ when $xgeqfrac{L}{10}$$~~~~~~c=0.3$ when $x<frac{L}{10}$

IC1 $u(x,0)=0$

IC2 $frac{partial u(x, 0)}{partial t}=0$

BC1=$u(L,t)=0$

BC2=$frac{partial u(0, t)}{partial t}=1$

If you can provide a detailed solution for $frac{partial u}{partial t}$, thank you very much!