The Leap$-$Frog scheme for the wave equation

Suppose we are given with the problem $($which is the classical wave equation$)$

$\frac{de{l}^{2}u}{del{t}^2}-b^2\frac{de{l}^{2}u}{del{x}^2}=0,0x1.$

Assume we have Dirichlet boundary conditions

$u(0,t)=u(1,t)=0,t>0$

and initial conditions

$u(x,0)=0\mathrm{.}5x(1-x),\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}(x,0)=0,0x1$

$($a$)$ Explain why two initial conditions are needed.

$($b$)$ Develop the Leap$-$Frog scheme for $(4),$ also known as Central$-$Time$-$Central$-$Space $($CTCS$).$

What is the scheme's order of accuracy?

$($c$)$ Analyze the stability of the scheme using Fourier Analysis and find a criterion for it$.$