$(10\%)$ When diffusion system approaches complete homogenization such as in the diffusion out both sides of a slab, $c(x,t)$ can be represented by the first few terms of an infinite trigonometric series. And for all time beyond some period $t^{\text{'}},c(x,t)$ is given by a single sine wave. $($a$)$ Please illustrate how to determine $t^{\text{'}}.($b$)$ For a slab of thickness $h,$ what will be the time in using the first term to represent $c(x,t)$ with the error less than $1\%,$ where the diffusion coefficient is $D?$