The flow rate in a rectangular channel with a width of $1$ m is $0\mathrm{.}5$ m$3/$s and the water depth is $0\mathrm{.}8$ m$.(1)$

Find the corresponding water depth that can flow with the same specific energy as the specific energy for this flow rate,

$(2)$ the limiting water depth for this flow, and $(3)$ the ratio of the specific energy after the diversion to the specific energy before the diversion. Here, the kinetic energy correction factor is $1\mathrm{.}005.$