  The constant vorticity {\bf two-layer water wave} in the $\beta$-plane
approximation with centripetal forces is investigated in this paper. Different
from the works (Chu and Yang\cite[JDE, 2020]{chu} and Chu and Yang \cite[JDE,
2021]{chu2}) on the singe-layer wave flows, we consider the two-layer water
wave model containing a free surface and an interface. The interface separates
two layers with different features such as velocity field, pressure and
vorticity. We prove that if the change in pressure in the $y$-axis direction is
bounded, then the pressure is a function only related to depth and the surfaces
of the water flows. And the inner wave will not affect the pressure function,
if the water flow densities in each layer are equal. Furthermore, the explicit
expressions of the velocity, pressure are given for the two-layer water flows.
It is interesting that our method and results are also valid for the
multi-layer water waves. Let the number of layers of water waves $n$ tend to
infinity, we prove that the squence of pressure in the lowest layer
$\{P_1^n(x,y,z,t)\}_{n\geq1}$ is uniformly convergent, if the density of each
layer is bounded and the each surface of wave flows is uniformly convergent.
