We consider the system of partial differential equations on bounded domains, known in the literature as the Whitham–Broer–Kaup system. The well-posedness of the problem, under suitable boundary conditions, is addressed, and it is shown to depend on the sign of the number $\varkappa = \alpha-\beta^2.$ In particular, existence and uniqueness occur if and only if $\varkappa > 0$. In which case, an explicit representation for the solutions is given. Nonetheless, for the case $\varkappa \leq 0$ we have uniqueness in the class of strong solutions, and sufficient conditions to guarantee exponential instability are provided.