Abstract:In the past two decades, a series of exact analytical solutions to the two kinds of depth-averaged equations in water waves, i.e., the linear long-wave equation and the mild-slope type equations, have been constructed. For the linear long-wave equation, if the bottom topography is idealized with the water depth being a power function, then the related analytical solution can be written in a closed form. If the bottom topography is quasi-idealized with the water depth being a power function plus a constant, then the related analytical solution can be expanded into a Taylor series or a Frobenius series. For the mild-slope type equations, a number of exact analytical solutions in the form of Taylor series are constructed recently, where the implicit modified mild-slope equation is successfully transformed into an explicit equation for both two-dimensional bathymetries and three-dimensional axisymmetric bathymetries with piecewise monotonicity and piecewise second-order smoothness. In this paper, these advances are summarized and reviewed. And some prospects of the research in the future are made