An analytical treatment is given of the problem of the establishment of the flow through a dam or levee with a horizontal underdrain, when the head behind it is raised and then kept at a constant value. The essential idea employed in the analysis is to consider the unsteady flow as a time-dependent perturbation of the final steady flow. The unsteady potential ϕ(x, y, t) is expanded in a power series of e−λt, of the form $\phi (x,y,t) = \phi_0(x,y) + \phi_1(x,y)e^ {\lambda t} + O(e^{-2\lambda t})$ where ϕ0(x, y) is the known steady-state potential, ϕ1(x, y) is a perturbation potential and O(e−2λt) = ϕ2(x, y) e−2δt + ϕ3(x, y) e−3λt + …. Each of the terms ϕn(x, y)e−nλt can be thought of as being a perturbation term of its precursor in the series, and the present approach is limited to the computation of the first perturbation term ϕ1(x, y)e−λt.

It is shown that ϕ1 satisfies Laplace's equation ∇2ϕ1 = 0 in a dimensionless hodograph plane. The free-boundary condition is linear but complicated, containing the eigenvalue λ, which is fixed by a determinantal equation. The amplitude of the displacement of the free surface is left undetermined; only the mode of the motion and the eigenvalue are computed. The results of a numerical example are summarized.