The linearized equations for wave motion of frequency ω in a shallow, viscous liquid of variable depth h are reduced to a partial differential equation. [Lscr ]Z = 0, for the complex amplitude Z of the free-surface displacement on the assumptions of no slip at the bottom and Kh, Kδ* [Lt ] 1, where K = ω2/g, and δ* = (ν/2ω)½ is a viscous lengthscale. It is shown that capillarity must be included in order to avoid an irregular singular point (which would imply the total absorption of an incoming wave) at h = 0. [Lscr ]Z then is fourth-order and has a regular singular point of exponents 2, 1, 0, 0 for h ∼ σx ↓ 0. The requirements that the free-surface displacement and the shear force be bounded as h ↓ 0 rule out the solutions of exponent 0 and imply a stationary contact line. This last prediction is supported by laboratory observation but is not consistent with the observed runup of long, non-breaking waves on real beaches (for which the condition of no slip presumably must be relaxed). The dissipation for sufficiently small capillarity and viscosity is equal to that calculated from a boundary-layer approximation (despite the violation of the assumption h [Gt ] δ* on which that approximation is based). The viscous modification of the Stokes edge wave on a uniform, gentle slope is calculated through matched asymptotic approximations to the solution of [Lscr ]Z = 0.