## 1. Introduction

[2] When water is applied over a long strip or furrow, we have essentially a two-dimensional infiltration process. The same scenario occurs with the application of water from closely spaced surface, drip emitters which results in a wetted strip. Early theoretical studies, for the most part, assumed a Gardner soil [*Gardner*, 1958] whose unsaturated hydraulic conductivity, *K* [L T^{−1}] is

with *K*_{s} [L T^{−1}] the saturated hydraulic conductivity, *α* [L^{−1}] the sorptive number and *h* [L] the pressure head which is assumed nonpositive. *Warrick and Lomen* [1976] and *Batu* [1977] simplified the mathematical difficulty by specifying the Darcian flow across the soil surface rather than as a mixed boundary condition with a specified water content or pressure along the strip and negligible flow to the outside. *Martinez and McTigue* [1991] used a numerical (boundary element) scheme to attack the same problem; *Basha* [1994] used Green's functions to write down relevant solutions. One of Basha's innovations was to predefine a velocity profile across the soil surface and use an inverse technique to evaluate parameters describing the velocity profile in order to match a desired constant pressure head. The emphasis has been on steady flow, although *Warrick and Lomen* [1976] and *Basha* [1994] did consider transient cases by use of a “linear” soil water characteristic [*Warrick*, 2003, p. 65].

[3] A closely related problem is infiltration from a disc. Interest in flow from disc sources has been heightened with the popularity of disc tension infiltrometers. *Haverkamp et al.* [1994] expressed the three-dimensional cumulative infiltration per unit source area *I*_{3D} [L] as a simple sum:

The *I*_{1D} [L] is the one-dimensional cumulative infiltration, *t* is time [T], *r*_{0} [L] the radius of the disc, *γ* a dimensionless “constant”, *S*_{0} [L T^{−0.5}] the sorptivity, θ_{0} (dimensionless) the volumetric water content at the disc source and θ_{n} (dimensionless) the initial water content in the profile. Haverkamp et al. rationalized that a reasonable bound on *γ* was 0.6 to 0.8. Note that once one-dimensional infiltration is described, results for the disc follow as a simple linear term with respect to time.

[4] The primary objective of the present study is to pursue a solution for the strip for which the cumulative infiltration is a sum of the one-dimensional term and an “edge effect”. The edge term will be taken proportional to time giving a solution analogous to (2). A second objective is to determine a range of values of the proportionality constant *γ* and determine how it is affected by texture, strip width, boundary conditions and initial conditions.