The boundary layer methods which have been used to describe wave propagation past obstacles in the regions of glancing incidence have particular relevance to the asymptotic treatment, for large s, of steady infiltration as modeled by Philip's quasi-linearization. Here s(=&agr;l/2) is defined as the ratio of the characteristic length l of the water supply surface (in fact, its radius) to the sorptive length 2&agr;-1 of the soil. In this paper the steady infiltration from surface sources which are either semicircular in cross section or hemispherical are considered. Unlike the cases of buried sources considered by Waechter and Philip (1985), where an exact solution is available, we do not have an exact solution here. However, we resolve the structure of the large s asymptotics using the advantages of the scattering analog with the elegance of boundary layer methods. In particular, we calculate the mean infiltration rates (4/&pgr;)(1+0.6954s-2/3) and (1+1.3909s-2/3) for trench and hemispherical pond respectively and compare these with those for the corresponding cases of buried sources. We also compare our dimensionless total flux for a trench with that for a shallow trench of Pullan and Collins (1987) and our dimensionless total flux for a hemispherical cavity with that for a large shallow pond of Weir (1986). ¿ American Geophysical Union 1993