Henry K. Harbury and Wolfgang Porod Recent scanning tunneling microscopy (STM) studies on the (111) faces of noble metals have directly imaged electronic surface-confined states, and dramatic standing-wave patterns have been observed in geometrically arranged ``quantum corral'' structures formed by Fe adatoms on Cu(111). We present a coherent elastic scattering theory for the electronic states in these ``leaky'' confining structures by seeking solutions of the two-dimensional Schr\"odinger equation compatible with Sommerfeld radiation conditions. A direct comparison to the reported experimental data reveals excellent agreement with our elastic scattering theory. We are led to the conclusion that the Fe adatoms are well modeled by finite-height potential barriers, as opposed to ``black dot'' absorbing scatterers, and that the adatoms do not significantly couple the surface-state electrons to the bulk metallic states.