STANDING-WAVE patterns in electron density have been seen recently(1-4) in images of the surfaces of noble metals obtained with the scanning tunnelling microscope (STM). These patterns are due to the scattering of surface electrons off impurities and step edges. By assembling specific enclosed structures of adatoms (’quantum corrals’) using the STM, one can generate standing waves of particular geometries(3). Here we describe a theory of the scattering process, which allows us to predict the standing-wave patterns of an arbitrary corral geometry with great accuracy. We can use the theory to examine the scattering properties of the atoms in the corral walls. We find that iron atoms assembled on the (111) surface of copper act as ’black dots’, soaking up all of the electron wave amplitude impinging on them. A scattered wave is generated nonetheless, but this behaviour means that the corral walls are only 25% reflective. In an acoustic analogy, the corral is therefore a rather quiet chamber.