  The adiabatic projection method is a general framework for studying
scattering and reactions on the lattice. It provides a low-energy effective
theory for clusters which becomes exact in the limit of large Euclidean
projection time. Previous studies have used the adiabatic projection method to
extract scattering phase shifts from finite periodic-box energy levels using
L\"uschers method. In this paper we demonstrate that scattering observables can
be computed directly from asymptotic cluster wave functions. For a variety of
examples in one and three spatial dimensions, we extract elastic phase shifts
from asymptotic cluster standing waves corresponding to spherical wall boundary
conditions. We find that this approach of extracting scattering wave functions
from the adiabatic Hamiltonian to be less sensitive to small stochastic and
systematic errors as compared with using periodic-box energy levels.
