  We give a revealing expose that addresses an important issue in scattering
theory of how to construct two asymptotically sinusoidal solutions of the wave
equation with a phase shift using the same basis having the same boundary
conditions at the origin. Analytic series representations of these solutions
are obtained. In 1D, one of the solutions is an even function that behaves
asymptotically as sin(x), whereas the other is an odd function, which is
asymptotically cos(x). The latter vanishes at the origin whereas the derivative
of the former becomes zero there. Eliminating the lowest N terms of the series
makes these functions vanishingly small in an interval around the origin whose
size increases with N. We employ the tools of the J-matrix method of scattering
in the construction of these solutions in one and three dimensions.
