  We study boundary effects in a linear wave equation with Dirichlet type
conditions in a weakly curved pipe. The coordinates in our pipe are prescribed
by a given small curvature with finite range, while the pipe's cross section
being circular. Based on the straight pipe case a perturbative analysis by
which the boundary value conditions are exactly satisfied is employed. As such
an analysis we decompose the wave equation into a set of ordinary differential
equations perturbatively. We show the conditions when secular terms due to the
curbed boundary appear in the naive peturbative analysis. In eliminating such a
secularity with a singular perturbation method, we derive amplitude equations
and show that the eigenfrequencies in time are shifted due to the curved
boundary.
