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We study boundary effects in a linear wave equation with Dirichlet typeconditions in a weakly curved pipe. The coordinates in our pipe are prescribedby a given small curvature with finite range, while the pipe's cross sectionbeing circular. Based on the straight pipe case a perturbative analysis bywhich the boundary value conditions are exactly satisfied is employed. As suchan analysis we decompose the wave equation into a set of ordinary differentialequations perturbatively. We show the conditions when secular terms due to thecurbed boundary appear in the naive peturbative analysis. In eliminating such asecularity with a singular perturbation method, we derive amplitude equationsand show that the eigenfrequencies in time are shifted due to the curvedboundary.

Amplitude equations for a linear wave equation in a weakly curved pipe