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

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

2009-10-03

arxiv.org/abs/0910.0549v1

Nonlinear Sciences

Pattern Formation and Solitons

To appear in J Phys A: Math. Theor

Scientific paper

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.

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