Propagation of a long wave in a curved duct (I) basic analysis of long wave propagation in a curved duct with variable cross section

The propagation of a long wave in a three-dimensional curved duct with variable cross section is studied in this paper. It is shown that a three-dimensional Helmholtz equation can be decomposed into a two-dimensional Laplace (or Poisson) equation and a one-dimensional Webster equation by the curvilinear orthogonal coordinate system, non-dimensionization of reduced wave equation and regular perturbation with small parameterka, wherek is the wave number anda is the characteristic radius of the duct. The influences of the duct's geometric parameters (the area variation of the cross section, the curvature and torsion of the central line) on the asymptotic expansion of the solution are analysed. It is concluded that the effects of the variation of the cross sectional area first appear in the first term of the asymptotic expansion, and when the cross section shape has certain symmetric properties, the effects of the curvature and torsion of the central line first appear in the third and the fourth terms, respectively. An example of long wave propagation in a curved circular duct is also given at the end of this paper.