Wave propagation through a slender curved tube

The transmission of an acoustic wave through a slender three-dimensional tubular region is investigated. The ratio of the reference length a of a cross section of the tube to the reference curvature of the center line of the tube is assumed to be small. The analysis is carried out for wavelengths ranging from the order of the reference curvature of the center line of the tube to wavelengths much smaller than the reference length of the cross-sectional area. It is shown that the leading solution inside the tube is governed by the same equation as for an equivalent straight tube except for the case where the wavelength is much smaller than the reference length a. For the latter the leading solution is governed by a partial differential equation in the cross section of the tube. The radius of curvature of the center line enters only into the amplitude of the leading order solution of the short waves.