It is proved (section 2.3) that there are only five duct shapes, forming two dual families, which have constant cut-off frequency(ies): namely, (I) the exponential duct, which is self-dual, and is the only shape with constant (and coincident) cut-offs both for the velocity and pressure (II) the catenoidal horns, of cross-section S∼cosh 2, sinh 2, which, with their duals (III) the inverse catenoidal ducts S∼sech 2, csch 2, have one constant cut-off frequency, respectively, for the acoustic pressure and velocity. The equipartition of kinetic and compression energies is shown (section 2.1) to hold at all stations only for (i) a duct of constant cross-section and (ii) an exponential horn these are the two cases for which the wave equations for the acoustic velocity and pressure coincide. The propagation of the fundamental, longitudinal acoustic mode in a duct of variable cross-section is considered, and the “Webster” wave equations for the sound pressure and velocity are used to establish some general properties of the exact acoustic fields.
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