Assume a duct like that in → Sect. B.6 , but whose duct lining is laterally (bulk) reacting, and whose outer wall (behind the lining) is hard. The field in the interior volume of the duct, with cross section A\({}_{{1}}\), is marked with an index i=(1), the field in the lining with an index i=(2), and its cross section is A\({}_{{2}}\). Let the characteristic wave number and wave impedance in A \({}_{{1}}\) be, respectively, k\({}_{{0}}\) and Z\({}_{{0}}\), and let the characteristic propagation constant and wave impedance of the lining material in A\({}_{{2}}\) be, respectively, \(\Gamma\) \({}_{{a}}\), Z\({}_{{a}}\). The transversal functions of a mode \(T_{m}^{{(i)}}(r)\) are different in the two areas; its axial function R\({}_{{m}}\)(x) is the same.
The modes are orthogonal over the cross section A\({}_{{1}}\)+A\({}_{{2}}\)with the mode norm in the case of a single homogeneous layer of the lining in a cylindrical duct:...
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(2008). Orthogonality of Modes in a Duct with Bulk Reacting Walls . In: Mechel, F.P. (eds) Formulas of Acoustics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76833-3_25
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