2. Acoustic Wave Propagation in an Elliptical Cylindrical Waveguide

The non-dimensional even and odd resonance frequencies of the vibration of a clamped elliptical membrane denoted by \(\left. {\left( {k_{0} \displaystyle\frac{{D_{1} }}{2}} \right)} \right|_{{(m,n){\text{ Even }}}}\) and \(\left. {\left( {k_{0} \displaystyle\frac{{D_{1} }}{2}} \right)} \right|_{{(m,n){\text{ odd }}}},\) respectively, were found by numerically solving Eqs. ( 2.62) and ( 2.63) for the parametric zeros \(q_{m,n}\) and \(\overline{q}_{mn} ,\) respectively, using the root-bracketing and bisection techniques explained in Sect.  2.5. Note that the parametric zeros computed here pertain to the Dirichlet boundary condition specified by Eqs. ( 2.62) and ( 2.63), and they are always interlaced with the parametric zeros corresponding to the Neumann or rigid-wall duct condition specified by Eqs. ( 2.49) and ( 2.50). The interlacing of parametric zeros is demonstrated by Fig.  2.3b which shows the variation of the functions \(Ce_{0} \left( {\xi = \xi_{0} ,q} \right)\) and \(\frac{{\text{d}}}{{{\text{d}}\xi }}Ce_{0} \left( {\xi = \xi_{0} ,q} \right)\) with the parameter q for \(D_{2} /D_{1} = 0.6.\) Table 2.29 presents the non-dimensional resonance frequencies of the radial mode \(Ce_{0} \left( {\xi ,q_{0,n} } \right)ce_{0} \left( {\eta ,q_{0,n} } \right)\) and the even circumferential modes \(Ce_{m} \left( {\xi ,q_{m,n} } \right)ce_{m} \left( {\eta ,q_{m,n} } \right)\) for the first seven orders, i.e., \(m = 1,2, \ldots 7\) , while Table 2.30 presents the non-dimensional resonance frequencies of the first eight odd circumferential modes up to four decimal places. For each mode type, the first four zeros are presented, i.e., \(n = 1,2,3,4\). Note that a complete range of aspect-ratio given by is considered in Tables 2.29 and 2.30, and the non-dimensionalization variable D ₁ is kept constant in these Tables. For certain aspect-ratio, the non-dimensional resonance frequencies shown in Tables 2.29 and 2.30 were compared with the values available in the literature for a clamped elliptical membrane [ 25, 26, 31] whereby an excellent agreement was observed. The graphical variation of the resonance frequencies presented in Tables 2.29 and 2.30 with aspect-ratio is not included for brevity; however, it may be mentioned that the graphs for both even and odd circumferential and cross-modes as well as radial modes have a concave nature. As anticipated, the concavity is more pronounced for the odd mode graphs as compared to their even mode counterparts. Here, we present a 9th degree least-squares interpolating polynomial using which one can accurately predict the non-dimensional resonance frequency values for any aspect-ratio within the range \(D_{2} /D_{1} = [0.3,1]\). It is given by where and the coefficients { b ₀, b ₁, b ₂, … b ₉} for the first few radial, even and odd modes are listed in Tables 2.31 and 2.32. Figure 2.12a–c presents the mode shapes of the first three radial modes given by \(Ce_{0} \left( {\xi ,q_{0,1} } \right)ce_{0} \left( {\eta ,q_{0,1} } \right),Ce_{0} \left( {\xi ,q_{0,2} } \right)ce_{0} \left( {\eta ,q_{0,2} } \right)\) and \(Ce_{0} \left( {\xi ,q_{0,3} } \right)ce_{0} \left( {\eta ,q_{0,3} } \right),\) respectively, for the aspect-ratio D ₂/ D ₁ = 0.5. Note here that the fundamental (0, 1) e mode exhibits a gradual variation across the elliptical cross-section and is characterized by a maximum at the center and zero displacement along the elliptical boundaries—indeed, this feature is in contrast with the fundamental or plane wave mode of a rigid-wall elliptical waveguide for which the acoustic pressure field is uniform across the cross-section. Therefore, if the wall of an elliptical waveguide is not rigid, rather, if it is lined with a sound absorbent or dissipative material, then the plane wave mode does not exist. Rather, the fundamental mode exhibits variation across the cross -section and has a dissipative part which implies it cannot propagate without attenuation [ 42]. In fact, by imposing the boundary conditions given by Eqs. ( 2.62) and ( 2.63), one obtains the pressure-release modes of an elliptical duct [ 47], while in case of lined ducts, these modes are more popularly referred to as the soft-wall modes [ 48]. The (0, 2) e and (0, 3) e radial mode shapes exhibit one and two pressure nodal ellipses, respectively, as their respective eigenfunctions are a solution of the S-L problem. The set of figures in the 2nd column, i.e., Fig. 2.12d–f, presents the first three even circumferential mode shapes given by \(Ce_{1} \left( {\xi ,q_{1,1} } \right)ce_{1} \left( {\eta ,q_{1,1} } \right),\) \(Ce_{2} \left( {\xi ,q_{2,1} } \right)ce_{2} \left( {\eta ,q_{2,1} } \right),\) and \(Ce_{3} \left( {\xi ,q_{3,1} } \right)ce_{3} \left( {\eta ,q_{3,1} } \right),\) respectively, while the set of figures in the 3rd column, i.e., Fig. 2.12g–i, presents the first three odd circumferential mode shapes \(Se_{1} \left( {\xi ,\bar{q}_{1,1} } \right)se_{1} \left( {\eta ,\bar{q}_{1,1} } \right),\) \(Se_{2} \left( {\xi ,\bar{q}_{2,1} } \right)se_{2} \left( {\eta ,\bar{q}_{2,1} } \right),\) and \(Se_{3} \left( {\xi ,\bar{q}_{3,1} } \right)se_{3} \left( {\eta ,\bar{q}_{3,1} } \right),\) respectively. The even circumferential mode-shapes shown in Fig. 2.12d–f exhibit pressure nodal hyperbolas which divide the elliptical cross-section in regions of alternating phase, and they are found to be similar to their counterpart mode shapes shown in Figs. 2.6a–c Similarly, the odd circumferential mode shapes shown in Fig. 2.12(g–i) resemble their counterpart mode shapes presented in Figs. 2.6d–f, respectively, as may be observed from the pressure nodal hyperbola pattern. An apparent difference between the Dirichlet and Neumann mode shapes is that the former is characterized by zero value of the function field along the elliptical boundaries and a local maximum is formed at the center of the same phase region, while the latter mode shape is characterized by formation of a local maximum near the elliptical boundary.