Asymptotic Analysis Of The Dispersion Characteristics Of Structural Acoustic Waveguides
In this work, we study the coupled dispersion characteristics of three distinct structural-acoustic waveguides, namely: -(1) a two-dimensional waveguide, (2) a ﬂuid-ﬁlled circular cylindrical shell and (3)a ﬂuid-ﬁlledelliptic cylindrical shell. Our primary interest is in ﬁnding coupled wavenumbers as functions of the ﬂuid-structure coupling parameter(µ). Using the asymptotic solution methodology, we ﬁnd the coupled wavenumbers as perturbations over the uncoupled wavenumbers of the component systems (the structure and the ﬂuid). The asymptotic method provides us with analytical expressions of the coupled wavenumbers for small and large values of µ. The dispersion curves obtained from these extreme values of µ help in predicting the nature of the continuous transition of the wavenumber branches over the entire range of µ. Since the coupled wavenumbers are obtained as perturbations over the uncoupled wavenumbers, the perturbation term characterizes the eﬀect of one medium over the other in terms of additional mass or stiﬀness. As is common in asymptotic methods, a particular form of the asymptotic expansion remains valid over a certain frequency range only. Hence, diﬀerent scalings of the asymptotic parameter are used for diﬀerent frequency ranges. In this regard, the method adopted uses principles of Matched Asymptotic Expansion (MAE). As mentioned above, we begin the study with a two-dimensional structural acoustic waveguide. Depending on the boundary condition at the top-edge of the ﬂuid-layer (rigid or pressure-release), two cases are separately analyzed. In both these cases, only a single perturbation parameter (µ) is used. This is followed by the study of the axisymmetric mode vibration of a ﬂuid-ﬁlled circular cylindrical shell. Here, in addition to , we include the Poisson’s ratio as another asymptotic parameter. The next problem studied is the beam mode (n =1)vibration of the same ﬂuid-ﬁlled circular cylindrical shell. Here, the frequency is used as an asymptotic parameter (in addition to ) and the derivations proceed in two separate parts, one for the high frequency and the other for the low frequency. Having completed the n = 0 and n = 1 modes of the cylindrical shell, the higher order shell modes are studied using the simpler shallow shell theory. For the ﬁnal system, viz., the elliptic cylindrical shell, another asymptotic parameter in the form of the eccentricity of the cross-section is used. Having derived the analytical expressions for the coupled wavenumbers and obtained the dispersion curves, a uniﬁed behavior of structural-acoustic systems is found to emerge. In all these systems, for small , the coupled wavenumbers are close to the in vacuo structural wavenumber and the wavenumbers of the rigid-walled acoustic duct. The measure of closeness is quantiﬁed by . As µ increases, these wavenumber branches get shifted continuously till for large µ, the coupled wavenumber branches are better identiﬁed as perturbations to the wavenumbers of the pressure-release acoustic duct. At the coincidence region, the coupled structural wavenumber branch transits to the coupled acoustic wavenumber branchand vice-versa. As a result, at coincidence frequencies, while the uncoupled wavenumber branches intersect, due to the coupling, there is no longer an intersection. These common characteristics are shared amongst all the systems despite the diﬀerence in geometries. This suggests that the above discussed features capture the essential physics of sound-structure coupling in waveguides.This workthus presents a novel uniﬁed view-point to the topic. Along the way, some additional novel studies are conducted which do contribute to the completeness of the work. The free wavenumbers determined from the asymptotic expressions are usedto calculate the forced response of the two-dimensional waveguide due to a δ forcing. Using this analysis, we are able to come up with a novel explanation of the observation that with coupling the dispersion curves cannot intersect. Additionally, the eﬀect of bulk ﬂow in the acoustic ﬂuid is also comprehensively studied for the easier case of the two-dimensional waveguide. Further, the well-known universal dispersion relation for the higher order circumferential modes of the in vacuo circular cylindrical shell is re-derived using a simpler method.
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