Asymptotic expansions for the structural wavenumbers of isotropic and orthotropic fluid-filled circular cylindrical shells in the intermediate frequency range

We consider wavenumbers in in vacuo and fluid-filled isotropic and orthotropic shells. Using the Donnell–Mushtari (DM) theory we find compact and elegant asymptotic expansions for the wavenumbers in the intermediate frequency range, i.e., around the ring frequency. This frequency range corresponds to the frequencies where there is a rapid change in the values of bending wavenumbers and is found to exist in isotropic and orthotropic shells (in vacuo and fluid-filled) for low circumferential orders n only. The same is first identified using the n  =0 mode of an orthotropic shell. Following this, using the expression for the intermediate frequency, asymptotic expansions are found for other cases. Here, in order to get compact expansions we consider slight orthotropy (ϵ⪡1ϵ⪡1) and light fluid loading (μ⪡1μ⪡1). Thus, the orthotropy parameter ϵϵ and the fluid loading parameter μμ are used as asymptotic parameters along with the non-dimensional thickness parameter ββ. The methodology can be extended to any order of ϵϵ, only the expansions become unwieldy. The expansions are matched with the numerical solutions of the corresponding dispersion relation. The match is found to be good.

► Found a compact formula for intermediate frequency range in an orthotropic shell.
► Found compact expansions for wavenumbers at this frequency in in vacuo shells.
► Found compact expansions for wavenumbers at this frequency in shells with fluid.
► Good comparison is found between the asymptotic and the numerical results.