Transient acoustic diffraction and radiation by an axisymmetrical elastic shell: a new statement of the basic equations and a numerical method based on polynomial approximations
Résumé
This work deals with an axisymmetrical shell composed of a cylindrical shell closed by two hemispherical shells made of the same material and with the same thickness. The shell is immersed into a homogeneous perfect fluid extending to infinity. The first part is devoted to the establishment of the equations governing the shell vibrations. The method used, which, in the authors' opinion, is not quite new, is based on the expansion of the elasticity equations into a Taylor series of the transverse variable: by using the same degree of expansion, the equations obtained for the cylindrical part and for the spherical parts are consistent (they correspond to the Donnell and Mushtari approximation). The first interest of this analysis is that the continuity conditions along the junction lines between the cylindrical and the spherical parts are immediately obtained. The main problem is to obtain the boundary conditions satisfied by the hemispherical shells displacement at the apexes. Indeed, due to the use of spherical co-ordinates—which is a quite natural choice—the coefficients of the equations become singular at the apexes and boundary conditions are required to express that an apex is a mechanically regular point. The method that is used here enables one to obtain such a result which, to the authors' knowledge, is new. The transient response of the system shell/external fluid is sought as a series of its resonance modes, that is its free oscillations. The main difficulty is to obtain a numerical approximation of the resonance modes: their calculation leads to solving the Fourier transform of the system of homogeneous equations. The numerical method for solving the problem is the following. The acoustic pressure is described by a hybrid layer potential, the density of which is approximated by a linear combination of orthogonal polynomials. Each component of the shell displacement is approximated by a linear combination of polynomial functions: these functions are chosen as linear combinations of orthogonal polynomials which satisfy the same continuity and boundary conditions as the shell displacement components. In the first step, the resonance frequencies are calculated. Then the coefficients of the corresponding resonance mode expansion are deduced. The validity and the efficiency of this approach will be shown in a second article through comparisons between numerical predictions and experimental results.