# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
###### Abstract
Although their use can be advantageous due to their ability to tackle complicated geometries, they involve a high computational cost in many cases. A common feature of these methods is Cartesian spatial periodicity. The Bloch-Floquet theorem, which was applied in 1946 by [PERSON] to solve the wave equation, relies, in fact, on the translational invariance of the problem formulation and is not rigorously applicable to polar coordinates systems [16]. In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship's deck, an aeroplane fuselage, a satellite antenna, a wall panel. In these cases, radial periodicity (e.g., as a sequence of annuli with alternating properties) may be used to reduce transmission of the vibration and structure-borne sound, and numerical approaches to study free and forced wave propagation in polar coordinates are desirable. The problem of wave propagation in radially periodic structures has been mainly formulated in optics within the theory of Bragg fibre [17]. Leaving aside numerous purely numerical studies (see, for instance, [18; 19; 20]), we notice that two approaches have been used so far to \"adjust\" the Bloch theorem for a cylindrically symmetric Bragg fibre. The first one is based on the use of far-field approximation of Bessel functions [21; 22], and the second one implies special radial varying of material properties [23; 24; 25]. Some recent works presented an approximation of the formulation of the Floquet theory for radially periodic membrane and plates [26]. However, studies of wave propagation in a polar periodic configuration in structural mechanics are rare. The paper aims to present a numerical method based on Floquet-theory and the FE discretisation of a small finite slice of a radial periodic structure. The method relies on adapting the Wave Finite Element (WFE) approach [14; 15] to radial periodic waveguides. In this method, a unit cell of the waveguide is discretised using standard FE elements. The FE mass and stiffness matrices are reduced using wave propagation theory in periodic structures, and wave characteristics are numerically evaluated from an eigenvalue problem. This allows a very substantial reduction in the number of DOFs involved in the computation, with a dramatic reduction of the computational time compared to other FE approaches. Forced wave propagation is then evaluated in the wave domain as described in [27; 28]. Amongst the numerical methods that could be used to investigate wave propagation problems, the WFE technique has several desirable features: it can be applied both to continuous and periodic structures; it exploits Bloch-Floquet theory and the versatility of standard FE analysis of a very small part of the structure; it allows the study of waveguideswith complex cross-sectional characteristics in a systematic manner, up to high frequency and with a low-computation cost. Applications of the technique to periodic, axisymmetric, helical, and slowly varying waveguides were presented in the literature, and the method has been validated through many benchmark cases, e.g., [29; 30; 31]. The approach rigorously assumes wave propagation in structures posed in Cartesian coordinates, and application to forced response in two dimensions was only possible by a semi-analytical computation via contour integration and the residue theorem [32]. Following a previous study by the same authors [33], this paper presents a simplified adaptation of this WFE technique to structures in polar coordinates exhibiting periodicity in the radial directions. Cylindrical wave propagation is thus estimated exploiting the Floquet theory formulation for an infinite periodic structure in one dimension. The approximation is achieved by taking only a very small slice of the structure and discretising the slice through piecewise Cartesian segments.