Resonant scattering of flexural waves in acoustic waveguide is analysed by using the recursive transfer method (RTM). Because flexural waves are governed by a fourth-order differential equation, a localized wave tends to be induced around the scattering region and dampening wave tails from the localized wave may reach the ends of a simulation domain. A notable feature of RTM is its ability to extract the localized wave even if the dampening tail reaches the end of the simulation domain. Using RTM, the enhanced reflection caused by a localized wave is predicted and the shape of the localized wave is explored at its resonance with the incident wave.

Flexural waves on a thin elastic plate are governed by the fourth-order differential equation, which is attractive not only from a harmonic analysis viewpoint but also useful for an efficient numerical method in the elastdynamics. In this paper, we proposed two novel ideas: (1) use of the tensor bases to describe flexural waves on inhomogeneous elastic plates, (2) weak form discretization to derive the second-order difference equation from the fourth-order differential equation. The discretization method proposed in this study is of preliminary consideration about the recursive transfer method (RTM) to analyse the scattering problem of flexural waves. More importantly, the proposed discretization method can be applied to any system which can be formulated by the weak form theory. The accuracy of the difference equation derived by the proposed discretization method is confirmed by comparing the analytical and numerical solutions of waveguide modes. As a typical problem to confirm the validity of the resultant governing equation, the influence of the spatially modulated elastic constant in waveguide modes is discussed.