The recursive transfer method (RTM) is a numerical technique that was developed to analyze scattering phenomena and its formulation is constructed with a difference equation derived from a differential equation by Numerov's discretization method. However, the differential equation to which Numerov's method is applicable is restricted and therefore the application range of RTM is also limited. In this paper, we provide a new discretization scheme to extend RTM formulation using the weak form theory framework. The effectiveness of the proposed formulation is confirmed by microwave scattering induced by a metallic pillar placed asymmetrically in the waveguide. A notable feature of RTM is that it can extract a localized wave from scattering waves even if the tail of the localized wave reaches to the ends of analyzing region. The discrepancy between the experimental and theoretical data is suppressed with in an upper bound determined by the standing wave ratio of the waveguide.

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.

Recently, we proposed a weak-form discretization scheme to derive second-order difference equations from the governing equation of the scattering problem. In this paper, under the scope of the proposed scheme, numerical expressions for the waveguide boundary conditions are derived as perfectly absorbing conditions for input and output ports. The waveguide boundary conditions play an important role in extracting the quasi-localized wave as an eigenstate with a complex eigenvalue. The wave-number dependence of the resonance curve in Fano resonance is reproduced by using a semi-analytic model that is developed on the basis of the phase change relevant to the S-matrix. The reproduction confirms that the eigenstate with a complex eigenvalue does cause the observed Fano resonance.

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.