Wave Motion 104 (2021) 102754 Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wamot Legendre orthogonal polynomial method in calculating reflection and transmission coefficients of fluid-loaded functionally gradient plates Gao Jie a , Lyu Yan a,∗ , Song Guorong a , Liu Mingkun a,b , Zheng Mingfang c , He Cunfu a , Lee Yungchun a,d a Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, China b School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing, China c School of Environment and Civil Engineering, Dongguan University of Technology, Dongguan 523808, China d Department of Mechanical Engineering, Cheng Kung University, Tainan, Taiwan, China article info Article history: Received 3 July 2020 Received in revised form 13 April 2021 Accepted 30 April 2021 Available online 8 May 2021 Keywords: Reflection and transmission coefficients Legendre orthogonal polynomials FGM plate Truncated order Displacement and stress distribution abstract Based on the Legendre orthogonal polynomial series expansion and the partial wave theory, a Legendre orthogonal polynomial method (LOPM) is proposed to calculate the reflection and transmission coefficients of plane waves at the liquid/solid inter- face of a liquid-loaded functionally gradient material (FGM) plate. The displacement solutions in FGM plate are fitted approximately by Legendre orthogonal polynomial series. The stresses and the governing differential equations of the FGM plate are derived. Based on the boundary conditions of the liquid/solid interface and the governing differential equations, the linearly independent equations are set up to calculate the reflection and transmission coefficients. Meanwhile, the expansion coefficients of the Legendre orthogonal polynomials are also obtained. The power function establishes the gradient model of the mechanical parameters along the thickness direction. The angular spectrums of reflection and transmission coefficients from LOPM are in well agreement with the calculation results from the transfer matrix method. By analyzing the convergence of the reflection coefficient spectrum, the critical value of the truncated order of Legendre orthogonal polynomials is determined. The mapping relationship between the FGM gradient models and the reflection coefficient angular spectrum, also the frequency spectrums and the displacement/stress distributions can be demonstrated simultaneously, which provides the theoretical fundamentals of the ultrasonic non- destructive testing for the mechanical properties of FGM and extends the application of the LOPM. © 2021 Elsevier B.V. All rights reserved. 1. Introduction Functionally gradient materials (FGMs) are widely used in aerospace, machinery, and biomedicine area, etc. The main feature of FGM is that its performance parameters (as elastic constants, density, etc.) vary continuously according to some specific functions along one particular direction. There is no distinct ‘‘interface’’ inside the material, so that it can avoid the stress concentration and the discontinuity induced by the interface. Moreover, the gradient models can be designed ∗ Corresponding author. E-mail address: lvyan@bjut.edu.cn (Y. Lyu). https://doi.org/10.1016/j.wavemoti.2021.102754 0165-2125/© 2021 Elsevier B.V. All rights reserved.