A novel method is presented for investigating elastic waves in functionally graded material (FGM) plates excited by plane pressure waves. The FGM plate is first divided into quadratic layer elements (QLEs). A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the frequency domain for an arbitrary FGM plate. The response of the plate to an incident pressure wave is obtained using the Fourier transform techniques. Results obtained by the present method are compared with an existing method using homogeneous layer elements. Numerical examples are presented to investigate stress waves in FGM plates. The relationship between the surface displacement response and the material property of quadratic FGM plates has been analytically obtained for the material characterization. A computational inverse technique is also presented for characterizing material property of an arbitrary FGM plate from the surface displacement response data, using present QLE method as forward solver and genetic algorithm as the inverse operator. This technique is utilized to reconstruct the material property of an actual SiC-C FGM.