This paper describes the extension of a wave and finite element (WFE) method to the prediction of noise transmission through, and radiation from, infinite panels. The WFE method starts with a conventional finite element model of a small segment of the panel. For a given frequency, the mass and stiffness matrices of the segment are used to form the structural dynamic stiffness matrix. The acoustic responses of the fluids surrounding the structure are modelled analytically. The dynamic stiffness matrix of the segment is post-processed using periodic structure theory, and coupled with those of the fluids. The total dynamic stiffness matrix is used to obtain the response of the medium to an incident acoustic pressure. Excitation of the structure by oblique plane waves and a diffuse sound field are considered. The response to structural excitation and the consequent radiation are determined. Since the size of the WFE model is small, computational times are small. Various example applications are presented to illustrate the approach, including a thin isotropic panel, an antisymmetric, cross-ply sandwich panel and a symmetric panel with an orthotropic core.