The paper presents results of numerical simulation for transverse elastic waves corresponding to A0 mode of Lamb waves propagating in a composite plate. This problem is solved by using the Spectral Finite Element Method. Spectral plate elements with 36 nodes defined at Gauss-Lobatto-Legendre points are used. As a consequence of selecting Lagrange polynomials discrete orthogonality guaranteed leading to a diagonal mass matrix. This results in a crucial reduction of numerical operations required for a chosen time integration scheme. Numerical calculations have been carried out for various orientations of reinforcing fibres within the plate as well as for various fibre volumes fractions. The paper shows that the velocities of transverse elastic waves in composite materials are functions of the fibre orientation and the fibre volume fraction.