The theoretical understanding of the linear standing or propagating magnetohydrodynamic (MHD) waves in a variety of solar coronal structures is far from complete since analytical solutions to the linearised MHD equations can only be found for very simple magnetic configurations. In this work, a numerical code is used to solve the linear MHD wave equations in a coronal magnetic arcade with a longitudinal magnetic field component and considering propagation in the longitudinal direction. In a cold plasma, the inclusion of these two effects leads to the coupling of fast and Alfvén modes. The numerical code provides us with solutions for these coupled fast and Alfvén modes in the form of the two-dimensional distribution of the perpendicular and normal velocity components together with the oscillatory frequency. Solutions have been computed in order to ascertain the effects of the longitudinal magnetic field component and of the longitudinal propagation on the properties of fast and Alfvén modes.