Summary: We establish existence of periodic travelling-wave solutions to a generalized Boussinesq system by using the topological degree theory for positive operators defined on a cone in an appropriate Banach space. Furthermore, we derive a high-accuracy pseudospectral solver based on a Fourier decomposition to construct numerical approximations of these stationary solutions. The numerical simulations are in perfect agreement with the theoretical results and new travelling-wave solutions of the system are computed which do not belong to the family of solutions proved to exist.