The Boussinesq system [see Y. Li and J. E. Zhang, Phys. Lett A 284, No. 6, 253–258 (2001; Zbl 0977.35114)] \[ \eta_t+\left[(1+\eta)u\right]_x + \frac14u_{xxx}=0,\quad u_t + uu_x + \eta_{x}=0, \] where \(\eta\) is the elevation of a water wave, and \(u\) is the surface velocity of water along the \(x\)-direction, is considered. With the help of Mathematica, using modified extended tanh-functions and Jacobi elliptic functions methods, the travelling wave and Jacobi doubly periodic wave solutions are obtained. Some illustrative figures are given.