A study of the Boussinesq equations in one dimension is presented. These equations describe the nonlinear wave propagation of a free surface under inviscid, incompressible, and irrotational constraints. Physically, they describe the motion of long waves (compared to the depth of the domain) which find applications in oceanography and coastal engineering. Dispersive properties are examined and numerical solutions are found using a hybrid Finite Volume / Finite Differencing Method. Since all of the numerical code is original (written in C++/Python), a detailed explanation of the numerical method is given. To validate the numerical model, convergence rates are computed using analytic solutions found by S. Ding and X. Zhao. Special attention is given to solitary waves (solitons) which can arise in systems exhibiting weakly non-linear and dispersive properties.