The Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ν=0${\nu=0}$and b=3${b=3}$. Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters b,α,ν${b,\alpha,\nu}$, and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific choices of the parameter β=0${\beta=0}$and β=1${\beta=1}$. Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ν=0${\nu=0}$and b=3${b=3}$depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant k1∈ℝ${k_{1}\in\mathbb{R}}$. In an appropriate limit β=1${\beta=1}$, the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.