This report is about the traveling wave solutions to the pseudo-parabolic equation, a kind of non-classical diffusion equation characterized by the mixed third-order derivative term. We demonstrate that the ratio of the mixed third-order derivative coefficient to the diffusion coefficient τ/D can serve as a bifurcation parameter. In detail, when τ/D≤1 , the equation possesses monotone traveling waves; when τ/D>1 , traveling waves are not monotonic and oscillate around the steady state u=1. The precise form of the minimal wave speed c^∗(τ,D) is also derived, exhibiting a monotonic increase with respect to τ and converging to 2√D as τ approaches 0. Numerical simulations confirm and support our theoretical results. They further show that the larger the value of τ is, the more non-monotonic the traveling waves become. Our findings regarding oscillating traveling waves predict saturation overshoot---a behavior that contradicts classical diffusion-like behavior yet is widely observed in unsaturated porous media. Mathematically, the threshold value of τ/D reveals the essential role of the dynamic capillary effect in the fundamental overshoot mechanism.