This study investigates the wave dispersion characteristics of Euler-Bernoulli beam situated on cubic nonlinear foundation. Employing the multiscale method, we conduct analytical formulation by utilizing a specific initial condition (fixed wavenumber) through the free wave approach to identify the frequency shift in the dispersion relation. The research delves into the combined influence of metadamping and stiffness nonlinearity. The findings reveal that, in the case of a damped system, the dispersion relation plot undergoes a transition from an undamped nonlinear state to a linear system with the progression of time. To validate our results, we execute numerical simulations using the 4th order Runge-Kutta method and the finite element formulation of the governing equation in MATLAB. Additionally, Fourier transform analysis of the displacement response at the beam's middle node is conducted, enabling the evaluation of the frequency response function and effectively eliminating the influence of wave reflections from the boundary. This study provides valuable insights into the metadamping-induced complexities, coupled with stiffness nonlinearity, in the wave dispersion of Euler-Bernoulli beams. It contributes to a broader understanding of the dispersion relation in nonlinear metamaterials