The Universe expansion rate is modulated around local inhomogeneities due to their gravitational potential. Velocity waves are then observed around galaxy clusters in the Hubble diagram. This paper studies them in a ~738 Mpc wide, with 2048^3 particles, cosmological simulation of our cosmic environment (a.k.a. CLONE: Constrained LOcal & Nesting Environment Simulation). For the first time, the simulation shows that velocity waves that arise in the lines-of-sight of the most massive dark matter halos agree with those observed in local galaxy velocity catalogs in the lines-of-sight of Coma and several other local (Abell) clusters. For the best-constrained clusters such as Virgo and Centaurus, i.e. those closest to us, secondary waves caused by galaxy groups, further into the non-linear regime, also stand out. This match is not utterly expected given that before being evolved into a fully non-linear z=0 state, assuming $\Lambda$CDM, CLONE initial conditions are constrained with solely linear theory, power spectrum and highly uncertain and sparse local peculiar velocities. Additionally, Gaussian fits to velocity wave envelopes show that wave properties are tightly tangled with cluster masses. This link is complex though and involves the environment and formation history of the clusters. Using machine learning techniques to grasp more thoroughly the complex wave-mass relation, velocity waves could in the near future be used to provide additional and independent mass estimates from galaxy dynamics within large cluster radii.