High penetration of converter-interfaced renewable energy resources will significantly change the swing dynamics between synchronous generators (SGs) in future power systems. This article examines the impact of high converter penetration on wave-like disturbance propagation arising from sudden generator and load losses in radial (1-D) and meshed (2-D) power systems. To keep the uniformity assumption as converters are introduced, the rating of each SG is decreased with a converter resource making up for the reduction. Numerical simulations demonstrate that as the penetration level of constant-power grid-following (GFL) converters increases, the speed of disturbance propagation increases due to the reduced system inertia. Naturally, converters with the capabilities to positively respond to disturbances would in turn reduce the propagation speed. Analytical studies based on continuum models are presented for the 2-D system with SGs and constant-power GFL converters in order to visualize the disturbance propagation and validate numerical simulations based on differential-algebraic equations. In addition, fast active power control of converters can slow down the electromechanical wave (EMW) propagation and even contain it. These concepts are illustrated on the idealized radial and meshed systems and a reduced model of the U.S. eastern interconnection.