The Scaled Boundary Finite Element Method and related semi-analytical approaches are widely used to analyze wave propagation in structures of infinite extent, such as layered soils, rail-foundation systems, and other waveguides. In the context of moving loads, an important quantity is the critical speed, associated with resonance‐like amplification when the load velocity coincides with characteristic propagation speeds of the structure. These critical conditions occur at points on the dispersion curves where phase and group velocities are equal. In practice, they are commonly identified by computing dispersion relations over a given range of frequencies or wavenumbers, followed by post-processing. This contribution presents a direct approach for computing critical speeds without explicit construction of dispersion curves. The method is formulated for systems described by a discretization of the structure's cross-section combined with harmonic wave propagation along the longitudinal direction, as commonly employed in SBFEM-based and related formulations. By enforcing the defining condition that phase and group velocities coincide, the governing parameter-dependent eigenvalue problem is transformed into a coupled quadratic two-parameter eigenvalue problem in the wavenumber and frequency. After adequate linearization, this formulation finally yields a standard multi-parameter eigenvalue problem. A key advantage of the proposed approach is that critical wavenumbers and frequencies are obtained directly as eigenvalues, avoiding root-finding or tracing of dispersion curves. The resulting systems can be solved using established methods for multi-parameter eigenvalue problems. The methodology is demonstrated on several representative examples, including analytical benchmark problems, a beam on an elastic foundation, layered soil media, and a three-dimensional structure. The results illustrate that critical speeds can be computed accurately and robustly, making the approach attractive for practical railway engineering applications and other wave-based dynamic analyses.