Constancy of the effective potential which governs the temporal evolution of a gravitational wave (GW) in an expanding Universe, leads to a critical comoving wave-number $(k_c)$, associated with a low-frequency cut-off in the corresponding energy-density. As a consequence, GW modes with $k \leq k_c$ remain outside the horizon for every $t$ and reduce to metric perturbations of constant amplitude. This property is met in a radiation model {\em contaminated} by a small fraction of cosmic strings. It is known that, the cosmological evolution gradually results in the {\em scaling} of the cosmic-string network and the Universe enters in the radiation era. Once the Universe becomes radiation-dominated, metric perturbations of $k \leq k_c$ are re-introduced inside the horizon, which now expands faster than their physical wavelength. Within the Hubble sphere, the energy-density associated with these waves makes up a logarithmically increasing fraction of the Universe matter energy-density.