The equation which governs the temporal evolution of a gravitational wave (GW) in curved space-time can be treated as the Schrodinger equation for a particle moving in the presence of an effective potential. When GWs propagate in an expanding Universe with constant effective potential, there is a critical value (k_c) of the comoving wave-number which discriminates the metric perturbations into oscillating (k > k_c) and non-oscillating (k < k_c) modes. The effective potential is reduced to a non-vanishing constant in a cosmological model which is driven by a two-component fluid, consisting of radiation (dominant) and cosmic strings (subdominant). However, the cosmological evolution (gradually) results in the scaling of any long-cosmic-string network and, therefore, after some time (Δτ) the Universe enters in the pure-radiation epoch. The evolution of the non-oscillatory GW modes during Δτ, results in the distortion of the low-frequency part of the stochastic GW power-spectrum, which, therefore, departs from scale invariance (anticipated in the pure-radiation case). As regards the corresponding high-frequency part (which is determined by the evolution of the oscillating modes), we find that the presence of cosmic strings gives rise to the quantum-gravitational creation of gravitons, leading to the amplification of the GW signal by (almost) two orders of magnitude.