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. As a consequence, if the non-oscillatory modes are outside the horizon they do not freeze out. 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 (sub-dominant). It is known that the cosmological evolution gradually results in the scaling of a cosmic-string network and, therefore, after some time (\Dl \ta) the Universe becomes radiation-dominated. The evolution of the non-oscillatory GW modes during \Dl \ta (while they were outside the horizon), results in the distortion of the GW power spectrum from what it is anticipated in a pure radiation-model, at present-time frequencies in the range 10^{-16} Hz < f < 10^5 Hz.