begin to be formed right after the formation of strings. Otherwise, the temperature at which kinks begin to appear is given by Eq. (14). In anyway, even if such kinks survive friction, they become extremely dense in the later period, so that they do not affect the observable part of the GW spectrum, as we will see later. On the other hand, if the reheating temperature is lower than T$_{c}$ , the kinks from the first matter era definitely continue to exist, and if the reheating temperature is extremely low, GWs from these kinks can be observed. Therefore, we concentrate on the situation where the reheating temperature is low enough so that the kinks generated in the first matter era survive without experiencing the friction domination. We denote the time when kinks starts to be formed by t ∗ . Then we can get the solution, but its precise form is very complicated. Here we assume that the matter-dominated epoch follows after inflation, and the reheating completes at t = t$_{r}$ after which the radiation dominated era begins. If we focus on only the dominant term and neglect O (1) numerical factor, we get in the first matter era, 7 ψ (RD) 2 ( t ) = t$_{r}$ t 2 ζ$_{r}$ ( t ∗ t$_{r}$ ) 2 ζ$_{m}$ , (18) ψ (RD) 1 ( t ) = ( t$_{r}$ t ) 2 ζ$_{r}$ , (17) in the radiation era where dN dψ ( t, ψ ) ∼    $^{ψ}$$^{−}$$^{β$_{m}$}$$^{/}$$^{2}$$^{ζ$_{m}$}$$^{t}$$^{−}$$^{1}$ for ψ > ( t ∗ t ) 2 ζ$_{m}$ ( t t ∗ ) β$_{m}$ + ζ$_{m}$ ψ 1 / $^{2}$t − 1 for ψ < t ∗ t 2 ζ$_{m}$ (15)