I. INTRODUCTION It is well known that cosmic strings are produced in the early Universe at the phase transition associated with spontaneous symmetry breaking in the Grand-Unified-Theories (GUTs) [1]. Although cosmic strings formed before inflation are diluted away, some GUTs predict series of phase transitions where symmetry such as U (1)$_{B}$ − L ( B and L denotes baryon and lepton number) breaks at low energy and produces strings after inflation. Cosmic strings are also produced at the end of brane inflation in the framework of the superstring theory [2, 3]. Once produced, they survive until now and can leave observable signatures. Thus, the cosmic strings provide us with an opportunity to probe unified theories in particle physics which cannot be tested in terrestrial experiments. Various cosmological and astrophysical signals of cosmic strings have been intensively studied for decades. Especially, many authors have studied gravitational waves (GW) emit- ted from the cosmic string network, in particular, GWs from cosmic string loops. Cosmic string loops oscillate by their tension and emit low frequency GWs corresponding to their size [4, 5]. Moreover, string loops generically have cusps and kinks, and these structures cause high frequency modes of GWs, i.e. GW bursts [6–9]. On the other hand, GWs from infinite strings (long strings which lie across the Hubble horizon) have attracted much less interest than those from loops. This is because there are only a few infinite strings in one Hubble horizon, so GWs from them are much weaker than those from loops whose number density in the Hubble volume is much larger. 1 Loops can emit GWs with frequency ω ≳ ( αt ) − $^{1}$, where α is the parameter which represents the typical loop size normalized by the horizon scale. Therefor, unless α is much less than 1, GWs from loops dominate in the wide range of frequency bands detected by GW detection experiments, e.g. pulsar timing or ground based or space-borne GW detectors. However, there are several reasons to consider GWs of infinite strings. First, recent papers suggest that α is much less than previously thought [10, 11], and hence loops cannot emit low frequency GWs which can be detected by pulsar timing experiments. If this is 1 For cosmic strings whose reconnection probability is small, such as cosmic superstrings, many infinite strings can exist in a Hubble horizon. For such kinds of cosmic strings, however, the number of loops in a horizon is accordingly large, hence the situation that the loop contribution is stronger than infinite strings remains unchanged. 2