Gravitational wave tails are produced by back-scattering of the outgoing gravitational radiation (emitted by an isolated system) off the curved spacetime associated with the total mass of the system. This paper investigates the spectral (or Fourier) decomposition of gravitational wave tails at large distances from the system, at the 1.5 post-Newtonian order in the wave field. It is shown that the effects of wave tails are (i) to increase the amplitude of the Fourier components of the (linear) waves by a factor linearly depending on the frequency, and (ii) to add to the phase of the waves a supplementary phase depending on the frequency as omega ln omega . The latter frequency-dependent phase introduces a new effect which should be observable in any radiation containing more than one frequency, for instance in the radiation emitted by a binary star system orbiting a Keplerian ellipse with non-zero eccentricity, or in the radiation emitted by an inspiralling (compact) binary star system. We propose in this paper to include the tail-induced effects (i) and (ii) in the matched filters of the future data analysis of inspiralling compact binary signals in laser interferometer gravity-wave detectors (at least in future, very sensitive, such detectors). In, this way, the filters will be highly correlated with the actual signal, and in particular will remain, as the frequency of the signal increases, in accurate phase with it. The contribution of the wave tail in the total gravitational energy emitted by a binary system is also calculated, and a numerical application to the binary pulsar PSR 1913+16 is presented. We find that the tail-induced relative correction in the orbital P_Th of the pulsar is equal to +1.65*10^-7 (too small to be observed).