At first-order of approximation a sea-state may be considered as an infinite sum of Airy components with angular frequencies ω i and wave-number vectors k i . A second-order analysis shows the co-existence of long waves appearing at the difference frequencies ω i — ω j with wave-number vectors k i —k j . In shallow water they become appreciable in amplitudes and may induce slow-drift motion of moored structures. For small values of ω i — ω j ,k i —k j may take all kinds of directions for an angular-spread wave system. Then it may be questioned how the in-line and transverse second-order accelerations compare to those obtained for a mono-directional wave-system. This analysis is carried out here by relating the spectra of the second-order horizontal accelerations to the directional wave-spectrum. Numerical applications are first performed for deep water. They show that at low frequencies, even for very narrowly spread wave systems, the transverse component is larger than the in-line component. In shallow water both components are dratically reduced as compared to the mono-directional case. As a consequence one may question the validity of model-testings or numerical models which take no account of the directionality of the wave-system.