For Alfv¿nic fluctuations in magnetohydrodynamics (MHD) the perturbed transverse velocity Vt and magnetic field Bt can be related by the Wal¿n relation, Vt=¿Bt/(&mgr;0&rgr;)1/2≡¿VAt, where &rgr; is the plasma density, VAt is the transverse Alfv¿n velocity, and the plus (minus) sign is for antiparallel (parallel) propagation. However, observations of Vt and Bt for Alfv¿n waves and rotational discontinuities in the solar wind and at the magnetopause showed an obvious deviation from the relation. In this paper, modifications of the Wal¿n relation for linear and nonlinear Alfv¿n waves and rotational discontinuities (RDs) are examined in the Hall-MHD formulation. Let Vit(≈Vt) be the transverse ion velocity and Vet be the transverse electron velocity. It is found that Vit=¿Bt(z)/(&mgr;0&rgr;1)1/2=¿(&rgr;(z)/&rgr;1)1/2VAt(z) and Vet=¿(&rgr;1/&mgr;0)1/2Bt(z)/&rgr;(z)=¿(&rgr;1/&rgr;(z))1/2VAt(z) for RDs in Hall-MHD, where &rgr;1 is the upstream plasma density. The ion and electron Wal¿n ratios are defined as Ai=Vit/VAt and Ae=Vet/VAt, respectively. It is found in Hall-MHD that Vit⋅Vet=VAt2, AiAe=1 and Ai1) for Alfv¿n waves and RDs with right-hand (left-hand) polarization. The Hall dispersive effect may modify the ion Wal¿n ratio by ΔAi≈¿0.14 for the magnetopause RDs and by ΔAi≈¿0.07 for the interplanetary RDs. ¿ 2000 American Geophysical Union