Dispersive magnetohydrodynamic (MHD) waves with short-wavelength modification have an important role in transforming energy from waves into particles. In this paper, based on the two-fluid mode, a dispersion equation, including the short-wavelength effect, and its exact solution are presented. The outcome is responsible for the short-wavelength modification versions of the three ideal MHD modes (i.e. the fast, slow and Alfvén). The results show that the fast and Alfvén modes are modified considerably by the shortwavelength effect mainly in the quasi-parallel and quasi-perpendicular propagation directions, respectively, while the slow mode can be affected by the short-wavelength effect in all propagation directions. On the other hand, the dispersive modification occurs primarily in the finite-β regime of 0.001 < β < 1 for the fast mode and in the high-β regime of 0.1 < β < 10 for the slow mode. For the Alfvén mode, the dispersive modification occurs from the low-β regime of β < 0.001 through the high-β regime of β > 1. It is well known that there are three ideal magnetohydrodynamic (MHD) modes: the fast, slow and Alfvén waves in the low-frequency limit (much lower than the ion cyclotron frequency ω ci ) and the long-wavelength limit (much longer than the ion Larmor radius ρ i ). When wavelengths reduce to the characteristic length scales, such as λ e and ρ i , where λ e is the electron inertial length, the ions are free while the electrons are tightly bound to magnetic field lines because their Larmor radius is much smaller than that of the ions (i.e. ρ e ρ i ). This leads to the presence of spatial charge and the dispersion of the MHD waves. In the case of the Alfvén wave, the dispersive wave is called the kinetic Alfvén wave (KAW). In the KAW, the perturbed electric fields have a non-zero component parallel to the background magnetic field that may cause efficient heating and acceleration of the plasma particles [1][2][3][4].KAWs have been extensively discussed because of their potential importance in the particle energization of plasmas.*Corresponding author (email: djwu@pmo.ac.cn) Also, they have been applied to laboratory, space and astrophysical plasmas, such as tokamak plasma heating [5][6][7], auroral electron acceleration [8][9][10][11], solar coronal plasma heating [12][13][14], and abnormal heating of heavy ions in the extended corona [15,16]. On the other hand, the other two modes, the fast and slow waves, have been given little attention. Some recent studies on the MHD turbulence in interstellar and interplanetary spaces show that the energy cascades primarily by developing small-scales structures perpendicular to the local field, with k ⊥ k [17][18][19][20]. Large values of k ⊥ could be the consequence of refraction [21,22], resonant absorption [23][24][25], and turbulent cascade [26]. This result is supported by numerical simulations of magnetized turbulence with a dynamically strong mean field [27,28]. In situ measurements of turbulence in the solar wind and observations of interstellar scintillat...