Based on a linearized hydrodynamic model and within the quasi-static approximation, the dispersion relation of electrostatic waves propagating through single-walled carbon nanotubes subject to an axial magnetic field is theoretically explored. In the classical limit, we obtain two main possible waves which in turn are divided into two branches, a low-frequency acoustical and a high-frequency optical plasmon branch. In the quantum case, we have found that the dispersion relation is substantially modified when the electron wavelength becomes large enough compared to the propagation wavelength of the electrostatic waves in the quantum plasma. We also show that the axial magnetic field manifest itself on the perturbed electron density through the quantum term and gives rise to the propagation of the electrostatic waves within the quantum plasma. As a result, the effect of the magnetic field is pronounced in the plasma dispersion relations in such a way that their curves approach to zero when the magnetic field is weak; and for the strong magnetic field, they asymptotically meet the constant lines.
