The vibration behavior of double-walled carbon nanotube (DWCNT) is investigated based on Donnell shell theory with van der Waals interaction taken into consideration. In addition, new results are obtained for the natural frequencies of a DWCNT based on a simplified version of Donnell shell theory by neglecting the tangential inertia terms. The resulting reduced characteristic equation for the natural frequency represents the radial mode of vibration. The factor of neglecting tangential inertia relative to the values of frequencies obtained by full Donnell shell theory is also obtained with attendant interesting results. Further possible simplifications of Donnell shell theory are introduced. For the first time in the literature, the effect of neglect of tangential inertia terms in DWCNTs is investigated. Accurate approximate analytical formulas are uncovered for the fundamental natural frequencies and compared with the exact values.