Advances in technology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices. Such gigahertz oscillating devices made from carbon nanotubes might be instrumental in the micro-computer industry, which is predominantly based on electron transport phenomena. There are many experiments and molecular dynamics simulations which show that a wave is generated on the outer cylinder by the oscillation of the carbon nanotubes and that the frequency of this wave is also in the gigahertz range. However, conventional applied mathematical modelling techniques are generally lacking. In order to analyse and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here, we find the van der Waals force using the Lennard-Jones potential to calculate the oscillation frequency using Newton's second law for double-walled carbon nanotubes of any length of the inner and the outer tubes, 2L1 and 2L2, respectively. These results are based on work by the present authors derived in (Baowan and Hill).