The axial and torsional wave propagation in a double-walled carbon nanotube (DWCNT) embedded on elastic foundations are investigated using nonlocal continuum shell theory. The effects of the surrounding elastic medium are considered using the spring constant of the Winkler-type and the shear constant of the Pasternak-type. The van der Waals (vdW) forces between the inner and the outer nanotubes are taken into account. The dynamic response of the carbon nanotube is formulated on the basis of nonlocal elasticity shell theory. The cut-off frequencies are obtained and it has been concluded that the cut-off frequencies are independent of small scale coefficient and shear modulus of the elastic medium. It has been found that the phase velocity sharply decreases by increasing the axial half wave number and approaches a constant value. It has also been concluded that the maximum phase velocity predicted by nonlocal theory is located between 5 and 10 nanometers while for local theories the phase velocity sharply decreases in this interval and approaches a constant value. Results show that the effect of Pasternak-type on phase velocity is significant but the effect of Winkler-type is not really considerable.