  Moir\'e patterns are additional, long-range periodicities in twisted
crystalline bilayers. They are known to fundamentally change the electronic
states of the layers, but similar effects on their mechanical and vibrational
properties have not been discussed so far. Here we show that the Moir\'e
potential shifts the radial breathing mode in double walled carbon nanotubes
(DWCNTs). It is expected that the frequency change is proportional to the shift
in optical transition energies, which are induced by the Moir\'e patterns. To
verify our model we performed resonant Raman scattering on the purified and
sorted semiconducting DWCNTs. We find that the radial breathing mode in the
Raman spectra of DWCNTs is up to 14 cm$^{-1}$ higher in energy and their
optical transitions energies up to 200 meV lower in energy than in the
corresponding single-walled tubes. The shift follows the expected linear
correlation with a slope -88 eV/cm$^{-1}$. We show how to identify the strong
coupling condition in DWCNTs from their phonon frequencies and construct a
Kataura plot to aid their future experimental assignment.
