We discuss how excitons can affect the generation of coherent radial breathing modes in the ultrafast spectroscopy of single-wall carbon nanotubes. Photoexcited excitons can be localized spatially and give rise to a spatially distributed driving force in real space which involves many phonon wave vectors of the exciton-phonon interaction. The equation of motion for the coherent phonons is modeled phenomenologically by the Klein-Gordon equation, which we solve for the oscillation amplitudes as a function of space and time. By averaging the calculated amplitudes per nanotube length, we obtain time-dependent coherent phonon amplitudes that resemble the homogeneous oscillations that are observed in some pump-probe experiments. We interpret this result to mean that the experiments are only able to see a spatial average of coherent phonon oscillations over the wavelength of light in carbon nanotubes and the microscopic details are averaged out. Our interpretation is justified by calculating the time-dependent absorption spectra resulting from the macroscopic atomic displacements induced by the coherent phonon oscillations. The calculated coherent phonon spectra including excitonic effects show the experimentally observed symmetric peaks at the nanotube transition energies, in contrast to the asymmetric peaks that would be obtained if excitonic effects were not included.