  We calculate the excitation modes of a 1D dipolar quantum gas confined in a
harmonic trap with frequency $\omega_0$ and predict how the frequency of the
breathing n=2 mode characterizes the interaction strength evolving from the
Tonks-Girardeau value $\omega_2=2\omega_0$ to the quasi-ordered, super-strongly
interacting value $\omega_2=\sqrt{5}\omega_0$. Our predictions are obtained
within a hydrodynamic Luttinger-Liquid theory after applying the Local Density
Approximation to the equation of state for the homogeneous dipolar gas, which
are in turn determined from Reptation Quantum Monte Carlo simulations. They are
shown to be in quite accurate agreement with the results of a sum-rule
approach. These effects can be observed in current experiments, revealing the
Luttinger-liquid nature of 1D dipolar Bose gases.
