  We calculate the energy spectrum of a k=2 flux tube that is closed around a
spatial torus, as a function of its length l. We do so for SU(4) and SU(5)
gauge theories in 2 space dimensions. We find that to a very good approximation
the eigenstates belong to the irreducible representations of the SU(N) group
rather than just to its center, Z_N. We obtain convincing evidence that the
low-lying states are, for l not too small, very close to those of the
Nambu-Goto free string theory (in flat space-time). The correction terms appear
to be typically of O(1) in appropriate units, much as one would expect if the
bosonic string model were an effective string theory for the dynamics of these
flux tubes. This is in marked contrast to the case of fundamental flux tubes
where such corrections have been found to be unnaturally small. Moreover we
find that these corrections appear to be particularly small when the `phonons'
along the string have the same momentum, and large when their momentum is
opposite. This provides information about the detailed nature of the
interactions in the effective string theory. We have searched for, but not
found, extra states that would arise from the excitation of the massive modes
presumably associated with the non-trivial structure of the flux tube.
