Yes, that is correct. It does not matter if the vibrations are longitudinal or transverse. The same equations have the same solutions.
I guess the reason for this is that if you start with a chain in equilibrium and you displace a mass in any direction, there is always a linear restoring force. So except for the fact that the restoring force (i.e., the effective spring constant) may be different in the two directions, the newton's equations that you write down are exactly the same.
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