Waves velocity on a cosmic string

I’m trying to understand a few things about cosmic strings in cosmology. Consider a single string of energy density $rho$ and tension $tau > 0$. I don’t assume $tau = rho$, as apparently most authors on cosmic strings seem to imply.

We could define at least two velocities on the string:
begin{align}
v_1 &= sqrt{frac{tau}{rho}}, tag{1} [1ex]
v_2 &= sqrt{frac{dtau}{drho}}. tag{2}
end{align}
If $tau = rho$, then $v_1 = v_2 = 1$ (speed of light, in units of $c equiv 1$). I believe (I may be flat wrong) that $v_1$ is the velocity of transverse waves on the string, while $v_2$ is the velocity of sound waves along the string, and not a phase velocity and group velocity, respectively. Is this true? How can we prove this?

Why most authors in cosmic strings are — apparently — assuming that $tau = rho$?

Do I really have to go on the scalar field route and studies the perturbations of the field to demonstrate the two perturbation velocities? Where can I find the calculations (Wikipedia or arXiv)?

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About the equation of state of a fluid of cosmic strings, please see this question :

Straight cosmic string energy-momentum tensor and the cosmic strings EoS