= Solution
The physical root of the dispersion relation is
$$
s^2=\frac{\sqrt{1+4A}-1}{2A}.
$$
For $A\ll1$,
$$
\boxed{c=c_0\left(1-\frac A2+O(A^2)\right).}
$$
This is a weakly localized, nearly grazing acoustic wave with speed close to the bulk sound speed.
For $A\gg1$,
$$
\boxed{
c=c_0A^{-1/4}\left(1-\frac1{4\sqrt A}+O(A^{-1})\right)
\sim\sqrt{\frac{\mu}{\rho_0k}}.
}
$$
Now $c\ll c_0$, the decay rate is $\alpha\simeq k$, and the fluid is effectively incompressible. A layer of depth $1/k$ supplies <added mass> per area of order $\rho_0/k$, whose balance with the spring stiffness gives the final scaling.
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