 Solution

The physical root of the dispersion relation isFor $A\ll 1$,This is a weakly localized, nearly grazing acoustic wave with speed close to the bulk sound speed.

For $A\gg 1$,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.