  We study the 2D linear stability analysis of a deformable solid of a finite
height $H$, steadily sliding on top of a rigid solid within a generic
rate-and-state friction type constitutive framework, fully accounting for
elastodynamic effects. We derive the linear stability spectrum, quantifying the
interplay between stabilization related to the frictional constitutive law and
destabilization related both to the elastodynamic bi-material coupling between
normal stress variations and interfacial slip, and to finite size effects. The
stabilizing effects related to the frictional constitutive law include
velocity-strengthening friction (i.e.~an increase in frictional resistance with
increasing slip velocity, both instantaneous and under steady-state conditions)
and a regularized response to normal stress variations. We first consider the
small wave-number $k$ limit and demonstrate that homogeneous sliding in this
case is universally unstable, independently of the details of the friction law.
This universal instability is mediated by propagating waveguide-like modes,
whose fastest growing mode is characterized by a wave-number satisfying $k
H\!\sim\!{\cal O}(1)$ and by a growth rate that scales with $H^{-1}$. We then
consider the limit $k H\!\to\!\infty$ and derive the stability phase diagram in
this case. We show that the dominant instability mode travels at nearly the
dilatational wave-speed in the opposite direction to the sliding direction.
Instability modes which travel at nearly the shear wave-speed in the sliding
direction also exist in some range of physical parameters. Finally, we show
that a finite-time regularized response to normal stress variations, within the
framework of generalized rate-and-state friction models, tends to promote
stability.
