  The Holmboe wave instability is one of the classic examples of a stratified
shear instability, usually explained as the result of a resonance between a
gravity wave and a vorticity wave. Historically, it has been studied by linear
stability analyses at infinite Reynolds number, $Re$, and by direct numerical
simulations at relatively low $Re$ in the regions known to be unstable from the
inviscid linear stability results. In this paper, we perform linear stability
analyses of the classical `Hazel model' of a stratified shear layer (where the
background velocity and density distributions are assumed to take the
functional form of hyperbolic tangents with different characteristic vertical
scales) over a range of different parameters at finite $Re$, finding new
unstable regions of parameter space. In particular, we find instability when
the Richardson number is everywhere greater than $1/4$, where the flow would be
stable at infinite $Re$ by the Miles-Howard theorem. We find unstable modes
with no critical layer, and show that despite the necessity of viscosity for
the new instability, the growth rate relative to diffusion of the background
profile is maximised at large $Re$. We use these results to shed new light on
the wave-resonance and over-reflection interpretations of stratified shear
instability. We argue for a definition of Holmboe instability as being
characterised by propagating vortices above or below the shear layer, as
opposed to any reference to sharp density interfaces.
