  We consider the linear stability of shear banded planar Couette flow of the
Johnson-Segalman fluid, with and without the addition of stress diffusion to
regularise the equations. In particular, we investigate the effect of
two-dimensional perturbations representing undulations along the interface
between shear bands. We demonstrate analytically that, for the linear stability
problem, the limit in which diffusion tends to zero is mathematically
equivalent to a pure (non-diffusive) Johnson-Segalman model with a material
interface between the shear bands, provided the wavelength of perturbations
being considered is long relative to the (short) diffusion lengthscale.
  For no diffusion, we find that the flow is unstable to long waves for almost
all arrangements of the two shear bands. Weak diffusion provides a small
stabilising effect, rendering extremely long waves marginally stable. However,
the basic long-wave instability mechanism is not affected by this, and where
there would be instability as wavenumber k tends to 0 in the absence of
diffusion, we observe instability for moderate to long waves even with
diffusion.
  This paper is the first full analytical investigation into an instability
first documented in the numerical study of Physical Review Letters 95 (2005)
134501 cond-mat/0501518. We discuss the relevance of this work to recent
experimental observations of complex dynamics seen in shear-banded flows.
