Maxwell shear-wave dispersion ($\tau {\sigma }^2+\sigma +\nu k^2=0$)

In a homogeneous Maxwell fluid, transverse momentum conservation and shear stress relaxation give $\tau u_{tt}+u_t=\nu u_{yy}$, where $\nu =\mu /\rho$ is the kinematic viscosity. A Fourier mode $e^{iky+\sigma t}$ has the displayed quadratic dispersion relation. For $4\tau \nu k^2>1$ its two exponents oscillate while decaying at rate $1/(2\tau )$; for smaller wavenumbers they decay without oscillating. At fixed $\nu$ and $k$, $\tau \to 0$ leaves the diffusion equation branch $\sigma \to -\nu k^2$ and removes the fast stress relaxation branch $\sigma \sim -1/\tau$.