= Maxwell shear-wave dispersion
{c}
{title2=$\tau\sigma^2+\sigma+\nu k^2=0$}
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 $k$, $\tau\to0$ leaves the <diffusion equation> branch $\sigma\to-\nu k^2$ and removes the fast <stress relaxation> branch $\sigma\sim-1/\tau$.
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