= Solution
Let $a_j=\nu\kappa_j^2t+\kappa_jx$. Differentiating the logarithm gives
$$
u=-2\nu\frac{\kappa_1e^{a_1}+\kappa_2e^{a_2}}{e^{a_1}+e^{a_2}}.
$$
For definiteness order $\kappa_1<\kappa_2$. Then $e^{a_1}$ dominates as $x\to-\infty$, while $e^{a_2}$ dominates as $x\to+\infty$. The limiting states are
$$
\boxed{u_L=-2\nu\kappa_1,\qquad u_R=-2\nu\kappa_2,\qquad u_L>u_R.}
$$
Without this ordering convention, $u(-\infty)=-2\nu\min(\kappa_1,\kappa_2)$ and $u(+\infty)=-2\nu\max(\kappa_1,\kappa_2)$. Thus the states $u_{1,2}$ correspond to $-2\nu\kappa_{1,2}$ after matching each label to its dominant exponential; the propagation speed does not depend on the naming of the two states.
Their exponent difference is
$$
a_2-a_1=(\kappa_2-\kappa_1)\left[x+\nu(\kappa_1+\kappa_2)t\right].
$$
Consequently, the <Viscous Burgers travelling shock> moves at
$$
\boxed{V=-\nu(\kappa_1+\kappa_2)=\frac{u_L+u_R}{2}=\frac{u_1+u_2}{2}.}
$$
Set $\Delta u=u_L-u_R=2\nu(\kappa_2-\kappa_1)>0$ and $\xi=x-Vt$. Factoring out the mean exponential gives the <hyperbolic tangent> profile
$$
\boxed{u(x,t)=V-\frac{\Delta u}{2}\tanh\left(\frac{\Delta u\,\xi}{4\nu}\right).}
$$
A precise measure of the width must be stated. The natural length in the <hyperbolic tangent> argument is
$$
\boxed{\ell=\frac{4\nu}{\Delta u}=\frac{2}{|\kappa_2-\kappa_1|}.}
$$
Thus the layer has thickness of order $\nu/\Delta u$. A different common convention, jump divided by maximum gradient, gives $\delta=\Delta u/|u_x(0)|=8\nu/\Delta u=2\ell$; the far-tail exponential e-folding length is $\ell/2$. These differ only by their stated definitions, rather than by different solutions.
Two distinct limits require $\kappa_1\ne\kappa_2$. The source permits arbitrary real constants, but in the degenerate case $\kappa_1=\kappa_2$ the solution is uniform, $u=-2\nu\kappa_1$, and there is no <shock wave> or finite shock width.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2022-q5-viscous-shock.png]
{title=Calculated travelling shock profile and its tangent width; the horizontal coordinate moves with the shock}
Back to article page
