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The Fay solution is a periodic solution of the negative-flux viscous Burgers equation for $Z>0$. Its Fourier series has exponentially decreasing coefficients at each positive $\delta Z$. Expanding $1/\sinh (ns)=2{\sum }_{j\ge 0}{e}^{-(2j+1)ns}$ and summing the sine series givesFor $s\gg 1$, the first harmonic gives $q\sim 4\delta e^{-s}\sin \theta$. The Fay shock-layer asymptotics describe its opposite, small-$s$ regime.

Put $s=\delta Z>0$. For the Fay solution, $1/\sinh (ns)=2e^{-ns}[1+O(e^{-2ns})]$. At $s\gg 1$ the first harmonic therefore giveswhich agrees with the late-time limit in (i).

For the opposite limit, expand the reciprocal hyperbolic sine as a geometric series and interchange absolutely convergent sums at every $s>0$:This exact positive-denominator form is useful near the shock. For small $s$ and $|\theta |$, expand the denominator of the terms with small $(2j+1)s$ and use $\sin \theta \sim \theta$. The leading sum isFor the shock-layer scaling $\theta =s\eta$, the passage to this sum follows from dominated convergence: $\cosh z-1\ge z^2/2$ and $1-\cos \theta \ge c{\theta }^2$ for small $|\theta |$ provide a summable bound proportional to $[(2j+1{)}^2+{\eta }^2{]}^{-1}$. The same leading expression matches the outer small-angle range $s\ll |\theta |\ll 1$. More generally, split the exact sum at $(2j+1)s=b$ with $\max (s,|\theta |)\ll b\ll 1$: the low terms have vanishing relative Taylor error, while the discarded exact and approximate tails are smaller than the leading expression by $O(\max (s,|\theta |)/b)$.

Now apply the partial-fraction identity for the hyperbolic tangent with $\eta =\theta /s$:The Fay shock-layer asymptotics are thereforeThe angular shock thickness is $O(\delta Z)$, and its matched states are $\pm \pi /Z$. At $\theta =0$, both formulas vanish; the limiting slopes agree. The denominator throughout this derivation is $\sinh (n\delta Z)$ as printed in the PDF, not the erroneous $\sin (n\delta Z)$ in the converted TeX.