= Solution
Write the negative-flux <Inviscid Burgers equation> as $f_Z+\partial_\theta F(f)=0$, with $F(f)=-f^2/2$. Along a <characteristic curve>,
$$
\frac{d\theta}{dZ}=-f,\qquad\frac{df}{dZ}=0,
$$
so a characteristic starting at $\theta_0$ has $\theta=\theta_0-f_0(\theta_0)Z$ and
$$
\boxed{f(Z,\theta_0-f_0(\theta_0)Z)=f_0(\theta_0)}.
$$
This formula is a classical solution only while the <method of characteristics> is one-to-one; after <characteristic crossing> one must select an <entropy solution>.
For a discontinuity with left and right values $f_-$ and $f_+$, the <Rankine-Hugoniot condition> follows by integrating conservation across a moving small interval:
$$
\theta_s'(f_+-f_-)=F(f_+)-F(f_-).
$$
For a nonzero jump, factor the difference of squares to obtain
$$
\boxed{\theta_s'=-\frac12(f_++f_-).}
$$
The negative flux makes increasing jumps compressive. Decreasing jumps spread into <rarefaction waves>; they cannot be retained as nonphysical expansion <shock waves>.
In the central ramp of each period, $f_0(\theta_0)=\theta_0-2m$, so its <method of characteristics> is $\theta-2m=(1-Z)(\theta_0-2m)$. For $0<Z<1$, the retained ramp therefore occupies $|\theta-2m|<1-Z$. At the odd boundary $r=2m+1$, the initial limiting values are $+1$ on the left and $-1$ on the right. Their <characteristic speeds> are $-1$ and $+1$, giving the fan $f=-(\theta-r)/Z$ on $|\theta-r|<Z$. Together these give the <periodic backward-sawtooth Burgers solution>
$$
\boxed{f(Z,\theta)=\begin{cases}
(\theta-2m)/(1-Z),&|\theta-2m|<1-Z,\\
-(\theta-(2m+1))/Z,&|\theta-(2m+1)|<Z,
\end{cases}\qquad0<Z<1,\quad m\in\mathbb Z.}
$$
These intervals tile the real line up to their matching endpoints, where both formulas agree at $\pm1$. The increasing ramp steepens, but the wave's maximum magnitude remains one before breaking. A steep continuous regularization of the original downward jump produces exactly the limiting fan, as suggested by the characteristic construction.
At $Z=1$, each increasing ramp collapses at $\theta=2m$. The fans on either side meet there with values $-1$ and $+1$, producing a compressive stationary <shock wave>. For all $Z\geq1$ the fan between successive <shock waves> remains centered at the odd point, giving
$$
\boxed{f(Z,\theta)=\frac{2m+1-\theta}{Z}\quad(2m<\theta<2m+2),\qquad
\theta_s=2m.}
$$
At a <shock wave> the limiting values are $f_-=-1/Z$ and $f_+=1/Z$. Their average is zero, so the <Rankine-Hugoniot condition> keeps the <shock wave> fixed. Their <characteristic speeds> satisfy $1/Z>0>-1/Z$, confirming compression into the <shock wave>. The arbitrary pointwise value at a <shock wave> does not affect the <weak solution>. The post-breaking amplitude decays as $1/Z$, despite the lack of explicit <viscosity>, because <shock wave> dissipate the wave.
For the requested sketches, $Z=1/3$ has ramp slope $3/2$ on $|\theta-2m|<2/3$ and fan slope $-3$ around odd points on width $2/3$. It is still continuous. At $Z=3$, the profile decreases linearly with slope $-1/3$ between even points and jumps from $-1/3$ to $+1/3$ at each even point:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-75-burgers-sawtooth.png]
{title=Periodic Burgers wave before breaking at Z equals one third and after breaking at Z equals three, with stationary shocks at even theta}
{height=440}
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