= Solution
Use the <shock frame>, with upstream and downstream fluid <velocities> $u_1=u$ and $u_2=u/4$ pointing in the same direction. These are the <Strong-shock Rankine-Hugoniot conditions> for a nonrelativistic <monatomic gas> with <adiabatic index> $5/3$. The two scattering media approach one another at relative <speed> $U=u_1-u_2=3u/4$, to first order in $u/c$. An energetic <electron> repeatedly crosses the <shock wave>, scatters from <magnetic turbulence> carried by the local <plasma>, and returns. Each scattering approximately conserves its <energy> in the local fluid rest frame while changing its direction; comparing successive fluid frames produces a systematic <energy> and <momentum> gain. The <energy> comes from the converging bulk flow, through the moving <magnetic field> irregularities and their associated <electric fields>. This is <diffusive shock acceleration>, a <first-order Fermi acceleration> process.
\Image[/past-exam-of-the-radiative-processes-and-high-energy-astrophysics-course-of-the-university-of-oxford/2024-shock-acceleration.png]
{title=Repeated shock crossings and the distinct upstream and downstream fluid speeds in the shock frame}
Let $\eta$ be the positive cosine of the crossing angle with the <shock wave>'s normal. The <Lorentz transformation> between fluid frames gives, for either crossing direction with the corresponding sign of the normal reversed,
$$
E'=E+Up\eta+O(U^2/c^2).
$$
For an almost <isotropic> population, the number crossing a unit area is weighted by its normal <velocity>. Thus the normalized crossing distribution is $2\eta\,d\eta$, rather than a uniform distribution in $\eta$, and
$$
\langle\eta\rangle=\frac{\int_0^1\eta^2\,d\eta}{\int_0^1\eta\,d\eta}=\frac23,
\qquad
\boxed{\langle\Delta E\rangle_{\rm crossing}=\frac23Up=\frac12up.}
$$
For an ultrarelativistic <electron>, $E\simeq pc$ and its <speed> $v\simeq c$, so
$$
\boxed{\frac{\langle\Delta E\rangle}{E}=\frac{u}{2c},\qquad
\frac{\langle\Delta p\rangle}{p}=\frac{u}{2c}.}
$$
More generally $dE/dp=v$ gives $\langle\Delta p\rangle/p=2U/(3v)$, while a nonrelativistic <kinetic energy> has $\langle\Delta K\rangle/K=4U/(3v)$. A complete upstream–downstream–upstream cycle contains \b[two] crossings and has ultrarelativistic fractional <energy> gain $u/c$.
The initially cold <electrons> need an injection stage: <shock wave> heating and small-scale <plasma> processes must first produce particles with <speed> much larger than $u$ that can return across the front. The $v\simeq c$ result is a description of that accelerated population; a cold <electron> cannot simply be assigned $E\simeq pc$ at its first encounter.
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