= Solution
{c}
Let $L$ be the junction length perpendicular to the width and $J_0$ be uniform. Put $\kappa=2\pi\Phi/(W\Phi_0)$. Integrating the local <current-phase relation of a superconducting weak link> over the junction gives
$$
\begin{aligned}
I_s(\Phi;\varphi_c)&=J_0L\int_{-W/2}^{W/2}\sin(\varphi_c-\kappa z)\,dz\\
&=J_0LW\sin\varphi_c\,\frac{\sin(\kappa W/2)}{\kappa W/2}.
\end{aligned}
$$
The odd-in-$z$ term integrates to zero. At fixed central phase $\varphi_c=\varphi(0)$, define $I(0)=J_0LW\sin\varphi_c$. Then
$$
\boxed{I_s(\Phi)=I(0)\frac{\sin(\pi\Phi/\Phi_0)}{\pi\Phi/\Phi_0}.}
$$
The value at zero flux is the continuous limit, and zeros occur at nonzero integer multiples of $\Phi_0$. This signed formula is the source's expression for a fixed phase. If the measured quantity is the maximum sustainable current, maximize over the central phase instead:
$$
\boxed{I_c(\Phi)=J_0LW\left|\frac{\sin(\pi\Phi/\Phi_0)}{\pi\Phi/\Phi_0}\right|.}
$$
For example, at $\Phi=3\Phi_0/2$ the signed coefficient is $-2/(3\pi)$, whereas the critical current is positive. The absolute-value envelope is the <Fraunhofer pattern of a Josephson junction>, shown in the right panel of the original tunnelling figure. Its simple form relies on uniform $J_0$ and the linear phase profile; it is not the two-junction cosine pattern of a <DC SQUID>.
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