= Solution
For $\mathbf r=R\widehat{\mathbf r}$ with the observation distance much larger than the scatterer, expand $|\mathbf r-\mathbf r'|=R-\widehat{\mathbf r}\cdot\mathbf r'+O(R^{-1})$. The <far-field pattern> in the <Born approximation for scalar wave scattering> is consequently
$$
\psi_s^{(1)}(R\widehat{\mathbf r})=\frac{e^{ikR}}R
f_\infty^{(1)}(\widehat{\mathbf r},\widehat{\mathbf r}_0)+O(R^{-2}),
$$
$$
\boxed{f_\infty^{(1)}=\frac{k^2}{4\pi}\int_V[n(\mathbf r')^2-1]
e^{-i\mathbf k_s\cdot\mathbf r'}\,d^3r',\qquad
\mathbf k_s=k(\widehat{\mathbf r}-\widehat{\mathbf r}_0).}
$$
This convention includes the factor $1/(4\pi)$ in $f_\infty$; moving that factor into the outgoing-wave definition would change the pattern's normalization.
The absolute <energy flux> depends on what physical amplitude $\psi$ represents. For acoustic <pressure> in a homogeneous fluid of <mass density> $\rho_0$ and <wave speed> $c_0$, the momentum equation gives the particle-velocity amplitude $\mathbf v=\nabla\psi/(i\rho_0\omega)$. The <time average of harmonic power> and <acoustic intensity> then give
$$
\langle\mathbf I\rangle=\frac12\operatorname{Re}(\psi\mathbf v^*)
=\frac1{2\rho_0\omega}\operatorname{Im}(\psi^*\nabla\psi).
$$
The radial derivative of the outgoing factor is $(ik-1/R)e^{ikR}/R$. Its imaginary current is therefore $k|f_\infty|^2/R^2$ to leading order. The incident unit-amplitude <plane wave> has imaginary current $k\widehat{\mathbf r}_0$. Hence, with $k=\omega/c_0$,
$$
\boxed{E_s(R)=\frac{k}{2\rho_0\omega R^2}|f_\infty^{(1)}|^2+O(R^{-3}),\qquad
E_i=\frac{k}{2\rho_0\omega}=\frac1{2\rho_0c_0}.}
$$
The scattered <energy flux> points along $\widehat{\mathbf r}$ at leading order, and the incident <energy flux> along $\widehat{\mathbf r}_0$. If $\psi$ is a velocity-potential amplitude instead, both leading expressions use the common prefactor $\rho_0\omega k/2$ in place of $k/(2\rho_0\omega)$. A normalized scalar current uses $k$ as the common prefactor. The paper does not specify this amplitude normalization; the ratio in part (c) is independent of that choice.
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