Let the incident plane wave be
$$
\psi_i(\mathbf r)
=e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r},
$$
and define the scattering vector
$$
\mathbf q=k_0
(\widehat{\mathbf r}-\widehat{\mathbf r}_0).
$$
For $r$ much larger than the diameter of $D$, the <far-field pattern> follows from
$$
G_0(\mathbf r-\mathbf r')
\sim\frac{e^{ik_0r}}{4\pi r}
e^{-ik_0\widehat{\mathbf r}\cdot\mathbf r'}.
$$
Writing
$$
\widetilde V(\mathbf q)
=\int_DV(\mathbf r')e^{-i\mathbf q\cdot\mathbf r'}\,d^3r',
$$
the Born result is
$$
\boxed{
\psi_B(\mathbf r)
\sim e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r}
+\frac{e^{ik_0r}}{4\pi r}
\widetilde V(\mathbf q)}.
$$

The corresponding Rytov logarithmic perturbation is
$$
\phi_1(\mathbf r)
\sim a(\mathbf r)\widetilde V(\mathbf q),
\qquad
a(\mathbf r)=
\frac{e^{ik_0(r-\widehat{\mathbf r}_0\cdot\mathbf r)}}
{4\pi r}.
$$
Therefore
$$
\boxed{
\psi_R(\mathbf r)
\sim e^{ik_0\widehat{\mathbf r}_0\cdot\mathbf r}
\exp\left[a(\mathbf r)\widetilde V(\mathbf q)\right]}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 335 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-335/2/ii/solution