Analysis of the Optical System:
The setup consists of a point source $S$ inside a reflecting cylindrical tube. The light rays emitted by $S$ can reach the hole directly or after undergoing multiple reflections from the inner surface of the tube.

The perfectly reflecting inner surface of the tube acts like a series of plane mirrors arranged in a cylinder. This creates a set of virtual images (virtual sources) of the original source $S$.

• The direct ray comes from the actual source $S$ on the axis.
• Rays undergoing one reflection appear to diverge from a virtual source ring at a radial distance $2r$ from the axis.
• In general, rays undergoing $n$ reflections appear to diverge from virtual sources located at a radial distance $2nr$ from the central axis.

Geometry of the Rings:
The opaque disc with the central hole acts as a pinhole camera aperture. We can analyze the formation of the pattern by tracing rays from the virtual sources through the center of the hole to the screen.

Let $y_n$ be the height of the $n$-th virtual source from the axis, and $R_n$ be the radius of the $n$-th ring formed on the screen.

Using similar triangles formed by the central axis, the virtual source, the hole, and the image on the screen: $$ \frac{y_n}{l} = \frac{R_n}{L} $$

The height of the $n$-th virtual source (corresponding to $n$ reflections) is: $$ y_n = 2nr $$ Where $r$ is the radius of the tube.

Substituting $y_n$ into the similarity equation: $$ \frac{2nr}{l} = \frac{R_n}{L} $$

Solving for the radius of the $n$-th ring, $R_n$: $$ R_n = \frac{2nrL}{l} $$