Elementary derivation for the minima of the interference pattern in the case of a thin wire without referring to the Babinet principle

For the single slit there is a well known elementary derivation along the following line: If the path difference $\delta$ between the elementary waves coming from the borders of the slit is one wave length $\lambda$ you imagine say 100 elementary waves evenly distributed starting from the slit. Then the path difference between the 1. and 51. wave is $\lambda/2$ so they interfere destructively, the same path difference occurs between the 2. and 52. wave, the 3. and 53, and so on. So you get a interference minimum for $\delta = \lambda$ . If $\delta = k\lambda$ ( $k \in \{1,2,3,\dots\}$ ) you just devide the ray bundle into $k$ bundles, where in each bundle the path difference between the "boundary rays" is $\lambda$ and you can argue as in 1. So for each bundle you get complete destructive interference. From geometry it follows then that you get minima for $\sin(\alpha) = \frac{k\lambda}{b}$ where $b$ is the width of the slit. From Babinets principle it follows that the same formula should apply for the interference pattern of a opaque thin wire of thickness $b$ . Is there any direct elementary derivation (like that one for the single slit as sketched out above) for the case of the thin wire?