Mode analogy between circular and square waveguides The TM01 mode in a circular waveguide is most closely analogous (in symmetry and nodal pattern) to which TM mode in a square waveguide?

Introduction / Context:
Although circular and rectangular (square) waveguides have different eigenfunctions (Bessel versus sines/cosines), useful analogies can be drawn based on symmetry and the number of field variations across the aperture. This helps engineers transfer intuition between geometries.

Given Data / Assumptions:

• Circular waveguide mode: TM01 (no azimuthal variation, single radial lobe with central maximum).
• Square waveguide modes: TMmn with m and n variations along the two transverse axes.

Concept / Approach:
TM01 in circular has an on-axis maximum and no angular variation. The closest square-guide analogue is a mode with one half-wave variation along one axis and none along the other, namely TM10 (or by symmetry TM01). From the provided options, TM10 best matches the symmetry and nodal structure of circular TM01.

Step-by-Step Solution:

Verification / Alternative check:
Mode charts show the lowest TM in circular is TM01; in rectangular/square, the simplest TM with a single variation is TM10/TM01, supporting the analogy.

Why Other Options Are Wrong:

• TM11: two transverse variations, not analogous in symmetry to TM01.
• TM20/TM02: higher-order patterns with two half-waves along one axis.

Common Pitfalls:
Assuming identical cutoff constants; analogies are qualitative, based on symmetry and nodal structure.

Final Answer:
TM10 mode in square waveguide