Mode identification in a rectangular waveguide: If the electric field shows one half-wave variation across the narrow wall and two half-wave variations across the broad wall, what is the resulting dominant mode notation?

Introduction / Context:
Rectangular waveguides label modes as TEmn or TMmn. The first index m counts half-wave variations across the broad dimension a, and the second index n counts half-wave variations across the narrow dimension b. Accurately mapping spatial variations to indices is a common exam task.

Given Data / Assumptions:

• Across the narrow side (b): one half-wave variation.
• Across the broad side (a): two half-wave variations.
• We are to provide the correct mode label.

Concept / Approach:

By convention, m corresponds to the broad dimension a and n to the narrow dimension b. Two variations across a imply m = 2; one across b implies n = 1. Because the statement refers to the electric-field pattern in a typical hollow guide, the appropriate family is TE. Therefore, the mode is TE21.

Step-by-Step Solution:

Verification / Alternative check:

Field sketches for TE21 exhibit two lobes along a and one along b, consistent with the description. Reference charts align with this notation.

Why Other Options Are Wrong:

• TM12 and TM25: wrong indices and family.
• TE13: implies three variations across b, not one.
• TE02: implies zero variations along a, which contradicts the given two.

Common Pitfalls:

Swapping a and b or forgetting which index maps to which dimension.

Final Answer:

TE21