Circular waveguide modal spectrum – True or false A hollow circular waveguide supports an infinite set of discrete TE and TM modes determined by Bessel function eigenvalues.

Introduction:
Waveguides impose boundary conditions that quantize field distributions into discrete modes with specific cutoff frequencies. Circular waveguides are solved with Bessel functions, producing an ordered spectrum of transverse electric (TE) and transverse magnetic (TM) modes that emerges naturally from the cylindrical symmetry.

Given Data / Assumptions:

• Perfectly conducting circular boundary of radius a.
• Air-filled guide with uniform cross-section and no dielectric loading.
• Linear, time-invariant fields with time-harmonic excitation.

Concept / Approach:

Solving Maxwell's equations with PEC boundary conditions yields radial dependencies governed by Bessel functions Jm and their derivatives. The cutoff wavenumbers correspond to zeros of Jm (for TM) or Jm' (for TE). Because there are infinitely many such zeros for each order m, the guide supports an infinite ladder of TE and TM modes, each with its own cutoff frequency and field pattern.

Step-by-Step Solution:

Verification / Alternative check:

Mode charts list TE11, TM01, TE21, etc., continuing without bound as frequency increases. Only modes with operating frequency above their cutoff propagate, reinforcing that the set is infinite but frequency-selective.

Why Other Options Are Wrong:

Restricting to only TE or only TM is incorrect; requiring dielectric filling is unnecessary for modal multiplicity. Claiming a finite number of modes contradicts the infinite sequence of Bessel zeros.

Common Pitfalls:

Confusing “infinite set of possible modes” with “all propagate at once.” At any fixed frequency, only a subset above cutoff will propagate; below cutoff, modes are evanescent and decay rapidly.

Final Answer:

True.