When multiple resonant cavities are coupled together (e.g., in filters or klystron intermediate cavities), how many distinct resonant frequencies typically result?

Introduction / Context:
Resonant cavities form the basis of many microwave components. When several cavities are coupled, the electromagnetic energy can distribute across them, producing collective normal modes whose frequencies are shifted relative to the uncoupled resonance. This is analogous to coupled mechanical oscillators.

Given Data / Assumptions:

• n nominally identical (or similar) cavities.
• Weak to moderate coupling through irises, slots, or transmission lines.
• Losses are small enough that resonances remain well-defined.

Concept / Approach:

Coupling lifts degeneracy: each cavity no longer resonates independently. The system supports n normal modes, each with a distinct field distribution and resonant frequency. The frequency set spans a passband whose width increases with coupling strength, a principle central to bandpass cavity filters.

Step-by-Step Solution:

Verification / Alternative check:

Measured S21 of n-cavity filters shows n poles within the passband; circuit models using coupled resonator theory predict the same pole count and spacing determined by coupling coefficients.

Why Other Options Are Wrong:

• One frequency only: Ignores mode splitting.
• Continuum: Requires distributed structures without discrete resonators or very strong losses.
• Conductivity-only dependence: Coupling, not just loss, sets mode count.
• Zero resonances: Non-physical for practical low-loss coupling.

Common Pitfalls:

Assuming identical frequencies remain unchanged with coupling; overlooking end-effects and loading which slightly detune edge cavities.

Final Answer:

n distinct resonant frequencies (mode splitting)