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Conducting cavities confine waves, permitting only discrete standing-wave normal modes at resonant frequencies determined by geometry. Each mode has specific field distribution and resonant frequency. Used as filters and oscillators in microwave engineering and particle accelerators.
You already understand waveguides: they confine electromagnetic waves to propagate along a single direction by imposing conducting boundary conditions on the transverse cross-section, which forces the fields to fit into discrete modes with frequencies above a cutoff. A cavity resonator takes this idea one step further — you close off the waveguide at both ends with conducting walls, trapping the wave entirely. The result is a fully enclosed 3D box for electromagnetic energy.
When you cap both ends of a waveguide, you introduce a third boundary condition: the fields must also satisfy the conducting-wall requirement in the propagation direction z. The fields in the cavity are now standing waves in all three directions, and only specific combinations of field patterns can fit inside the box while satisfying the boundary conditions everywhere simultaneously. These are the normal modes (or resonant modes) of the cavity. For a rectangular cavity of dimensions a × b × d, the resonant frequencies are f_mnp = (c/2)√[(m/a)² + (n/b)² + (p/d)²], where m, n, p are non-negative integers (not all zero) labeling the mode. Each distinct triple (m,n,p) corresponds to a unique standing-wave field pattern.
The key physics is the energy trapping: unlike a waveguide where power flows continuously along the guide, a cavity stores energy. At resonance, energy sloshes back and forth between the electric field (concentrated when charges are maximally separated) and the magnetic field (concentrated when currents flow). This is the electromagnetic analogue of a mass-spring oscillator trading potential and kinetic energy — a connection you can make precise through the circuit analogy of an LC resonator. The ratio of stored energy to power dissipated per cycle is the quality factor Q, which can be extremely large in metal cavities (Q ~ 10⁴–10⁵) because the only loss is resistive heating in the small skin-depth layer at the cavity walls.
These properties make cavity resonators indispensable wherever precision frequency control is needed. In particle accelerators, microwave cavities at precisely tuned resonant frequencies impart energy to charged particles on each pass — the cavity's high Q means the driving source must supply only the small energy lost to the walls, while the cavity itself stores the bulk of the field energy. In radar and telecommunications, cavities act as narrow-band filters: only signals near a resonant frequency couple efficiently to the cavity, rejecting everything else. The mode index (m,n,p) determines both frequency and field geometry; choosing which mode to excite is a design decision that controls where the fields concentrate and how efficiently energy is transferred.