Analyses of time-harmonic problems

In this chapter, we specifically study the time-harmonic Maxwell equations. They derive from the time-dependent equations by assuming that the time dependence of the data and fields is proportional to exp (− ıωt), for a pulsation ω ≥ 0 (the frequency is equal to ω∕(2π)). When the pulsation ω is not known, the time-harmonic problem models free vibrations of the electromagnetic fields. One has to solve an eigenproblem, for which both the fields and the pulsation are unknowns. On the other hand, when ω is part of the data, the time-harmonic problem models sustained vibrations. Generally speaking, we refer to this problem as a Helmholtz-like problem, for which the only unknown is the fields.