Dispersion curves of electrostatic electron cyclotron harmonic waves with a phase exp {i(k. r – ωt)}, where ω and k are the angular frequency and wave vector, respectively, are calculated numerically for a case of real ω and complex k that is pertinent for a case of pulsed short dipole excitation of plasma waves. Oblique propagation with respect to the ambient magnetic field direction is considered. There are positive κi and negative κi modes, where k is the imaginary part of the wave number κ = ∣κ∣ in the solutions. The energy propagates in opposite directions in the two modes, and the waves are always subjected to thermal damping. The positive κi mode usually indicates a large κi value (i.e. a large attentuation rate); and it is here called the ‘short-life’ mode.