Second harmonics have been observed in various magnetospheric waves. Although particle simulations are able to produce the similar features as reported in satellite observations, the exact physical mechanism for the generation of second harmonics remains uncovered yet. In this letter, we first give wave equations describing the fundamental and second harmonics via multiscale analysis. Then, we apply a kinetic theory approach to obtain the conductivity tensor, nonlinear current density, and the necessary conditions for the second harmonic generation of electromagnetic emissions in a magnetized plasma. To verify our theoretical results, we also perform hybrid simulations to study the second harmonic generation of electromagnetic ion cyclotron waves for three cases in which the second harmonic deviates from the inherent modes by different degrees. Our results suggest the important role of the linear dispersion relation in the second‐harmonic generation.

Electromagnetic ion cyclotron (EMIC) wave is one of the most important ion-scale plasma waves in the inner magnetosphere, which often appear near the magnetic equator with frequencies below the local proton gyrofrequency (Anderson et al., 1992;Jordanova et al., 1997;Loto'aniu et al., 2005). EMIC waves are often observed with left polarizations and the direction of propagation is along the background magnetic field when the source region is near the magnetic equator (

Electromagnetic ion cyclotron (EMIC) waves are frequently found to be naturally generated with frequencies below the proton gyrofrequencies in the terrestrial magnetosphere (

Electromagnetic ion cyclotron (EMIC) waves and fast magnetosonic (MS) waves play important roles in the evolution of the terrestrial magnetosphere. EMIC waves have typical characteristic frequencies below proton cyclotron frequencies. Due to the presence of heavy ions, EMIC waves can be divided into three common bands: the H + band, He + band, and O + band (

Observations of non‐linear second harmonic (SH) in various waves have recently been reported by several studies. However, the quantitative relation of the amplitudes between the fundamental wave (FW) and the SH has not been derived and verified yet. In this letter, the relation equations for the FW and the SH are simplified under the cold plasma assumption, which is reasonable in the inner magnetosphere. The quantitative relation of the amplitudes between the FW and the SH of the electromagnetic ion cyclotron wave is theoretically derived and verified through 1‐D hybrid simulations. The quantitative influences of the FW frequency and wave normal angle on the SH excitation mechanism are obtained.