On the decay of a dipole pump wave to half-harmonics

Starting from a three-wave interaction coefficient for a homogeneous Vlasov plasma, we show that the parametric decay to half-harmonics has zero growth rate if the wavenumber of the pump wave vanishes. Earlier proofs of this fact for electrostatic decay modes are thus extended for a general case. As a specific example, we study the decay of a fast magnetosonic wave to two slow waves.