Klein–Gordon Equation Under Random Pulse Forcing: Exact Statistics and the Cut-Off Spectral Peak

We study slow magnetoacoustic waves in a thin, gravitationally stratified flux tube filled with a strongly ionised two-fluid plasma. The tube is driven at its base by a train of pulses that arrive at random times and carry random amplitudes. Arrival times form a Poisson process. Because the governing Klein–Gordon equation is linear, every realisation has an exact Green’s function representation. Campbell’s theorem then gives the mean, the cumulants, the autocovariance and the power spectrum of the ion velocity in closed form. The stratified tube acts on the driver spectrum as a filter that attenuates frequencies below the ion cut-off Ωion and passes the rest unchanged. We prove that the spectral maximum sits at exactly Ωion above a critical height z*, and below it the maximum is at zero frequency. For a gamma-shaped pulse of width τ we find z*=(cT/Ωion)ln(1+Ωion2τ2). The autocovariance oscillates at Ωion and decays as s−3/2 at large lag s, so a random driver produces a quasi-periodic signal at the cut-off with algebraically decaying oscillatory correlations. The neutrals, dragged by the ions, carry the same signal. Monte Carlo simulations and a finite-difference solver confirm the analytical results.