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We study the excitation of a damped harmonic oscillator by a random force as a model for the stochastic excitation of a solar p-mode by turbulent convection. An extended sequence of observations is required to separate different p-modes and thus determine the energies of individual modes. Therefore, the observations yield time-averaged values of the energy. We apply the theory of random differential equations to calculate distribution functions for the time-averaged energy of the oscillator. The instantaneous energy satisfies a Boltzmann distribution. With increasing averaging time the distribution function narrows, and its peak shifts toward the mean energy. We also perform numerical integrations to generate finite sequences of time-averaged energies. These are treated as simulated data from which we obtain approximate probability distributions for the time-averaged energy. A comparison of our calculated distributions with those determined observationally should help to resolve whether the solar p-modes are stochastically excited. If they are, modes of the same frequency with degree l ≾ 200 should have identical values for the products of their mean energies, line-widths, and masses. If, in addition, turbulence or radiative dissipation provides the principal damping mechanism, the mean energies should be independent of angular order, l.

Distribution functions for the time-averaged energies of stochastically excited solar p-modes