Acoustic wave propagation in the solar atmosphere 1. Rediscussion of the linearized theory including nonstationary solutions

The normal dispersion analysis for linear adiabatic wave propagation in stratified atmospheres adopts a real frequency and solves for the complex vertical wavenumber. We show that an exponentially stratified atmosphere does not have any spatially bounded normal modes for real frequencies. The usual treatment involves a representation where the imaginary part of the vertical wavenumber yields a rho(sup -1/2) dependence of the velocity amplitude which diverges as the absolute value of z approaches infinity. This solution includes a cutoff frequency below which acoustic modes cannot propagate. The standard dispersion analysis is a local representation of the wave behavior in both space and time but which is assumed to represent the motion throughout - infinity is less than t is less than infinity and 0 is less than infinity. However, any solution which has a purely sinusoidal time dependence extends through this full domain and is divergent due to the rho(sup -1/2) dependence. We show that a proper description is in terms of a near field of a boundary piston which is driven arbitrarily as a function of space and time. The atmosphere which responds to this piston is a semi-infinite layer which has an initially constant sound speed but which has the usual gravitational stratification. In a restricted domain of space and time above this boundary, the wavelike behavior of the medium may be described by frequencies and vertical wavenumbers which are both complex. When both parameters are allowed to have imaginary components, a new range of solutions is found for which there is virtually no cutoff frequency. We show that vertical energy propagation can take place through the solar atmosphere as a result of oscillations below the nominal cutoff frequency. Previously, the largest amplitude oscillations which generally have low frequencies were dropped from the calculation of energy flux becuase their frequencies are below the cutoff frequency. This new family of near-field waves permits these modes to carry energy vertically outward and raises the possibility that the largest amplitude 5 minute oscillations play a substantial role in the transport of acoustic energy to the chromosphere.