"Dispersion" in Time-Distance Helioseismology

In time-distance helioseismology, travel time is the time taken by a wave packet to travel between two spatially separated locations on the surface of the Sun. It is computed by cross-correlating oscillation signals at the two locations and identifying the position of the envelope peak of the cross-correlation function, or the position of one of its phase peaks, as the travel time. The wave packet spectrum is a subset of the signal spectra. Adding more frequencies to the wave packet spectrum is shown to not necessarily narrow the width of the envelope of the cross-correlation function. "Dispersion" in the travel time across the spectrum restricts the minimum width of the cross-correlation function and shifts the position of the envelope and phase peaks as a function of the central frequency and width of the wave packet spectrum. Wave packets at the surface of polytropes show no dispersion in travel time; hence, Gaussian spectra yield Gaussian envelopes, and envelope widths at constant central frequency go to zero with increasing spectral width, showing no shift in the envelope peak or phase peaks. In the Sun, however, dispersion is inherent: Envelope and phase peaks are functions of the central frequency and width of the spectrum, and Gaussian spectra do not yield Gaussian envelopes and can even conspire to resemble a sum of two or more Gaussians.