The number of jet and rocket noise studies has increased in recent years as researchers have sought to better understand aeroacoustic source and radiation characteristics. Although jet and rocket noise is finite-amplitude in nature, little is known about the existence of shock formation and coalescence close to the source. A numerical experiment is performed to propagate finite-amplitude noise and determine the extent of the nonlinearity over short distances with spherical spreading. The noise is filtered to have a haystack shape in the frequency domain, as is typical of such sources. The effect of the nonlinearity is compared in both the temporal and frequency domains as a function of distance. Additionally, the number of zero-crossings and overall sound pressure level is compared at several distances. The results indicate that the center frequency plays a particularly important role in the amount of coalescence and spectral redistribution that occurs. The general applicability of these results to actual near-field finite-amplitude jet and rocket noise experiments is also presented.