Field statistics of two vectorially superposed wave populations

Field statistics of observed waves and radiation constrain the physics of the emission process and source region. However, data often contain two or more superposed signals or a signal superposed on a noise background, creating difficulties for interpretation. Here, the combined probability distribution of the field formed by vector superposition of two signals, each with specified statistics, is written as a double integral with integrable singularities. The analytic result and its numerical solutions for combinations of Gaussian and lognormally distributed signals show that these predictions differ from those for field or intensity convolution and from the individual wave distributions. At high fields, the combined distribution takes the qualitative form of the dominant individual distribution (which is localized or otherwise extends to larger fields) but develops a significant tail at low fields due to vector superposition of almost antiparallel fields with similar magnitudes. It is shown that very nearly power-law distributions can develop in significant field domains, despite neither component distribution being power law. This is relevant to alternative interpretations in terms of self-organized criticality and certain modulational wave instabilities. The formalism is then applied to observations of the Vela pulsar, resulting in greatly improved fits to data and different interpretations. Specifically, the results are strong evidence that stochastic growth theory (SGT) is relevant and that the approximate power-law statistics found at some phases are not intrinsic but rather due to vector convolution of a Gaussian background with a lognormal; the latter is interpretable in terms of SGT. The field statistics are consistent with the emission mechanism being a direct linear instability or indirect generation via linear mode conversion of nonescaping waves driven by a linear instability. They are inconsistent with nonlinear self-focusing instabilities generating the observed pulsar radiation. This formalism and its results should also be widely applicable to other types of wave growth in inhomogeneous media.