The uses of truncated series in describing the propagation of high-intensity broadband noise

The behavior of the high-frequency part of the spectrum of a broadband noise signal propagating through the atmosphere cannot, according to observation, be accounted for solely by linear mechanisms. Truncated series expansions of the initial signal can afford information on how these linear mechanisms, such as atmospheric attenuation and spherical spreading, might interact with weak nonlinearity to produce the effects observed in the evolved spectrum. Through a Taylor expansion we obtain expressions relating the evolved power spectral density solely to the initial power spectral density and other quantities available directly from measurement, regardless of initial shape. The energy cascade characteristic of broadband noise propagation can be explicitly linked to convolution terms in these expressions. It is also shown that attenuation and spherical spreading affect linear and nonlinear terms in the evolution equation in the same way, producing the same rate of decay. Numerical simulation confirms the validity of these results for distances before shock formation. In the absence of attenuation we find that the range of validity of Taylor expansions is greatly increased by construction of Padé approximants, yielding extremely accurate predictions of the evolved waveforms for distances only slightly shorter than the expected shock formation distance.