Development of advanced methods for microseismic monitoring of hydraulic fracturing

Review of longitudinal perturbation formalism

“…Substituting into (1), moving to the right side, and using orthogonal polynomial expansion, we get [2,7], where N (m) is called the frequency dispersion matrix, whose elements are, Note that M (m) is proportional to the beam intensity, therefore, at very low intensity, the system is dominated solely by N (m ). This matrix is real and symmetric, which decomposes the original system into several lossless resonators with slightly difference inherent oscillation frequencies, and introduces no instability mechanism.…”

“…The transition of the two different status shows the property of Landau damping. The equation (20) can be used for study of Landau damping in the process of antidamping, damping, and the combined situation, with a reasonable computational load [7]. To include mode coupling, is substituted into the equation (15).…”