A simplified method for the analysis and synthesis of dispersed wave trains

Abstract

A disturbance at one point of a dispersive medium resulting from an impulse applied at another point may be represented as a superposition of traveling plane waves. The phase and period of the disturbance at any instant are related by the principle of stationary phase to the phase and period of a traveling wave component. For the instantaneous phase of that traveling wave component the following equation may be written. Ct‐x=(N‐ϕ0/2π)CT where C is the phase velocity, x the distance, T the period, t the travel time, N an integer, and ϕ0 the initial phase of the traveling wave component. Since t and T may be measured from a record of the disturbance and x may be determined, the equation may be used to compute the phase velocity as a function of period, if the initial phases are known. If distance and the dispersion are known, initial phases may be determined. From distance, initial phases, and phase velocities the disturbance at any point may be constructed. The practical use of the method is demonstrated by application to antisymmetric waves in a cylindrical rod, Rayleigh waves from United States and Russian nuclear explosions, Rayleigh waves from the Hudson Bay earthquake of January 30, 1959, and Love waves from the Fairview Peak and Fallen, Nevada, earthquakes of 1954.

Anisotropy and the structural evolution of the oceanic upper mantle

“…The technique for measuring phase velocities using only one station was originally developed by Brune et al (1960). Early studies using this method (Kuo et al, 1962) were hampered by the necessity of choosing an arbitrary initial phase which was independent of period.…”

“…The two-station method of measuring phase velocity (Brune et al, 1960) is inadequate for the purposes of this study.…”

Modeling anomalous surface - wave propagation across the Southern Caspian basin

“…In order to extend the dispersion curves to higher frequencies we use the method of Brune, et al (1960) to determine the phase velocity from events occurring within and on the boundary of the basin. The phase velocity curve may be obtained for a dispersed wave train provided the initial phase is known.…”

Source and Propagation Effects of Rayleigh Waves from Central Asian Earthquakes

“…Early surface wave studies made extensive use of focal phase through the phase equalization method (Aki, 1960a-d;Brune et al, 1960;Brune and.Pomeroy, 1962). Studies by Aki (1960a) The study of a small intra-plate earthquake in Southeastern…”

“…T.. refers to the time interval between the origin time of the jth earthquake and the start of the digitization window, tj, on the ith record. The integer, n, represents the order number (Brune et al, 1960).…”