By J M T Thompson; G W Hunt
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Additional info for A general theory of elastic stability
Fourier transform of a pulse function. Origin of Earthquakes and Seismic Waves 35 If a series of pulses repeating each other at equal intervals of time are generated in the earthquake’s focus they form a periodical function. According to Fourier’s theorem any periodic or semi-periodic function can be presented as a sum of sines and cosines—a Fourier series. Each sine and cosine comes with their own amplitude and phase—Fourier coefficients. 14) T For a series of pulses the Fourier’s coefficients are: 1 2 am = 0; m = 1, 2,3,...
A longitudinal elastic wave displaces the ground particles along the direction of its propagation, while a shear elastic wave moves the ground particles along some of the directions that are perpendicular to the direction of propagation of the wave. Two independent shear waves and one longitudinal wave can propagate in each direction in an isotropic infinite solid. In planewave approximation (the medium of propagation and the wave-front radius are considered to be infinite) the two independent shear waves have flat fronts and orthogonal polarizations.
Medium’s particles do not move on straight lines but follow complex trajectories that make the polarization a mixture of P- and S-modes which are not orthogonal to each other as it is in the case of a pure-mode plane wave. In the hypocenter the bulk elastic wave may start its propagation as a longitudinal elastic wave. Through multiple reflections, refractions, and mode conversions at the boundaries between domains this P-wave generates a multitude of new P- and S-waves. Because of the complex boundary conditions the fronts of the waves get distorted to the point where they cannot anymore be classified as plane P-waves or S-waves.
A general theory of elastic stability by J M T Thompson; G W Hunt