Open AccessA phase‐front method ‐ III. Acoustic waves, P ‐ and S ‐waves
Author(s) -
Haines A. J.
Publication year - 1984
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1984.tb01926.x
Subject(s) - curvilinear coordinates , conservative vector field , mathematical analysis , love wave , isotropy , physics , diffraction , acoustic wave equation , wave propagation , envelope (radar) , mechanical wave , phase (matter) , acoustic wave , phase velocity , mathematics , classical mechanics , longitudinal wave , geometry , optics , mechanics , quantum mechanics , telecommunications , radar , computer science , compressibility
Summary. A new approach to modelling the propagation of short‐period seismic waves is further developed by presenting the theory for acoustic waves and for 3‐dimensional systems involving P ‐ and S ‐waves in isotropic media. The method is based on the fast, numerical solution of approximate, parabolic forms of the equation of motion in heterogeneous media, which are formulated in the time domain and are derived by, first, choosing a curvilinear space coordinate so that the surfaces of constant phase roughly coincide with the surfaces on which the coordinate is constant, and then ordering the terms in the equation of motion according to magnitude. The parabolic approximations, which are only valid for waves whose wavelengths are small compared with the scale‐length of variations in material properties, resemble the equations of ray theory, except that the parabolic approximations allow for narrow‐angle diffraction. To the first level of approximation P ‐ and S ‐waves propagate independently. The principal component of each type of wave can be approximated by the solution to a parabolic equation which involves the appropriate wave speed. The additional component, which is defined in terms of the principal component, is such that acoustic waves and P ‐waves are essentially irrotational and S ‐waves are essentially equivoluminal. Numerical solutions to a simple 2‐D problem are presented for two initial disturbances with the same time‐varying envelope but different periods. The results for the two frequencies are quite different. The higher‐frequency disturbance has foci near where they are predicted by ray theory, whereas the lower‐frequency disturbance does not. The approximate solution for the lower‐frequency disturbance, which, like the solution for the higher‐frequency disturbance, is valid for P ‐ and S ‐waves as well as for acoustic waves, since the approximate equations have effectively the same form, is compared with the exact solution to the corresponding problem for time‐harmonic acoustic waves, and is in good agreement.