00518507

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ハドウ ホウテイシキ カイセキジョウ ノ モンダイテン Klein-Gordon Equation

On the Solution of Wave Equations : Klein-Gordon Equation

Japanese (日本語)

神戸大学都市安全研究センター研究報告

11

17-34

神戸大学都市安全研究センター

2007-03

2008-07-09

Wave equations with initial and boundary conditions have been investigated following two different schemes. The one relies on Green's functions and concepts of delta functions (DG scheme) for inhomogeneous equations. The other uses Laplace transforms and general solutions of homogeneous equations (Sommerfeld,1917; Stratton,1941; S scheme) with some initial and boundary conditions. In seismology the former corresponds to potential methods and the latter Bessel function methods, respectively. In previous studies, we argued some problems concerning these two schemes and pointed out that in many cases of DG schemes only particular solutions (Love, 1944) are investigated and general solution are not included, while in S schemes both solutions for homogeneous and inhomogeneous equations are considered. The importance of general solutions in constructing Green functions is emphasized in text books. Here, we call waves (solutions) obtained from inhomogeneous equations with sources or singularities forced waves and those of general solutions of homogeneous equations free waves. In ordinary waves equations, solutions of homogeneous and inhomogeneous ones are similar and thus this problem has been little considered. However, in the Klein-Gordon (KG) equation that has a term proportional to the variable (displacement and potential), distinctions appear in solutions. Deutch and Low (1993) investigated wave propagation of gaussian through a barrier region using S scheme for KG equation. They revealed evanescent solutions and intriguing phenomena of tunneling waves, which have been confirmed in many experiments of optic and electromagnetic waves. The results are in contrast to those obtained by DG scheme that show propagation of modulated waves. In the analysis of wave equations, we have to take into consideration not only particular solutions of inhomogeneous equations (forced waves) but also general solutions of homogeneous equations (free waves). However, in many cases, only forced waves from inhomogeneous equations are evaluated. Of importance is conversion and interference of these two type waves. Indeed, if some boundaries or heterogeneities that interfere with wave fields exist in the media, we should consider the influence of free waves as well as forced waves. The results of Deutch and Low using KG equation show physical situations represented by homogeneous and inhomogeneous equations are different. In seismology, this problem is important, particularly, in earthquake source studies. If we study only situations of inhomogeneous systems and attributes features of observed waves to assumed distribution of body forces in the source area, we might introduce some artifacts and fail to understand actual source processes correctly.

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