Annual Report 2000 - WIT

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137
Triggering fronts for the case of a quasi-harmonic pressure perturbation
In the following we shall consider relaxation of a harmonic component of a pressure
perturbation. By analogy with (4) we will look for the solution of (1) in a similar form:
Ÿ=ì Å
×Yí0î
åjæÄçDC ÷ éFE¦š{õƒŸ G0• (15)
蚊õ!•.ˆ=Ÿt‘¯1ä0šŠõ
We also will assume 1ä0š{õƒŸ that E!šŠõƒŸ , »½×…Ú(šŠõ°Ÿ and are functions slowly changing õ with .
Substituting (15) into (1), accepting é as a large parameter and keeping only terms
with largest powers é of (these are terms of the H’š é Ÿ order ; the other terms, which are
and Hšé ÇI ÒÆŸ are neglected) we obtain the following equation:
Ÿ
of the orders Hšé
È (16)
¬­ºW‘»½×…Ú
Ê1×
ÊÛÚ
Considering again the homogeneous-medium solution (4) we conclude that the
frequency-independent E quantity is related to the frequency-dependent phase travel
time as follows:
È (17)
Note, that in turn KJ
E‘³šŠº¦¬Ï̦Ÿ ÷ é
̦ ÷ é . Substituting equation (17) into equation (16) we obtain:
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È (18)
Ì´‘n“¦éA»P×]Ú
In the case of an isotropic poroelastic medium this equation is reduced to the following
one:
È (19)
Ê1×
ÊÛÚ
Thus, we have obtained a standard eikonal equation. The right hand part of this equation
is the squared slowness of the slow wave. One can show ( jCerveny, 1985) that
equation (19) is equivalent to the Fermat's principle which ensures the minimum time
(stationary time) signal propagation between two points of the medium. Due to equation
(17) the minimum travel time corresponds to the minimum attenuation of the signal.
Thus, in this sense, equation (19) describes the minimum-time maximum-energy
front configuration.
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