Annual Report 2000 - WIT
Annual Report 2000 - WIT
Annual Report 2000 - WIT
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¬<br />
’ £ }}¤† à £6 ¥ ‹ £ ##¦† à £ Ú ¥ ‹ £ [[¦† à £ Ú ¨§<br />
©<br />
<br />
£ }}¤† £ ##¤† à £ Ú ¥ † à £S ¥ £ [[¦†ª‘ £ ÚÚ « £S « É<br />
’<br />
¬‡ ¾ †<br />
µ<br />
Ò š · <br />
¥<br />
¬ š<br />
Ã<br />
¾ <br />
<br />
¬<br />
š»º<br />
¿<br />
Ú<br />
<br />
²<br />
Ú<br />
¸<br />
¾<br />
¬<br />
š¯º<br />
ÿ<br />
· Â<br />
À<br />
Ú<br />
’Ä ‘¦Ã<br />
õ<br />
Ù<br />
Ú<br />
Ú<br />
¸<br />
¬ š<br />
š»º °<br />
<br />
<br />
Ú<br />
²<br />
º<br />
Ú<br />
Œ<br />
‹<br />
F<br />
¸<br />
¾ <br />
<br />
¬<br />
H š»º<br />
¼<br />
<br />
245<br />
symmetry if non-zero elastic moduli satisfy:<br />
£ Ú 6’ ²<br />
‘ £ ÚÚ £S Ã £ Ú ’S§ (4)<br />
I«’ £ }}¤† £ ##¤† £ [[¦†ª‘ £S « £6¥¥ « É<br />
‘ £S S£S¥¥ Ã £ ¥ ’ †<br />
£6 ¥6’ ²<br />
The case is most simple but not useful for the purpose of approximating the slowness<br />
surface of one type of wave since the ellipsoidal symmetry appears as a result<br />
of crossing slowness surfaces of different type of waves. The solution for this case is<br />
given by (Ettrich et al., <strong>2000</strong>).<br />
We consider the case © . Here there are five independent parameters £ ÚÚ<br />
, £6 , £S¥¥ ,<br />
£ }}<br />
and £ Ú<br />
. These parameters must be adjusted to best approximate an anisotropic<br />
medium by an ellipsoidal one. Taking the physics of wave propagation into account,<br />
we minimize the average of the sum of the differences between the Christoffel matrices<br />
of the ellipsoidal medium. Here,<br />
† –<br />
³ÖẼš®ÖS®¡˜<br />
¾ <br />
of the ³¯š<br />
³š<br />
anisotropic medium and ¬<br />
is the -component of the unit vector that points into the direction of ®`³ the wave front<br />
propagation. For details, see (Fedorov, 1968) and (Ettrich et al., Ô <strong>2000</strong>).<br />
For the case ©<br />
¾ <br />
with (4) ³¯š the Christoffel matrix simplifies to:<br />
¬<br />
£ Ú ¥6’ ²<br />
£ Ú 6’ ²<br />
‘ £ ÚÚ £6 Ã £ Ú ’<br />
†ª‘ £ ÚÚ « £ z(z « É<br />
‘ £ ÚÚ £6¥¥ Ã £ Ú ¥ ’ †e‘ £ ÚÚ « £6 « É<br />
M+, £ ÚÚ<br />
0/T<br />
® ¥<br />
® Ú « £ [[<br />
® « £ }}<br />
‘ £ Ú « £ [[ ’ ®<br />
®<br />
Œ<br />
°t±² and<br />
Œ £ }}¨‘<br />
® Úz« ® ’ « £6¥¥ ® ¥<br />
Œ<br />
‘ £ Ú « £ [[ ’ ®<br />
® £ [[<br />
® Ú « £S ® « £ }}<br />
® ¥<br />
5E¾<br />
5£¾<br />
5£¾<br />
†´³<br />
½¼ †e³<br />
¼ Ã É ³<br />
¼»¿ ³<br />
¬ ¾ <br />
Ú¹¸<br />
has to be minimized with respect £ ÚÚ<br />
to £6 , £6¥¥ , £ }}<br />
, £ Ú<br />
and .<br />
a function ° :<br />
denotes averaging<br />
?À<br />
ÿ<br />
m › m(n ‹<br />
Ç ~ § †<br />
¾ <br />
· Á<br />
where m ‹ n¡’ is a wavefront normal. This averaging is used in order to remove<br />
the dependency of the function on the direction ((Fedorov, 1968)). Since parameter<br />
ÃņGÃÆ‘ [[<br />
is a complicated function of £ ÚÚ<br />
, £S and £ Ú<br />
it is more convenient to consider £ [[<br />
as<br />
£<br />
an independent parameter and to seek the minimum of<br />
°È±²<br />
¸ U<br />
with respect £ ÚÚ<br />
to £S , £S¥¥ , £ }}<br />
, £ Ú<br />
, £ [[<br />
and U . is Lagrange's factor.<br />
Ç †<br />
‘ £ ÚÚ ¿<br />
¿ É<br />
£6 <br />
£ Ú 6’EÉ<br />
‘ £ ÚÚ £S Ã £ Ú ’ Ã £ [[
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