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A model of the Earthquake surface waves
V.K.Ignatovich. FLNP JINR
STI2011 June 8
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This report is along the papersV.K. Ignatovich and L.T.N. Phan.
Those wonderful elastic waves. Am.J.Phys.
v. 77, n. 12, pp. 1093-I17, (2009)
A.N. Nikitin, T.I. Ivankina, and V.K. Ignatovich The Wave Field Patterns of the Propagation
of Longitudinal and Transverse Elastic Waves in Grain-Oriented Rocks
Physics of the Solid Earth, 2009, v. 45, n. 5, pp. 424-436
And a little bit more
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A theory of elastic waves In isotropic media
ljlj tu 22 )()( uu jjj u
Usually solution of this equation is represented as a sum
][ φu is a scalar potential
φ is a vector potential
however why not to do differently?)exp(),( tiit rkAru
22)(2 ijuF
u
i
j
j
iij x
uxuu
21
ijijijij uuF 2)( u
)()(22 uuuu t
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)exp(),( 0 tiiut rkAru 10 Au
)()( 22 AkkAAkkA k
)()(22 AκκAAκκA k
2: k
kkκ κttA 32211
κtt ,, 21
02,122 k
02 322 k
22222,1 tck 2tc
2
2223 2 lck
22 2tl cc
)()(22 uuuu t
2/ lt ccAll this is trivial. Reflection from interfaces is less trivial
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Reflection from a free surface
2A 2A3A
)exp()exp()exp()( 3332222222 zikrzikrzikz rri AAAu
ln
)()exp(),( |||| ztiit urkru
2||
22 kck nn tl cccc 23
03 kcAt such a critical angle A Longitudinal Surface wave appears
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Calculations of reflection amplitudes0 jiji n ijijij u 2)( u
0)()( unununΣ tiizik
rzik
rzik
i eereret ||||3223322222),( rkAAAru
03322222 rri rr ΣΣΣΣ
22222 )( iiii k AAnkΣ 22222 )( rrrr k AAnkΣ
333333 )( rrrr kk AAnknΣ
2||2 kki nlk 2||22 kkki nlA
:
2
||22||
22
2
2k
kkkki
nlΣ
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2
||22||
22
2
2k
kkkkr
nlΣ
3
2||
22||3
3
2k
kkkkr
nlΣ
03322222 rri rr ΣΣΣΣ
2
||22||
22
2
2k
kkkki
nlΣ
2||22
||3
3
23222
21
kkkk
kkrr
||2
2||
22
3
23222 2
1kkkk
kkrr
2
||3222
||22
2||
22||2
2
332
4
22
kkkkk
kkkkkkr
2
||3222
||22
22||
22
2||32
224
4
kkkkk
kkkkkr
nl,lc
k 3 2
||233 kkk
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2
||3222
||22
2||
22||2
2
332
4
4
kkkkk
kkkkkkr
22
||22
2||32
22||
22
2||32
224
4
kkkkk
kkkkkr
22
2
3 sinsin ck
k
22232
sinsin)sin()2sin(2)2(cos
)2cos()2sin(sin2
c
cr
-- angle of incidence
sin2|| kk
cos22 kk
222sin ltc cc
)2tan(sin2)(32 cccr
1)( 222 cr
2||
2222
2||
233 kcckkkk lt
1)(22 cr
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sincos2 nlA r lA 3r
cccr 2tansin2)(32
sincos2 nlA i
llA 8.612tantancos2 ccc
65.0sin ltc cc
462 A
1)(22 cr
lAAAA cccrri rr cos22tansin23322222
71)(| 22 Atliss ccEEQ
c
l12tantancos2 ccc
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ctl rrccQ sin)()( 232
232
2
222
232
sinsin)sin()2sin(2)2(cos
)2cos()2sin(sin2
c
cr
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Tomas Lokajicek, Vladimir Rudajev
V.K. Ignatovich. A proposal of a UCN experiment to check an earthquake waves model.Europhys. Lett. 92 (69002-p1-4) 2010.
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Experiments byLokajicek Tomas, Rudajev Vladimir
4E-005 6E-005 8E-005 0.0001time of flight [s]
-0.0 8
-0.0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
-0 .0 8
-0 .0 4
0
0.04
0.08
90 deg., 30 dB
80 deg., 30 dB
70 deg., 30 dB
65 deg., 30 dB
60 deg., 30 dB
55 deg., 30 dB
50 deg., 30 dB
40 deg., 36 dB
30 deg., 36 dB
20 deg., 36 dB
10 deg., 36 dB
5 deg., 36 dB
S5_S5_signal
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4E-005 6E-005 8E-005 0.0001tim e of flight [s]
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
90 deg., 30 dB
80 deg., 30 dB
70 deg., 30 dB
65 deg., 30 dB
60 deg., 30 dB
55 deg., 30 dB
50 deg., 30 dB
40 deg., 36 dB
30 deg., 36 dB
20 deg., 36 dB
10 deg., 36 dB
5 deg., 36 dB
S5_S5_reference
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90 deg.
0 deg.
113,5 mm
90 deg.
recieverS-wave transducer5 MHz resonant frequencydiameter 5 mm
transmitterS-wave transducer5 MHz resonant frequencydiameter 5 mm
material:
in 90 deg. P-wave time propagation: 41,8 s
perpsexthickness: 20 mmP-wave velocity: 2,72 km/sS-wave velocity: 1,37 km/s
[ ]S-wave time propagation: s82.9 [ ]
reference transducerP-wave transducer1 MHz resonant frequencydiameter 10 mm
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012tantancos2 lA ccc
5.0sin l
tc cc
0)30cos()60cos()60sin()30sin(12tantancos
ccc
62tansin4 2 cc
232rQ
02 A
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57.0sin l
tc cc
llA 4.112tantancos2 ccc
steel
So, to observe an effect we need a material with
ct/cl>0.6
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Anisotropic media
jlljut
2
2
ijklc -- a set of phenomenologocal constants
klijklij uc
j
l
l
jjl x
uxu
u21
In general 21 constants
222 )(2 jljljll uauuF
lmjmmlmjjllljljl
jl auaauauuuF
22
)()(22 AkkAA k
)]())[(()])(()([ 2 AakAakakAkakAaa k
But anisotropy means a vector and an additional constant. So we can define
)exp(),( tiit rkAru
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kkκ cbκA |][|][ κaκac
κa
aκaκκcb
][
))(())(()( 222 AaakAkkAak k
))(()(2 AkakAaa k
0)( 222 akk
κaab
0)(2 22 kakka
0)(2)(42 222 akkaakk
All we need is a linear vector algebra
κbc ,,
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0)(2 22 kakka
0)(2)(42 222 akkaakk
)cos4)(1(2sin 2222 czz
22sin4))cos41(1()cos41(1
12222222
cc
z
22 kz 222 2 tl ccc
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2cos1)( vt
)(cos κa
22sin4))cos41(1()cos41(1
1)(2222222
cc
vql
22sin4))cos41(1()cos41(1
1)(2222222
cc
vqt
1tc
58.1 tl ccc
5.0
)()( vqlcc tql
)()( vqtcc tqt
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It is important to saythat we cannot exclude
by averaging of values over alldirections of propagation,
because all the values depend on
22 )()(cos aκ
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Polarization of waves
babaκκabaκ
babaκκA ))((2
)()(41
))((2122222
2
EV
V
ql
qlql
κabaκbabaκ
babaκκA ))((2
)()(41
))((2122222
2
EV
V
qt
qtqt
122 qtql AA 0qtqlAA
))((212 abaκ VbκA
2
)()(16)(411)(411 22222222 abaκaκaκ
cc
κ a
b
22
c
12,qtqlV
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In an anisotropic medium propagate plane waves of only 3 modes
• transverse with Аt~[kxa] and ct2=ct0(1+)
• quasi transverse with Аqt in the plane [k,a]
• quasi longitudinal with Аql in the plane [k,a]
quasi longitudinalquasi transverse
)exp(),( tiit krAru
a akk
2Atransverse
a k1A 3A
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Reflection of a quasi transverse wave from a free surface
0,,, rqlltrqtttiqt rr ΣΣΣΣ
)()()( AknAnkAknΣ
)()()())(())(( AakAkaanAnkaAakna
One can find an analytical solution
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of two reflected waves
2.0)cos( a)(
sin)(
sin)(
sin
qtrqlrql
rql
rqtrqt
rqt
VVV
5.222
c
5.0
nl a
a rqt
rql
quasi longitudinal wave becomes surface one at 6.0
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It seems possible to find such a direction of vector a
that for given elastic parameters the amplitude of the
surface longitudinal wave becomes maximal.
2)2()2(
2
2
ql
qt
VV
For instance
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Summary• Reflection of elastic waves from free surfaces is
accompanied by beam splitting.• At some critical angle of the incident shear
wave polarized in the incidence plane a longitudinal surface wave is created.
• Its amplitude and energy can be large, and its polarization along the surface is alike to devastating earthquake waves.
• For observation of such waves the materials with ratio ct/cl>0.6 are needed.
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Thanks