A model of the Earthquake surface waves

V.K.Ignatovich. FLNP JINR

STI2011 June 8

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

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

u

x

uu

2

1

ijijijij uuF 2)( u

)()(22 uuuu t

)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 cc

All this is trivial. Reflection from interfaces is less trivial

Reflection from a free surface

2A 2A

3A

)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

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

2

k

kkkki

nlΣ

2

||22||

22

2

2

k

kkkkr

nlΣ

3

2||

22||3

3

2

k

kkkkr

nlΣ

03322222 rri rr ΣΣΣΣ

2

||22||

22

2

2

k

kkkki

nlΣ

2||22

||3

3

23222

21

kk

kk

k

krr

||2

2||

22

3

23222 2

1kk

kk

k

krr

2

||32

22||

22

2||

22||2

2

332

4

22

kkkkk

kkkk

k

kr

2

||32

22||

22

22||

22

2||32

224

4

kkkkk

kkkkkr

nl,lc

k

3 2||

233 kkk

2

||32

22||

22

2||

22||2

2

332

4

4

kkkkk

kkkk

k

kr

22

||22

2||32

22||

22

2||32

224

4

kkkkk

kkkkkr

22

2

3 sinsin ck

k

22232sinsin)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)(2

22 cr

2||

2222

2||

233 kcckkkk lt

1)(22 cr

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)(|2

2 Atliss ccEEQc

l12tantancos2 ccc

ctl rrccQ sin)()(2

32

2

32

2

222

2

32sinsin)sin()2sin(2)2(cos

)2cos()2sin(sin2

c

cr

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.

Experiments by

Lokajicek Tomas, Rudajev Vladimir

4E-005 6E-005 8E-005 0.0001time of flight [s]

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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

4E-005 6E-005 8E-005 0.0001tim e of flight [s]

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0

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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

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

012tantancos2 lA ccc

5.0sin l

tc c

c

0)30cos()60cos(

)60sin()30sin(12tantancos

ccc

62tansin4 2 cc

2

32rQ

02 A

57.0sin l

tc c

c

llA 4.112tantancos2 ccc

steel

So, to observe an effect we need a material with

ct/cl>0.6

Anisotropic media

jlljut

2

2

ijklc -- a set of phenomenologocal constants

klijklij uc

j

l

l

jjl x

u

x

uu

2

1

In general 21 constants

222 )(2 jljljll uauuF

lmjmmlmjjllljljl

jl auaauauuu

F

22

)()(22 AkkAA k

)]())[(()])(()([ 2 AakAakakAkakAaa k

But anisotropy means a vector and an additional constant. So we can define

)exp(),( tiit rkAru

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 ,,

0)(2 22 kakka

0)(2)(42 222 akkaakk

)cos4)(1(2sin 2222 czz

2

2sin4))cos41(1()cos41(11

2222222

ccz

22 kz 222 2 tl ccc

2cos1)( vt

)(cos κa

2

2sin4))cos41(1()cos41(11)(

2222222

ccvql

2

2sin4))cos41(1()cos41(11)(

2222222

ccvqt

1tc

58.1 tl ccc

5.0

)()( vqlcc tql

)()( vqtcc tqt

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

Polarization of waves

babaκκabaκ

babaκκA ))((

2

)()(41

))((21

22222

2

EV

V

ql

qlql

κabaκbabaκ

babaκκA ))((

2

)()(41

))((21

22222

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

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

2A

transverse

a k

1A 3A

Reflection of a quasi transverse wave from a free surface

0,,, rqlltrqtttiqt rr ΣΣΣΣ

)()()( AknAnkAknΣ

)()()())(())(( AakAkaanAnkaAakna

One can find an analytical solution

of two reflected waves

2.0)cos( a)(

sin

)(

sin

)(

sin

qtrqlrql

rql

rqtrqt

rqt

VVV

5.222

c

5.0

n

l a

a rqt

rql

quasi longitudinal wave becomes surface one at 6.0

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

V

VFor instance

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.

Thanks