3. harmonic ((, , y gp, s, rayleigh & love)...

3. HARMONIC

(P, S, Rayleigh & LOVE)( , , y g )

WAVESWAVES

George BouckovalasP f f N T U AProfessor of N.T.U.A.

October, 2016

The Seismic motion is due to …….

Pressure (P), Shear (S), Rayleigh (R) and other wave typespropagating through the ground (soil & bedrock) ...GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.1

3.1 ELASTIC WAVESELASTIC WAVES ((P & S)P & S)in UNIFORM MEDIAin UNIFORM MEDIA

P WAVE S (SH SV) WAVE

in UNIFORM MEDIAin UNIFORM MEDIA

P- WAVEwith planar front

S (SH, SV) - WAVEwith planar front

ww

SV

SH

v v

u u

Compression - extension

P-waves

Wave length

S-wavesshearing

S waves

Wave lengthGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.2

Wave equation dx

Wave equation . . .A

x dxx

2

x dxxx

u¶2

2

tu

x ux x

uΌμως σ Dε D

x

¶= =-

¶x, u

2 2x

2 2

σ u uκαι D ρ

x x t

¶ ¶ ¶=- =-¶ ¶ ¶x x t¶ ¶ ¶

DC

General solutionGeneral solution...

( )( )u f Ct x=

why;

Y Ct x=

( ) ( ) ( )22

2 2f Yu ¶¶

( ) ( ) ( )2 2

2 2

f YuC C f ''

t Y

¶¶ = =¶ ¶

¶ ¶2 22u u

άρα πράγματι

2 2

2 2

u f (Y )ενώ f ''

¶ ¶= =

¶ ¶=¶ ¶

22 2

u uC

t x

2 2f

x Y¶ ¶GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.3

Physical meaning...Physical meaning...

The problem variables are NOT two independent The problem variables are NOT two independent ones (‘x’ and ‘t’) but a combined one: Y= x±Ct

If Y=Ct-x If Y=Ct+x

then, t=0 Y= -xo

t≠0 Y=Ct-xt=-xo

then, t=0 Y= xo

t≠0 Y=Ct+xt=xot 0 Y Ct xt xo& xt = xo + Ct

f(Ct-x) f(Ct+x)

t 0 Y Ct xt xo& xt = xo - Ct

tt=0

f(Ct x)

o)

t t=0

( )

f(-x

o

xo+Ctxo x xo-Ct xo x

P & S – wave propagation velocities& p p gin soil & rock formations

EGS- waves: 12

EG,GVs

P- waves:

21

1G2

211

1ED,DVp

21211

( / )VS (m/s) VP (m/s)dry saturated

loose-recent SOIL deposits <400 <1000 ≈1500

stiff SOILS – soft ROCKS 400-800 800-1600 1500-2000

Rocks >800 >1600 >2000GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.4

Vibration velocity (at a point) VVibration velocity (at a point) V

f ( Ct )u f ( x Ct )

* *u u X u x XV u

* *V u

t t x tX X

xV C x

xά V C CD

D

Vibration velocity (at a point) V

f ( Ct )

Vibration velocity (at a point) V

u f ( x Ct ) ATTENTION:

* *u u X u x XV u

The velocity of VIBRATION is different

(2 to 3 orders of magnitude lower!)* *

V ut t x tX X

than the velocity of PROPAGATION

xV C x

xά V C CD

DGEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.5

Special Case: Harmonic wavesp m

( )( ) ( )

ωi Ct x i ωt kxCf Ct x Ae Ae

+ ++ = =( )af Ct x Ae Ae+ = =

( ) ( ) ( )ω

i Ct x i ωt kxCbf Ct x Ae Ae

- -- = =( )bf Ct x Ae Ae

where

k /C 2 /(Τ C) 2 /λ bk =ω/C = 2π/(Τ•C) = 2π/λ = wave number

Ground VIBRATION at a given POINT (i e x=x )

titiikxf * Harmonic vibration

(i.e. x=xο)

tio

tiikxba exueAef o *, with frequency

ω=2π/Τ

fa,b

T=2π/ω

tku*(xo) =Ae+ikxo

fa,b u*(xo) =Ae+ikxo

xxo

GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.6

Ground DISPLACEMENT at a given INSTANT OF TIME (i e t=t )at a given INSTANT OF TIME (i.e. t=tο)

ikxikxti etueAef o * oba etueAef o ,

Όπως προηγουμένως αλλά τώρα τον ρόλο As before, but now the role of t ς ρ γ μ ς ρ ρτης συχνότητας τον παίζει ο κυματικός

αριθμός k

,frequency is played by

the wave number kΤ=2π/ωx=xkλ=2π => λ=2π/k

Τ=2π/ω

to

x=xo

[k= ω/C λ = 2π/ωC = T·C]u(xo,t)

xo

u(x) t=to

λx

EXAMPLE: Two rods in contact

x k kxti

oeuu 1

xkti2 eu

kxtiB kxtiBeu 1

12ρ C

ρ1,C1

ρ2,C2

i t kx i t kx i t kxou e u u e Be 2 1

x xu uM M M Mx x

2 12 1

i t kx i t kx i t kxoM ke Mu ke MBke GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.7


Boundary conditions:

x=0 u1=u2 d l & x=0 u1=u2

σ1=σ2

displacement & stress compatibility

tititioo MBkekeMuekM o21 MBkekeMuekM

22

o C

C

M

M

k

kBu

h k /C M C2 d kΜ C

11CMk

where k=ω/C, M=C2ρ and kΜ=ωCρ


Buouou 22C1

o21

C

Buouou o

22

11 u

C

C1

C1

B

11

22o C

CBu 11C

S i l

o22

u

C

C1

2

Special cases:

F d C 0 B & Γ 2

11C

Free end: ρ2C2=0 B=uo & Γ=2uo

Fixed end: ρ2C2=∞ B=-u & Γ=0Fixed end: ρ2C2 B uo & Γ 0


3.2 ELASTIC WAVES IN nonELASTIC WAVES IN non--UNIFORM UNIFORM MEDIA ( ith i t f ) MEDIA ( ith i t f ) MEDIA (with interfaces) MEDIA (with interfaces)

SNELL’s LAW:SNELL’s LAW: 2121

C

sin

C

sin

C

sin

C

sin

2121 sspp CCCC

C1 > C2

VIBRATION AMPLITUDE of reflected & transmitted refracted wavesof reflected & transmitted- refracted waves

I th l d k im l f 1 D ti :

medium 1: ρ1 C1 medium 2: ρ2 C2

In the already known, simple case of 1-D wave propagation:

transmitted incident wave

medium 1: ρ1, C1 medium 2: ρ2, C2

wave

reflected wave Cwave 2 2

1 1. .

2 2

1

1refl inc

CC

u uC

Vibration amplitude u:

2 2

1 1

.

1

1 refl

C

uu u

. .

.

1

2

ftransm inc

inc

u uu

.2 2

1 11

incuCC


VIBRATION AMPLITUDE of reflected & transmitted refracted waves

The amplitude of vibration of the reflected and refracted of reflected & transmitted- refracted waves

pwaves in 2-D problems, is computed based on stress equilibrium and strain compatibility (continuity) at the interface. In this case, the amplitude is also a function of the angle of wave incidence.

EXAMPLE: SH waves (Richter 1958)

2 2

. 1 1

cos1

cosrefl

Cu C

θδ

2 2.

1 1

cos1

cosinc

CuC

2

1 1

. . 21refr reflu u

C

θπρ θαν

1

2 2. .

1 1

cos1

cosinc inc

Cu uC

C1>C2SHinc

Criticalrefraction angle: 121 C

sinsinsinsin

refraction angle:1

221

21

1C

i1i

Csinsin

CC

2

2

121

Cii

1C

sin1sin

21

1

2cr2

C

Csinsin

1

21cr2 C

Csin

C1>C2What happens for θ2 > θcr ?

Application:Application:

Seismic refraction method for geophysical exploration geophysical exploration ……….


3.3 SURFACE WAVESSURFACE WAVESWave length

αδιατάραχτο μέσοαδιατάραχτο μέσο

P-wave+

SV wave RayleighRayleighSV-wave+

FREE C/CS

Rayleighwave

SURFACECR=0.94CS

Ground displacements due to R-waves

( ) ( )RR Rz z2 5.8 i ωt k xλ λu u z x t iB 0 55 e e e

- - -ì üï ïï ï= = -í ý( )

( ) ( )RR Rz z2 5.8 i ωt k xλ λ

u u z ,x ,t iB 0.55 e e e

w w z x t B 1 66e 0 92e e- - -

= = í ýï ïï ïî þì üï ïï ïí ý( ) ( )RR Rw w z ,x ,t B 1.66e 0.92e e= = -í ýï ïï ïî þ

k / b w/wo & u/uo

/

kR=ω/CR wave numberλR=CRT wave lengthΒ = constant u/uoΒ = constant

λR z/λR

w/wo

R

u

w|w(0)||w(0)|

|u(0)|=1.6


LOVE WAVES (in layered soil profile)( y p )

They are SH waves “trapped” within the upper soft layer

of a non-uniform soil profile.of a non uniform soil profile.

CS11 S11

CS22

LOVE WAVES (in layered soil profile)( y p )

They are SH waves “trapped” within the upper soft layer

of a non-uniform soil profile.of a non uniform soil profile.

Ground displacement variation with depth

CS11 S11

CS22


( )1 L 1 L Lir k z ir k z i( ωt k x )1u Ae Be e- -= +

C

L

( )2 L Lir k z i( ωt k x )

2u Γe e- -=CS1

C

L Lk ω / C=

CS2

2 2L L

1 22 2

C Cr 1, r 1

C C= - = -

2 2s1 s2C C

( )1 L 1 L Lir k z ir k z i( ωt k x )1u Ae Be e- -= +

C( )2 L Lir k z i( ωt k x )

2u Γe e- -=CS1

C

1( ) : 0z H ύ ά

CS2

1 2 1 20 ( ) : ,z ά u u

1 1 0

0

ir kH ir kHAe Be substituting for

A B

1 1 1 1 2 2

0 , ,

0 :

A B

A G r B G r G r gives

2LC1 «Dispersion»

relationship

L22S 2L 2

2 2S1 1

1CC GωH

tan 1C C G C

-é ùê ú- =-ê úê úë û relationshipL S1 1 L

2S1

C C G C1

Cê úë û -


Observe that, since tan(*) must be a real number, the following condition must be satisfied for the previous following condition must be satisfied for the previous equation to have a real solution:

C C CCS1 < CLOVE < CS2

CCs2

2LC1

C

L22S 2L 2

2 2

1CC GωH

tan 1C C G C

-é ùê ú- =-ê úê úë ûCs1

0

L S1 1 L2S1

C C G C1

Cê úë û -

0 ΩΗ/CL

In addition, LOVE waves are not possible unless the surface layer is softer than the underlying mediumy y g(i.e. CS1 < CS2).

Dispersion: Code name for LOVE (and other) waves h h h b f d d l which exhibit a frequency dependent propagation velocity

(even for a uniform medium).

Implications of dispersion:

The propagation velocity CLOVE decreases with increasing frequency ω (decreasing period Τ).q y g pAs a result….. the low-frequency components of the excitation get separated from the high-frequency excitation get separated from the high-frequency components, as they propagate faster. Thus:

Th d ti f h ki i ith di t f The duration of shaking increases with distance from the source

Th f t t f th ti ( th l ti The frequency content of the motion (e.g. the elastic response spectrum) changes with distance from the sourcesource


Special Case:Soft soil layer upon BEDROCK (i.e. CS2=∞)

CLOVE

CSH2122

LOVE

H

HC

bedrock

2

sC

bedrock

Fi ll LOVE

CLOVE

Finally, LOVE waves cannot exist for f i l th th frequencies lower than the fundamental vibration frequency of the soft soil

CS

frequency of the soft soil layer, i.e. when

Critical frequency

π/2 ωΗ/CS

ω < ωC= (π/2) Cs/HCritical frequencyωc= (π/2) CS/H


PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3PROBLEM SOLVING FOR CHAPTER 3

HWK 3.1:HWK 3.1:The free end of an infinite rod is displaced according to the The free end of an infinite rod is displaced according to the following relationship:

U=0 for t<0U=t (cm) for 0<t<0.1sU=0 2 t for 0 1s<t<0 2sU=0.2-t for 0.1s<t<0.2sU=0 for 0.2s<t

Find the displacement variation along the rod at t=0.3s. Assume that the wave propagation velocity is 300m/s.

(aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα)


HWK 3 2:HWK 3 2:HWK 3.2:HWK 3.2:

To isolate a precision measurement laboratory (e.g. a geotech lab.) from traffic vibrations it is proposed to Α traffic vibrations, it is proposed to construct the trench ΑΑ’. Which of the following two fill materials is Lab

Α

gpreferable :(α) concrete, with ρ=2.5 Mgr/m3 &

Lab

C=2000 m/s, or(b) pumice, with ρ=0.8 Μgr/m3

groundρ=2.0 Mgr/m3

C=100 m/s ?ρ gC=500 m/s

Α’

HWK 3.3:HWK 3.3:D h H h h h h l f l Draw the SH wave propagation path through the soil profile shown below, and compute the horizontal acceleration applied t h il lto each soil layer.

± 0mCs=300m/s ρ=1.6Mgr/m3

- 20mCs=600m/s ρ=1.8Mgr/m3

- 40mCs=1200m/s ρ=2.2Mgr/m3

- 60m

Cs=2400m/s ρ=2.4Mgr/m3

SH50o

SH

(uo=1.6cm, T=0.40s)GEORGE BOUCKOVALAS, National Technical University of Athens, 2016 3.17

HWK 3.4:HWK 3.4:

Draw the reflected and refracted waves

(type and propagation di ti ) direction)

in the special cases h n in th fi shown in the figure .

.

HWKHWK 3 53 5HWKHWK 3.53.5::To get a feeling of the depth (from the free ground surface) which is affected by R-waves,

1) Compute the normalized depth z/λR where ground displacements are reduced to 10% of the value at the free ground surface.

2) What is the value of the above critical depth (range of variation) in the case of soft soil, stiff soil-soft rock and rock;rock;

(Aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα)


3. harmonic ((, , y gp, s, rayleigh & love)...

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