3. HARMONIC. (P, S, Rayleigh & LOVE) WAVES

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1 3. HARMONI (P, S, Rayleigh & LOVE) WAVES George Bockovalas Professor of N.T.U.A. October, 00 Steven Kramer: Sggested Reading: hapter (for a general pdate on seismic waves, falts and other basic concepts of Engineering Seismology) & hapter 5 (yo may skip section 5.5) Γιώγος Γκαζέτας: Κεφάλαιο 4 Any other book on wave propagation theory and/or Engineering Seismology GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

2 The Seismic motion is de to. Pressre (P), Shear (S), Rayleigh (R) and other wave types propagating throgh the grond (soil & bedrock)... GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

of propagation v v P-waves ompression - etension S-waves Wave length shearing

3 3. ELASTI WAVES WAVES (P P & S) in UNIFORM MEDIA P- WAVE with planar front S (SH, SV) WAVE with planar front w direction of propagation w SH SV direction of propagation v v P-waves ompression - etension S-waves Wave length shearing Wave length GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.3

4 t d t d + + σ σ σ σ t D D D Ό σ και ε σ μως οπου D t t σ d d + σ σ, Wave eqation... General soltion... ( ) t f ± why; ( ) ( ) ( ) f ώ f t f t ± εν t παγματι αα άα πάγματι GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.4

5 Physical meaning... The problem variables are NOT two independent ones ( and t ) bt a combined one: X* ±t f(-t) If X*-t then, for X* -t o o +t If X*+t Then, for X* +t o o -t f(+t) t0 t t t0 f( o ) o o +t o -t o GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.5

6 HWK 3.: The free end of an infinite rod is displaced according to the following relationship: U0 for t<0 Ut (cm) for 0<t<0.s U0.-t for 0.s<t<0.s U0 for 0.s<t Find the displacement variation along the rod at t0.3s. Assme that the wave propagation velocity is 300m/s. (aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα) GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.6

7 P & S wave propagation velocities in soil & rock formations S- waves: P- waves: V V s p G, D, G D ( + ν ) E( ν ) ( + ν )( ν ) E ( ν ) G ν loose-recent SOIL deposits stiff SOILS soft ROKS Rocks V S (m/s) < >800 V P (m/s) dry satrated < >600 >000 Vibration velocity (at a point) V f( ± t ) V * * X X & * * t X t X t V ( ε )( ) ± σ m D ά V α GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.7

8 Vibration velocity (at a point) V f( ± t ) ATTENTION: The velocity of VIBRATION is different * * ( to 3 orders of magnitde lower!) X X V & than the velocity of PROPAGATION * * t X t X t V ( ε )( ) ± σ m D ά V α Special ase: Harmonic waves f a ( + t ) Ae ω i ( + t ) i ( ωt + k Ae ) f b ( t ) Ae ω i ( t ) i ( ωt k Ae ) where k ω/ π/λ wave nmber alternatively: i ( ωt ± k f ) a Ae, b GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.8

9 Grond VIBRATION at a given POINT (i.e. ο ) t ( ) e i ± ik i t fa b Ae o ω * ω, e o Harmonic vibration with freqency ωπ/τ f a,b *( o ) Ae +iko t f a,b *( o ) Ae +iko o Grond DISPLAEMENT at a given INSTANT OF TIME (i.e. tt ο ) ik ( t ) e iωt ik fa b Ae o ± * ±, e o o t o Τπ/ω Όπως As ποηγουμένως before, bt now αλλάthe τώα role τον of όλο της συχνότητας freqency τονis παίζει played ο by κυματικός the wave αιθμός nmber k k kλπ > kπ/λ ω/ π/λ > ( o,t) o λ π/ω T () tt o λ GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.9

10 EXAMPLE: Two rods in contact διεχ i( ωt+ k ) Γe ποσπ i ( ωt + k ) o e, ανακλ i ( ωt k Be ), ( + ) ( + ) ( ) i ωt k i ωt k i ωt k o Γe e + Be σ Mε M σ Mε M ( + ) ( + ) ( ) i ωt k i ωt k i ωt k o MΓke M ke + MBke EXAMPLE: Two rods in contact Bondary conditions: 0 σ σ πλήης συμβατότητα σ ( o) ( o) σ MΓke iωt M o ke iωt + MBke iωt o B k k M M Γ Γ where kω/, M and kμω GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.0

11 Special cases: Free end: 0 B o & Γ o Fiedend: B- o & Γ0 ( ) ( ) Γ Γ o o B B o o + o o B Γ + + EXAMPLE: Two rods in contact GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

HWK 3.: Lab grond.0 Mgr/m 3 500 m/s Α Α To isolate a precision measrement laboratory (e.g. a geotech lab.) from traffic vibrations, it is proposed to constrct the trench ΑΑ.

12 HWK 3.: Lab grond.0 Mgr/m m/s Α Α To isolate a precision measrement laboratory (e.g. a geotech lab.) from traffic vibrations, it is proposed to constrct the trench ΑΑ. Which of the following two materials fill materials is preferable : (α) concrete, with.5 Mgr/m 3 & 000 m/s, or (β) pmice, with 0.8 Μgr/m 3 00 m/s? GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

$3. ELASTI WAVES IN non-uniform MEDIA (with interfaces) SNELL s LAW: sinθ p sinθ sinϑ sinϑ p s s > VIBRATION AMPLITUDE of reflected & transmitted- refracted waves In the already known, simple case of$

13 3. ELASTI WAVES IN non-uniform MEDIA (with interfaces) SNELL s LAW: sinθ p sinθ sinϑ sinϑ p s s > VIBRATION AMPLITUDE of reflected & transmitted- refracted waves In the already known, simple case of -D wave propagation: incident wave medim :, medim :, transmitted wave reflected wave Vibration amplitde : + refl. inc. + ανακλ transm. inc. π inc. GEORGE BOUKOVALAS, National Technical University of Athes, + Greece, 0 3.3

14 VIBRATION AMPLITUDE of reflected & transmitted- refracted waves The amplitde of vibration of the reflected and refracted waves in -D problems, is compted based on stress eqilibrim and strain compatibility (continity) at the interface. In this case, the amplitde is also a fnction of the angle of wave incidence. EXAMPLE: SH waves (Richter 958) SH inc θ δ θ π θ αν > refl. inc. refr. refl. inc. + cosθδ cosθπ cosθδ cosθ + inc. cosθδ + cosθ π π GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.4

15 HWK 3.3: Draw the SH wave propagation path throgh the soil profile shown below, and compte the horizontal acceleration applied to each soil layer. ±0m s300m/s.6mgr/m 3 s600m/s.8mgr/m 3-0m -40m s00m/s.mgr/m 3-60m s400m/s.4mgr/m 3 ( o.6cm, T0.40s) SH 50 o GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.5

$3: Fill the reflected and refracted waves (type and propagation direction) in the special cases$

16 HWK 3.3: Fill the reflected and refracted waves (type and propagation direction) in the special cases shown in the figre.. GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.6

17 Note that in the previos special cases, there are no : SV or SH waves refracted in the water P, SV & SH waves refracted throgh the free sarface (in the air) What does this epression ( there are no ) specifically mean? That the corresponding vibration amplitde is nll, or something else? Reminder.. rod :, rod :, Α Β Γ ob oγ + + oa oa ritical refraction angle: sinθ sinθ sinθ sinθ sinθ sinθ cr sin sinθ sinθ cr η θ θ > What happens for θ > θ cr? Εφαμογή: Μέθοδος επιφανειακής διάθλασης για την γεωφυσική διασκόπηση του εδάφους GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.7

3.3 SURFAE WAVES Wave length αδιατάαχτο μέσο P-wave + SV-wave +

8z λ i ωt π λr R e i ωt π λr λ R R /f (wave length) Β constant /

18 3.3 SURFAE WAVES Wave length αδιατάαχτο μέσο P-wave + SV-wave + FREE SURFAE Rayleigh wave R 0.94 S / S Grond displacements de to R-waves w w ( z,, t ) ( z,, t ) ib 0. 55e B. 66e z λ z λ R R 5. 8z λ e 0. 9e R e 5. 8z λ i ωt π λr R e i ωt π λr λ R R /f (wave length) Β constant / o w/w o & / o λ R w/w o z/λ R w(0) (0).6 w GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.8

19 HWK 3.4: To get a feeling of the depth (from the free grond srface) which is affected by R-waves, ) ompte the normalized depth z/λ R where grond displacements are redced to 0% of the vale at the free grond srface. ) What is the vale of the above critical depth (range of variation) in the case of soft soil, stiff soil-soft rock and rock; (Aπό «Σημειώσεις Εδαφοδυναμικής» Γ. Γκαζέτα) GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.9

S S ( ir kz ir kz iω ( t ) Ae + Be ) e / Γe ir kz e iω ( t / ) z H ( ελεύθεη επιϕάνεια): τ 0 z 0 (

20 LOVE WAVES (in layered soil profile) They are SH waves trapped within the pper soft layer of a non-niform soil profile. S S ( ir kz ir kz iω ( t ) Ae + Be ) e / Γe ir kz e iω ( t / ) z H ( ελεύθεη επιϕάνεια): τ 0 z 0 ( διεπιϕάνεια): τ τ, 0 A + B Γ 0 Α, Β, Γ A( G r ) B ( G r ) ( Gr) 0 + Γ gives : irkh irkh Ae Be sbstitting for ωh tan S G G S «Dispersion» relationship S GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.0

21 Observe that, since tan(*) mst be a real nmber, the following condition mst be satisfied for the previos eqation to have a real soltion: S < LOVE < S In other words, LOVE waves are not possible nless the srface layer is softer than the nderlying medim (i.e. S < S ). ωh tan S G G S S «Dispersion» relationship Graphic variation of the ( LOVE ω) relationship s s Grond displacement variation with depth 0 ω GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

22 Special ase: Soft soil layer pon BEDROK (i.e. S ) LOVE LOVE ωh s ωh π H S bedrock LOVE S π/ ωη/ S ritical freqency ω c (π/) S /H LOVE waves cannot eist for freqencies lower than the fndamental vibration freqency of the soft soil laye, i.e. when ω < ω (π/) s /H Dispersion: ode name for LOVE (and other) waves which ehibit a freqency dependent propagation velocity (even for a niform medim). Implications: The propagation velocity LOVE decreases with increasing freqency ω (decreasing period Τ). This means that the low-freqency components of the ecitation get separated from the high-freqency components, as they propagate faster. As a reslt: The dration of shaking increases with distance from the sorce The freqency content of the motion (e.g. the elastic response spectrm) changes with distance from the sorce GEORGE BOUKOVALAS, National Technical University of Athes, Greece, 0 3.

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