12.510 Introduction to Seismology
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MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 12.510 April, 8, 2008 Continents: Quick review. Surface waves Ground roll-acoustic π = π. ∇. Love waves – SH π’π¦ = π. ∇. 1 p 1 π ∇π . where p is the pressure ∇uy Rayleigh waves P-SV Quick review (refer to April, 4, 2008 for details) Surface wave S P Ground roll T x ω – k domain ω c2=ω/k1 cfix c1=ω/k2 kfix 3rd higher mode 2nd higher mode 1st higher mode ω2 ω1 ω0 ωfi Fundamental mode k2 k1 x k0 k 1 12.510 April, 8, 2008 Ground roll dispersion relationship π Pre-cretical π© < π π R T π2 = 1 − π2 ∈ β π2 2 (1) π Post-critical π© > π π Head wave π2 = i π2 − 1 = iπ2 ∈ β π2 2 2 (2) 12.510 April, 8, 2008 \ π i j / π \ π π ππ(π) π ππ(π) = =p π½ πΌ (3) if β < α, there can be critical reflection and horizontal propagating p-wave. If j > jc then there will be evanescence in the p-wave ( p > 1/ α). ‘P S P S’ Both p-wave and s-wave horizontal propagation if p οΎ 1 οΎ 1 ο½ 1 . If a wave comes in with a c ο’ ο‘ 1/c that is larger than local 1/α and 1/β, the above will occur. This will also happen if the source emits a horizontal energy (rare). π: ∅ = π΄ππ₯π −πππΌ π§ ππ₯π ππ(ππ₯ − π‘ ), ππΌ = π 1 2 π − 1 2 πΌ 1 πΌ2 1 − π2 = π π2 − πΌ 2 = πππΌ = 1 πΌ π> 1 ; where π = ππ = π (ππ : phase velocity for Rayleigh waves) π: π = π½ππ₯π −πππ½ π§ ππ₯π ππ(ππ₯ − π‘ ), ππ½ = 1 1 − π2 = π π2 − 2 = πππ½ 2 π½ π½ 1 π> π½ =π 1 π 2 1 − π½ 2 3 (4) (5) 12.510 April, 8, 2008 We follow the same “Recipe” we used before: 1. Potentials 2. Boundary Conditions (Kinematic and dynamic) 3. Zoeppritz equatios. Boundary conditions π’ π₯, π‘ = π»∅ + π» × π (6) In this case π / \ , π / \ , π / \ , π / π π π π π π \ π π (7) After some work we get: π΄ π + 2π ππΌ2 + ππ2 + 2ππππ½ = 0 π΄ 2πππΌ + π΅ π2 − ππ½2 = 0 (9) Zoepprtiz π + 2π ππΌ2 + ππ2 2ππππ½ 2πππΌ π2 − ππ½2 0 π΄ = 0 π΅ (10) Trivial solution is π¨=π©=π Non-trivial solution leads to: π + 2π ππΌ2 + ππ2 π2 − ππ½2 − 2πππΌ (2ππππ½ ) = 0 (11) This expression; Rayleigh wave denominator (Rayleigh, 1887), can be written looking at wave speeds, but usually done numerically assuming a Poisson’s medium (λ=μ) and ο‘ ο½ 3ο’ . This scaling will help to simplify the above equation. π π − π πΆ π π πΌπΆ = π − ππ = πΆπ π π − π·π π π πΌπ· = π − ππ = π·π 4 (12) 12.510 πΆ= April, 8, 2008 π + ππ π ο° ππΆπ = π + ππ π·= π π ο° ππ·π = π πΆπ πΌππΆ + π − ππ·π ππ π− πΌππ· ππ ο° − ππ·π πΌπΆ πΌπ· ππ π· πΆ Assumptions Poisson medium:π = π π Poisson ratio: π½ = π(π+π) = π. ππ πΆ = ππ· πππΉ π·π π πππΉ −π π π· π + ππ πππΉ π π·π − ππ =π π Three solutions 1. π π π· π π π = π ο¨ π = ππ·, π > π·, π = π < π, π < πΆ j \ π i / π \ π 5 =π 12.510 2. April, 8, 2008 π π π· π π π π = π + π π, πΆ < π < π·, ππ = (π + π π)π·π \ π / π 3. π π π· = (π − π π π), π πΆ \ π π <π< ,π= π· π+ π π π π π π· π, π < π·, π = > π > , (πΌπΆ = ππΌπΆ , πΌπ· = ππΌπ· ) πΆ \ π \ π / π Note: cR ~ 0.92β which makes it about 10% slower than the shear wave velocity. This explains why the Rayleigh wave is slower than the love wave. The Love wave, at low frequency ο c2 and at high frequencies Love wave (at it’s slowest) is c1…Rayleigh wave is about 90% of the Love wave at the most. exponential decay with depth ∅ = π΄ππ₯π −πππΌ π§ ππ₯π ππ(ππ₯ − π‘ ) π = π½ππ₯π −πππ½ π§ ππ₯π ππ(ππ₯ − π‘ ) (13) π’ = π»∅ + π» × π ux = ∂∅ ∂x ∂ψ − ∂z ; uz = ∂∅ ∂z ∂ψ − ∂x (14) ux ~A sin kx − ωt … π ππ , ππ are π out of phase uz ~A cos kx − ωt … Updated by: Sami Alsaadan Sources: April 13, 2008 by Patricia Gregg. April 8, 2008 lecture. “An Introduction to Seismology, Earthquakes, And Earth Structure” by Stein &Wysession (2007). 6
