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Seismic Waves: Surface Waves and Ground Roll, Study notes of Geology

An in-depth analysis of various types of seismic waves, focusing on surface waves and ground roll. It covers the fundamental modes, dispersion relationships, and boundary conditions for rayleigh and love waves. The document also includes equations for wave propagation and the zoeppritz equations.

Typology: Study notes

2012/2013

Uploaded on 07/19/2013

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Continents: Quick review. Surface waves
Ground roll-acoustic 𝑝 =𝑉.. 1
𝑝∇𝑝 . where p is the pressure
Love waves SH 𝑢 𝑦=𝜇.. 1
puy
Rayleigh waves P-SV
Quick review (refer to April, 4, 2008 for details)
ω k domain
ω
k
c2=ω/k1
c1=ω/k2
Fundamental mode
1st higher mode
2nd higher mode
3rd higher mode
ωfi
x
k0
k1
k2
kfix
ω0
ω2
ω1
x
Surface wave
S
P
Ground roll
T
Docsity.com
pf3
pf4
pf5

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1

Continents: Quick review. Surface waves

Ground roll-acoustic 𝑝 = 𝑉. ∇. (^1) 𝑝 ∇𝑝. where p is the pressure Love waves – SH 𝑢𝑦 = 𝜇. ∇. (^) p^1 ∇uy Rayleigh waves P-SV Quick review (refer to April, 4, 2008 for details)

ω – k domain

ω

k

c 2 =ω/k 1 c 1 =ω/k 2

Fundamental mode

1 st^ higher mode

2 nd^ higher mode

3 rd^ higher mode

ω fi x

k 2 k 1 k 0

kfix

ω 0

ω 2 ω 1

cfix

x

Surface wave

S

P

Ground roll

T

2

Ground roll dispersion relationship

Pre-cretical 𝐩 < (^) 𝐜𝟏𝟐

− 𝑝^2 ∈ ℝ (1)

Post-critical 𝐩 > (^) 𝐜𝟏 𝟐

𝜂 2 = i 𝑝^2 −

= i𝜂 2 ∈ ℂ (^) (2)

T

R

Head wave

4

We follow the same “Recipe” we used before:

  1. Potentials
  2. Boundary Conditions (Kinematic and dynamic)
  3. Zoeppritz equatios.

Boundary conditions

𝑢 𝑥, 𝑡 = 𝛻∅ + 𝛻 × 𝜓 (6)

In this case

𝑅/ 𝑠 \ 𝑠 , 𝑅/ 𝑠 𝑝 \ , 𝑅/ 𝑝 \ 𝑠 , 𝑅 𝑝/ \ 𝑝 (7)

After some work we get:

𝐴 𝜆 + 2 𝜇 𝜂𝛼^2 + 𝜆𝑝^2 + 2 𝜇𝑝𝜂𝛽 = 0 𝐴 2 𝑝𝜂𝛼 + 𝐵 𝑝^2 − 𝜂𝛽^2 = 0 (9)

Zoepprtiz

𝜆 + 2 𝜇 𝜂𝛼^2 + 𝜆𝑝^2 2 𝜇𝑝𝜂𝛽 2 𝑝𝜂𝛼 𝑝^2 − 𝜂𝛽^2

𝐵 =^

Trivial solution is 𝑨 = 𝑩 = 𝟎

Non-trivial solution leads to:

𝜆 + 2 𝜇 𝜂𝛼^2 + 𝜆𝑝^2 𝑝^2 − 𝜂𝛽^2 − 2 𝑝𝜂𝛼 ( 2 𝜇𝑝𝜂𝛽 ) = 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   3 . This scaling will help to simplify the above equation.

𝜶𝟐^

− 𝒑𝟐^ =

𝜶𝟐^

𝟐

𝜷𝟐^

− 𝒑𝟐^ =

𝜷𝟐^

𝟐

5

 𝝆𝜶𝟐^ = 𝝀 + 𝟐𝝁

 𝝆𝜷𝟐^ = 𝝁

𝜶𝟐^

𝜼𝜶^ 𝟐

𝝆𝟐^

+ 𝟏 − 𝟐𝜷𝟐^ 𝟏 −

𝜼𝜷^ 𝟐

𝝆𝟐^

𝝆𝟐^

Assumptions Poisson medium: 𝝀 = 𝝁

Poisson ratio: 𝑽 = (^) 𝟐(𝝀𝝀+𝝁) = 𝟎. 𝟐𝟓

𝜶 = 𝟑𝜷

𝒄𝑹^ 𝟐

𝟑 − 𝟖

𝒄𝑹^ 𝟐

𝟐

𝒄𝑹^ 𝟐

𝜷𝟐^

Three solutions

  1. (^) 𝜷𝒄

𝟐 = 𝟒  𝒄 = 𝟐𝜷, 𝒄 > 𝜷, 𝒑 = 𝟏𝒄 < 𝟏𝒑, 𝒑 < 𝟏𝜶

𝑆

/ 𝑆

\

𝑃


j

i