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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.
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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 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
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:
Boundary conditions
In this case
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
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.
𝟐
𝟐
5
Assumptions Poisson medium: 𝝀 = 𝝁
Poisson ratio: 𝑽 = (^) 𝟐(𝝀𝝀+𝝁) = 𝟎. 𝟐𝟓
𝜶 = 𝟑𝜷
𝟑 − 𝟖
𝟐
Three solutions
𝟐 = 𝟒 𝒄 = 𝟐𝜷, 𝒄 > 𝜷, 𝒑 = 𝟏𝒄 < 𝟏𝒑, 𝒑 < 𝟏𝜶
𝑆
/ 𝑆
\
𝑃
j
i
