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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 ๐ = ๐. โ. (^1) ๐ โ๐. where p is the pressure Love waves โ SH ๐ข๐ฆ = ๐. โ. (^) p^1 โuy Rayleigh waves P-SV Quick review (refer to April, 4, 2008 for details)
ฯ โ k domain
ฯ
k
c 2 =ฯ/k 1 c 1 =ฯ/k 2
Fundamental mode
1 st^ higher mode
2 nd^ higher mode
3 rd^ higher mode
ฯ fi x
k 2 k 1 k 0
kfix
ฯ 0
ฯ 2 ฯ 1
cfix
x
Surface wave
S
P
Ground roll
T
2
Ground roll dispersion relationship
Pre-cretical ๐ฉ < (^) ๐๐๐
Post-critical ๐ฉ > (^) ๐๐ ๐
๐ 2 = i ๐^2 โ
= i๐ 2 โ โ (^) (2)
T
R
Head wave
4
We follow the same โRecipeโ we used before:
Boundary conditions
In this case
After some work we get:
๐ด ๐ + 2 ๐ ๐๐ผ^2 + ๐๐^2 + 2 ๐๐๐๐ฝ = 0 ๐ด 2 ๐๐๐ผ + ๐ต ๐^2 โ ๐๐ฝ^2 = 0 (9)
Zoepprtiz
๐ + 2 ๐ ๐๐ผ^2 + ๐๐^2 2 ๐๐๐๐ฝ 2 ๐๐๐ผ ๐^2 โ ๐๐ฝ^2
Trivial solution is ๐จ = ๐ฉ = ๐
Non-trivial solution leads to:
๐ + 2 ๐ ๐๐ผ^2 + ๐๐^2 ๐^2 โ ๐๐ฝ^2 โ 2 ๐๐๐ผ ( 2 ๐๐๐๐ฝ ) = 0 (11)
This expression; Rayleigh wave denominator (Rayleigh, 1887), can be written looking at wave speeds, but usually done numerically assuming a Poissonโs medium (ฮป=ฮผ) and ๏ก ๏ฝ 3 ๏ข. This scaling will help to simplify the above equation.
๐
๐
5
Assumptions Poisson medium: ๐ = ๐
Poisson ratio: ๐ฝ = (^) ๐(๐๐+๐) = ๐. ๐๐
๐ถ = ๐๐ท
๐ โ ๐
๐
Three solutions
๐ = ๐ ๏จ ๐ = ๐๐ท, ๐ > ๐ท, ๐ = ๐๐ < ๐๐, ๐ < ๐๐ถ
๐
/ ๐
\
๐
j
i
