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Lecture notes on the reflection and refraction of p and sv waves in seismology. It includes equations for potentials, boundary conditions, and the solution of zoeppritz's equations to obtain reflection and conversion coefficients. The document also discusses the dependence of coefficients on the incidence angle and frequency, as well as the concept of critical and post-critical reflection.
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P-Sv CASE
A P wave incident on a free surface generates both reflected P and Sv waves (Fig. 1)
f
z
\ P Sv
/ P
i 1 (^) j i 2
α
β
ree surface
Fig. 1
sin i 2 , α 〉β ⇒ j 〈 i α β β
1
We follow the same approach as the SH case:
Potentials / /
\ \
\
ϕ (^) R = C exp{ i ω ( px +η (^) β z − t )} Sv wave
Boundary Conditions
The traction on the free surface must vanish
T 3 = 0 = ( σ (^) xz , σ (^) yz , σ (^) zz ) = 0
σ
As before
ij =^ λδ^ ij ∆^ +^2 με^ ij 2 2 σ =λ∇ 2 φ + 2 μ
∂ φ
∂ ϕ zz ∂ z^2 ∂ x ∂ z
∂^2 φ ⎛ ∂^2 ϕ ∂^2 ϕ ⎞ σ = 2 μ x z
∂ x^2
∂ z^2 ⎟ ⎠
xz^ ⎟ ∂ ∂ ⎜
2 nd^ March 2005
Then solve
\
\
Which gives
2
p λ C μ p η β 0
After a bit of work we can get expressions of the reflection coefficient (Rpp ) and the
conversion/refraction coefficient (Rps).
\ (^2 2 2 2 2) − 1
α
\ (^2 ) / \ (^) C ⎛ (^4) p ηα ⎫ ⎟
A ⎝ β^ ⎠⎩^ α^ α^ β^ ⎭
(See figure on overhead of P and Sv reflection coefficients versus angle of incidence)
There are several things that should be noted about these equations:
cos j sin i 1 ηα = α
β β
and p = α
1 1 ηα = ; η = α
β β so R (^) pp = − 1 , a full reflection with a change in polarity
RPS = 0 , there is no P → Sv conversion /
RSP = S P = 0 /
RSS = S S = 1
nd (^) March 2005
ρ 1 β 1 cos i 1 − ρ 2 β 2 cos i 2 S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2
2 ρ 1 β 1 cos i 1 T (^) \ \ = S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2
If we increase i 1 to the critical angle, i c (Fig. 4).
1 S
/ 1
Figure 4
1 i 1 β 1
i 2
β (^2) π i 1 = i (^) c , i 2 = 2
p =
⇒ sin i =
β (^1) c β 1 β 1 β 2 β 2 β (^2)
β
ηβ =
= 0 therefore there is no vertical propagation, k (^) z= 2
2 β (^2)
2
\ / R^1 1 2 \ / =^ S^ S =
ρ β cos i 1 − ρ β cos i 2 = 1 , a full reflection S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2 \
T^1 \ \ =^ S^ S =^
2 ρ β cos i 1 = 2 (also known as the Head Wave) S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2
When the incidence angle is increased to post-critical, i.e. i 1 〉 i (^) c , i 2 cannot increase to an angle greater than 90º.
sin i 1 sin i (^) c 1
β (^1)
β (^1)
= p = β (^2)
For the incoming wave p 〉 β (^2)
nd (^) March 2005
2 2 2
2 2 2 2
η β β
η = − p = i p − = i
η 2 is now a complex number
For an SH wave the displacement vector is U = (0,U (^) y,0)
In a postcritical situation when η 2 = i ηˆ
Where the first term on the RHS describes an exponential decay in the z- direction (there is no propagation in the z-direction). The frequency, ω, controls the rate of decay. The second term on the RHS describes a harmonic function ( x , t), therefore the wave propagation is in the x -direction.
The property of decreasing wave amplitude with depth based on the frequency of the wave is known as Evanescence. 2 Π ωη = k (^) z = λ 2 Π η = λω
A wave with a short wavelength (λ), high frequency (ω), will decay more quickly. At infinite frequency the decay is instantaneous and the wave becomes a ray.
If the properties of a medium change with depth, for example there is a body which allows a wave to pass through it more quickly at depth only the low ω, long λ, waves will sample it as the high ω waves will have been stripped out. A wave with a frequency dependence is a Dispersive wave. Often a wave can be dispersive and evanescent.
(See figure on overhead of post critical reflection and transmission coefficients)