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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 1
sin j
sin i 2 , α 〉β ⇒ j 〈 i α β β
1
We follow the same approach as the SH case:
- Potentials
- Boundary conditions
- Solve system of equations (often in the matrix form of Zoeppritz’s Equations)
Potentials / /
φ I = A exp{ i ω ( px −η α z − t )}
\ \
φ R = B exp{ i ω ( px +η α z − t )} P wave
\
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
A = ⎜⎜
\
\
C
B
Which gives
( A + B ){ ( λ+ 2 )
2
p λ C μ p η β 0
( A − B ) 2 p η α − C ( p^2 −η 2 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
/ \ B (λ + 2 μ )η + p λ + 4 μ p η η α β( p −η β)
R / \ = P P = / =
α
p p − {(λ + 2 μ )η 2 + p^2 λ } + 4 μ p^2 η η^ ( p^2 −η^2 −^1
A α α β β )
\ (^2 ) / \ (^) C ⎛ (^4) p ηα ⎫ ⎟
⎧⎪ (λ + 2 μ )η α+ p λ ⎪
R / \ = P S = = ⎜ 2 2 2 2 2 2 2 − 1 ⎬
P S /^ ⎜^ p −η ⎟⎪ (λ + 2 μ )η + p λ − 4 μ p η η ( p −ηβ ) ⎪
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:
- The coefficients are dependent on the incidence angle, through cos i 1 , η =
cos j sin i 1 ηα = α
β β
and p = α
- For normal incidence the case is more simple i 1 = 0 → p = 0
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
- Notice that RPP and R (^) SS do not depend on frequency, ω. This is only true so long as :
nd (^) March 2005
R / =
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).
S
1 S
/ 1
Figure 4
1 i 1 β 1
i 2
S 2
β (^2) π i 1 = i (^) c , i 2 = 2
p =
sin i 1
sin ic
sin i 2
⇒ 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)
U y , 2 = exp{ i ω ( px + η 2 z − t )}
In a postcritical situation when η 2 = i ηˆ
U y , 2 = exp{ iw ( px + i ηˆ 2 z − t )}
U y , 2 = exp{− ωηˆ^2 z }exp{ i ω( px − t )}
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.
U y , 2 = A ( z ) exp{ i ω( px − t )}
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)
