Critical and Post-critical Reflection

P-Sv CASE A P wave incident on a free surface generates both reflected P and Sv waves (Fig. 1) Fig. 1 α free surface i1 z β j i2 / P sin i1 α = \ P \ Sv sin j β = sin i2 β , α 〉 β ⇒ j〈i1 We follow the same approach as the SH case: 1. Potentials 2. Boundary conditions 3. Solve system of equations (often in the matrix form of Zoeppritz’s Equations) Potentials / / φ I = Aexp{iω ( px − ηα z − t )} \ \ φ R = B exp{iω ( px + ηα z − t )} P wave ϕ R = C exp{iω ( px + η β z − t )} Sv wave \ \ Boundary Conditions The traction on the free surface must vanish T3 = 0 = (σ xz , σ yz , σ zz ) = 0 As before σ ij = λδ ij ∆ + 2µε ij ∂ 2φ ∂ 2ϕ + 2 µ ∂x∂z ∂z 2 2 2 ⎛ ∂ ϕ ∂ 2ϕ ⎞ ∂ φ = 2µ + µ ⎜⎜ 2 − 2 ⎟⎟ ∂x∂z ∂z ⎠ ⎝ ∂x σ zz = λ∇ 2φ + 2µ σ xz Then solve ⎛1⎞ ⎛ A⎜⎜ ⎟⎟ = ⎜⎜ ⎝1⎠ ⎝ / Which gives \ ⎞⎛⎜ B ⎟⎞ ⎟⎟ \ ⎠⎜⎝ C ⎟⎠ ( A + B ){(λ + 2µ )ηα 2 + p 2 λ }+ C (2µpη β ) = 0 ( A − B )2 pηα − C ( p 2 − η β2 ) = 0 After a bit of work we can get expressions of the reflection coefficient (Rpp) and the conversion/refraction coefficient (Rps). / R/ \ / \ pp = PP = R/ \ = PS = PS \ (λ + 2µ )ηα2 + p 2 λ + 4µp 2ηαη β ( p 2 − η β2 )−1 = / 2 2 2 2 2 −1 A − {(λ + 2µ )ηα + p λ }+ 4µp ηαη β ( p − η β ) \ B \ ⎛ 4 pη = ⎜ 2 α2 / ⎜ A ⎝ p −ηβ C ⎫⎪ ⎞⎧⎪ (λ + 2µ )ηα2 + p 2 λ ⎟⎨ 2 2 2 2 2 − 1 ⎟⎪ (λ + 2 µ )η + p λ − 4 µp η η ( p − η ) ⎪⎬ α α β 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: 1. The coefficients are dependent on the incidence angle, through cos i1 sin i1 cos j ηα = and p = ,η β = α β α 2. For normal incidence the case is more simple i1 = 0 → p = 0 ηα = 1 ;η β = α 1 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 3. Notice that RPP and RSS do not depend on frequency, ω. This is only true so long as : 1 − p 2 > 0 so that η = 2 c 1 − p 2 is a real number. 2 c We can now do the same for the incoming Sv wave RSP and repeat the exercise for waves coming from above (Fig. 2). \ / P1 S1 Figure 2 / P1 1 welded interface \ P2 2 \ S2 This gives the scatter matrix (or “dispersion” matrix) ⎛/ \ / \ \ \ \ \⎞ ⎜ P/ P\ S/ P\ S\ P\ P\ P\ ⎟ ⎜ ⎟ S = ⎜ P/ S/ S/ S/ S\ S/ P\ S/ ⎟ ⎜ P P S P S P P P⎟ ⎜/ / / / \ / \ /⎟ ⎝PS S S S S PS⎠ 4. Poisson Medium, when λ = µ , leads to Rayleigh wave system (more later in course). CRITCAL AND POST-CRITICAL REFLECTION Figure 3 shows the pre-critical situation. \ / S H1 S1 1 2 i1 Figure 3 µ1 ρ 1 → β 1 i2 µ2 ρ2 → β2 \ S2 β 2 〉 β1 R ρ1 β 1 cos i1 − ρ 2 β 2 cos i2 ρ1 β 1 cos i1 + ρ 2 β 2 cos i2 2 ρ1 β 1 cos i1 = ρ1 β 1 cos i1 + ρ 2 β 2 cos i2 = / SS T\ \ SS If we increase i1 to the critical angle, ic (Fig. 4). / \ Figure 4 S1 S1 i1 1 β1 \ S2 i2 2 β2 i1 = ic , i2 = p= ηβ = 1 (β 2 ) 2 − 1 (β 2 ) 2 sin i1 β1 = sin ic β1 = sin i2 β2 = 1 β2 π 2 ⇒ sin ic = β1 β2 = 0 therefore there is no vertical propagation, kz=0 ρ1 β 1 cos i1 − ρ 2 β 2 cos i2 = 1 , a full reflection SS ρ1 β1 cos i1 + ρ 2 β 2 cos i2 \ \ 2 ρ 1β1 cos i1 = 2 (also known as the Head Wave) T =SS = SS ρ1 β1 cos i1 + ρ 2 β 2 cos i2 \ / R\ / = S S = \ \ When the incidence angle is increased to post-critical, i.e. i1 〉 ic , i2 cannot increase to an angle greater than 90º. sin i1 sin ic 1 〉 = p= β1 β1 β2 For the incoming wave p〉 1 β2 1 η2 = (β 2 ) 2 − p2 = i p2 − 1 ( β 2 )2 = iηˆ 2 η2 is now a complex number For an SH wave the displacement vector is U= (0,Uy,0) U y , 2 = exp{iω ( px + η 2 z − t )} In a postcritical situation when η 2 = iη̂ U y , 2 = exp{iw( px + iη̂ 2 z − t )} U y , 2 = exp{− ωη̂ 2 z}exp{iω ( 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{iω ( px − t )} The property of decreasing wave amplitude with depth based on the frequency of the wave is known as Evanescence. 2Π ωη = k z = λ η= 2Π 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)