Problem Set 2 Please don’t just copy the results from books: show full derivations! 1. For (i) SH wave incidence on the solid-solid interface and (ii) P wave incidence on the free surface: (a) Give the potentials, boundary conditions, Zoeppritz’ equations and all the elements of the scattering matrix. (20 points) (b) In some cases an angle exists where the reflection coefficient is zero; the angle for which that happens is called the intramission angle: compute – for the wavespeeds given – the intramission angle. (10 points) (c) Make plots of reflection and transmission coefficients as function of incidence angle (from i = 0 o through i = ic to i = 90 o ). (10 points) (d) Explain the plots. (10 points) SH incidence ( ρ1 = 2.5 g ⋅ cm −3 ρ 2 = 3g ⋅ cm −3 β1 = 3km ⋅ s −1 β 2 = 4km ⋅ s −1 ) P incidence ( α = 5.5km ⋅ s −1 β = 3.2km ⋅ s −1 ) \ / u1 u1 i ρ1 , β1, µ1 Free surface i ρ 2 , β 2, µ 2 j z SH incidence \ u2 x α ,β x j i \ φ1 / φ1 \ z P incidence ψ1 2. Give the Zoeppritz’ equations of the P-SV system (P incidence) across a solid-solid interface. (20 points) / ψ1 \ φ1 / φ1 ρ 1 , α 1 , β 1 , µ1 x ρ 2 ,α 2 , β 2 , µ 2 \ φ2 \ z ψ2 P-SV system (P incidence) 3. Suppose we have an incidence plane wave SH as depicted in Figure below: prove that the total response has the same amplitude as that of the incidence wave. (30 points) Free surface ρ1 , β1, µ1 ρ 2 , β 2, µ 2 SH