2. SH Reflection and Transmission Coefficients (25 pts)

(a) Write a MATLAB code to calculate the SH-wave reflection and transmission coefficients (equations 6.46 and 6.47). Use this code to generate a duplicate of the plot in Figure 6.5. The plot in Figure 6.5 uses two layers across the MOHO with densities: ?1 = 2.9 g/cm3 and ?2 = 3.38 g/cm3 and S-wave velocities: ?1 = 3.9 km/s and ?2 = 4.49 km/s. Note that beyond the critical angle one must make a substitution of Eqn 6.59, but that this substitution works for pre-critical angles as well, so you should just make that substitution from the beginning. Note that there is a sign ambiguity in the imaginary part. An additional boundary condition requires that we take the negative of the imaginary part. For example, the plot in the top left corner of Fig. 6.5 shows the reflection coefficient, R. If your MATLAB variable for the Reflection coefficient is R, then in MATLAB you can simply calculate the real part, imaginary part, and magnitude of R by:

>> Rreal = real(R);

>> Rimag = -imag(R);

>> Rmag = abs(R);

Where, I note that we have to take the negative of the imaginary part. Similarly, the phase shift can then be determined by:

>> phaseR = atan2(-imag(R),real(R)).*(180/pi);

(b) Redo the plots for a SH-wave interacting with the D” discontinuity. Assume the discontinuity is located 300 km above the core-mantle boundary and that there is a 2% increase in both S-wave velocity and density. In the PREM model, at a depth of 2591 km, ?1 = 7.2 km/s and ?1 = 5.4 g/cm3. For SH-wave arrivals that are closer to vertical incidence do you expect to observe a reflected or refracted arrival associated with this discontinuity?