The denominator of the depth-dependent Green’s function for the fluid–elastic halfspace problem described in the previous problem also has a symmetric pair of complex roots which become important for the propagation characteristics in certain cases.
a. Employ a numerical root finding scheme (e.g., a complex Newton–Raphson scheme) to determine the complex root with positive real value. (Warning: take care how you choose the branch cuts for the square root).
b. Assuming the sound speed in water to be 1500 m/s and the density 1000 kg/m3, compressional speed 5000 m/s and density 2500 kg/m3 in the solid, map the position of the root as a function of shear speed in the range 1500–3500m/s.
c. Discuss the physical significance of the real and imaginary part of the root.