Show that (3.46) satisfies (3.45) for r = 0.
(COMPUTER) In the case of plane-wave propagation in the x direction within
a uniform medium, the homogeneous momentum equation (3.9) for shear waves
can be expressed as
where u is the displacement. Write a computer program that uses finite differences
to solve this equation for a bar 100 km in length, assuming β = 4 km/s. Use
dx = 1 km for the length spacing and dt = 0.1 s for the time spacing. Assume a
source-time function at u (50 km) of the form
Apply a stress-free boundary condition at u (0 km) and a fixed boundary condition at u (100 km). Approximate the second derivatives using the finite difference
scheme:
Plot u(x) at 4 s intervals from 1 to 33 s. Verify that the pulses travel at velocities
of 4 km/s. What happens to the reflected pulse at each endpoint? What happens
when the pulses cross?
Hint: Here is the key part of a FORTRAN program to solve this problem: