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7. (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 azu = B2 a²u ax²' 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 B = 4km/s. Use dx = 1 km for the length spacing and dt = 0.1s for the time spacing. Assume a source-time function at u (50km) of the form uso(t) = sin2(tt/5),0 <t <5s.="<br">Apply a stress-free boundary condition at u (0km) and a fixed boundary condi- tion at u (100km). Approximate the second derivatives using the finite difference 3.10 EXERCISES 63 scheme: alu = Plot u(x) at 4s intervals from 1 to 33s. 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: (initialize t, dx, dt, tlen, beta and ul,u2,u3 arrays) 10 t=t+dt do i=2,100 rhs=beta* 2* (u2 (i+1)-2. *u2 (i) +u2 (i-1)) /dx**2 u3(i)=dt**2*rhs+2.*u2(i)-ul(i) enddo u3(1)=u3(2) u3 (101) =0. if t.le.tlen) then u3 (51) (3.1415927*t/tlen) **2 end if do i=1, 101 ul (i) =u2(i) u2 (i)=u3(i) enddo (output u2 at desired intervals, stop when t is big enough) go to 10 </t>

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% Given:
clear all;
close all

t=0; dt = 0.1; dx = 1;
tlen = 5;
beta = 4;
tol = 1e-4;

Xlen = 100;
tend = 33;

n = (Xlen/dx)+1; %n is the number of nodes

%initializing u1,u2

u1 = zeros...

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