A wave is propagating in a closedend tube Compute the wave p

A wave is propagating in a closed-end tube. Compute the wave propgation up to t=0.15 sec by solving the first-order wave equation partial u/partial r = -a partial u/partial x. Assume the speed of sound to be 200 m/sec. The wave has triangular shape as show in the figure, which is to be used as the initial condition at t=0.0 sec. And Delta x = 1.0 amp Delta t = 0.005 Write a computer program to Solve the problem using Lax method, and the numerical result at (t = 0.15) must be printed in the form of table or plot. Show all statements in your answer. Where the Lax method is given as: u_i^n + 1 = 1/2 (u_i + 1^n + u_i - 1^n) - a Delta t/2 Delta x (u_i+1^n - u_i-1^n) The method is stable when C = a Delta t/Delta x lessthanorequalto 1

Solution

The matlab script would be as follows:

dx=1;
dt=.005;
a=200;
%%%%intial condition at t=0;
t(1)=0;
x=0:dx:70;
for n=1:71
u(n,1)=0;
end
for n=6:16
u(n,1)=2*x(n)-10;
end
for n=17:26
u(n,1)=-2*x(n)+50;
end
%iterations over t
m=0.15/dt;
for n=1:(m-1);
t(n+1)=t(n)+dt;
for a=2:70
u(1,(n+1))=0;
u(71,(n+1))=0;
u(a,(n+1))=1/2*(u((a+1),n)+u((a-1),n))-a*dt/2/dx*(u((a+1),n)+u((a-1),n));
end
end