How to Visualize the Wave Equation Solution in Mathematica?

[Dustinsfl]

$$
u(x,t) = \frac{1}{2}\int_{x - t}^{x + t}g(s)ds = \begin{cases}
t, & (x,t)\in R_1\\
\frac{1}{2}(1 - x + t), & (x,t)\in R_2\\
\frac{1}{2}(x + t + 1), & (x,t)\in R_3\\
1, & (x,t)\in R_4\\
0, & (x,t)\in R_5,R_6
\end{cases}
$$
where
\begin{alignat*}{3}
R_1 & = & \{(x,t):-1 < x - t < 1\text{ and } -1 < x + t < 1\}\\
R_2 & = & \{(x,t):-1 < x - t < 1\text{ and } x + t > 1\}\\
R_3 & = & \{(x,t):x - t < -1\text{ and } -1 < x + t < 1\}\\
R_4 & = & \{(x,t):x - t < -1\text{ and } x + t > 1\}\\
R_5 & = & \{(x,t):x + t < -1\}\\
R_5 & = & \{(x,t):x - t > 1\}
\end{alignat*}
How do I plot this in the Mathematica?
I tried this but it doesn't work.

u[x_, t_] = 
  Piecewise[{{t, -1 < x - t < 1 And - 1 < x + t < 
      1}, {1/2 (1 - x + t), -1 < x - t < 1 And x + t > 
      1}, {1/2 (x + t + 1), x - t < -1 And - 1 < x + t < 1}, {1, 
     x - t < -1 And x + t > 1}, {0, x + t < -1}, {0, x - t > 1}}];
Manipulate[Plot[u[x, t], {x, -10, 10}], {t, 0, 1, .01}]

 

[mighty2000]

Hello,
To plot this function in Mathematica, you can use the "Piecewise" function and define the different regions as separate functions. Here is an example code that you can use:

u1[x_, t_] := t
u2[x_, t_] := 1/2 (1 - x + t)
u3[x_, t_] := 1/2 (x + t + 1)
u4[x_, t_] := 1
u5[x_, t_] := 0
u6[x_, t_] := 0

u[x_, t_] :=
Piecewise[{{u1[x, t], -1 < x - t < 1 && -1 < x + t < 1}, {u2[x, t],
-1 < x - t < 1 && x + t > 1}, {u3[x, t],
x - t < -1 && -1 < x + t < 1}, {u4[x, t],
x - t < -1 && x + t > 1}, {u5[x, t], x + t < -1}, {u6[x, t],
x - t > 1}}]

Manipulate[Plot[u[x, t], {x, -10, 10}, PlotRange -> {0, 1}], {t, 0, 1,
0.01}]

In this code, the different regions are defined as separate functions and then combined using the "Piecewise" function. The "Manipulate" function allows you to change the value of "t" and see the plot change in real time. You can also adjust the plot range to better visualize the function. I hope this helps!