
Wave Equation Problem
Hey guys,
I have the following question on this problem:
Utt = Uxx, 0<x<infinity, 0<t
Ux(0,t) = 0
U(x,0) = f(x)
Ut(x,0) = 0
f(x) is given as a graph: a triangle with left bottom corner at x = 2
right bottom corner at x = 4
height of the triangle at y = 2.
I am asked to give a detailed picture for different time values to show the periods: 1) during separation of the wave into 2 smaller waves
2.) before bounding from x = 0, 3.) during bouncing from x = 0
and 4.) long time after bouncing from x= 0.
I know I have to use the D'Alembert's Eqn: U(x,t) = [f(xct) + f(x+ct)]/2, but I don't have any clue as to how to get the times, or how to obtain the picture at that time. Can anybody get me started/show me an example as I am seriously confused on this. Thanks.

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I would first define the oddextension $\displaystyle f_0(x)$ into the interval $\displaystyle (\infty,0)$ which below I'm using Piecewise in Mathematica:
f_0[x_] := Piecewise[{{0, 2 <= x <= 2}, {2 (x  2),
2 <= x <= 3}, {2 (x  3) + 2,
3 < x < 4}, {2 (x + 3)  2, 3 <= x <= 2}, {2 (x + 4), 4 <=
x <= 3}, {0, Abs[x] > 4}}];
You read that as: zero in the interval (2,2), 2(x2) in the interval (2,3), and so on.
and then convert it to a problem on the infinite string in which the solution is then:
$\displaystyle 1/2(f_0(x+t)+f_0(xt))$.
You can then draw this piecewise function at various time periods or use Manipulate in Mathematica:
Manipulate[
Plot[1/2 (f_0[x + t] + f_0[x  t]), {x, 10, 10},
PlotRange > {{10, 10}, {2, 2}}, PlotPoints > 50], {t, 0, 10}]

Thanks for the reply Shawsend.
I don't have mathematica, and I think I am supposed to use an even extension on this problem, but I still have another question that I guess I still don't understand. In the equation 1/2[f(xt) + f(x+t)], how exactly is it used?
For example, how do I know which x coordinates to use?
or which values for time (t) to pick?
I know I am making it more complicated than it is, but I just don't get this part yet.

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My updated version then:
f[x_] := Piecewise[{{0, 2 <= x <= 2}, {2 (x  2),
2 <= x <= 3}, {2 (x  3) + 2,
3 < x < 4}, {2 (x + 3) + 2, 3 <= x <= 2}, {2 (x + 4), 4 <=
x <= 3}, {0, Abs[x] > 4}}];
Not sure though how to figure out at what time the wave will be in a particular form, for example when it bounces off the wall entirely like in the fourth picture.

Thanks. I ll continue working on it and use the graphs you posted to try and figure out the time value that should be used.