1. ## 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(x-ct) + 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.

2. I would first define the odd-extension $f_0(x)$ into the interval $(-\infty,0)$ 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(x-2) 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:

$1/2(f_0(x+t)+f_0(x-t))$.

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}]

3. 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(x-t) + 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.

4. 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.

5. 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.