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Math Help - Wave Equation Problem

  1. #1
    Junior Member
    Joined
    Apr 2009
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    70

    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.
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  2. #2
    Super Member
    Joined
    Aug 2008
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    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}]
    Attached Thumbnails Attached Thumbnails Wave Equation Problem-manip1.jpg   Wave Equation Problem-manip2.jpg   Wave Equation Problem-manip3.jpg  
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  3. #3
    Junior Member
    Joined
    Apr 2009
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    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.
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  4. #4
    Super Member
    Joined
    Aug 2008
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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.
    Attached Thumbnails Attached Thumbnails Wave Equation Problem-manipb.jpg   Wave Equation Problem-manipb2.jpg   Wave Equation Problem-manipb3.jpg   Wave Equation Problem-manipb4.jpg  
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  5. #5
    Junior Member
    Joined
    Apr 2009
    Posts
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    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.
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