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Last wave problem ever

  1. Jan 7, 2015 #1
    1. The problem statement, all variables and given/known data
    The question is to sketch for values of t = 0, t =0.25, t = 1, t=1.25 and t = 1.5 for the wave equation

    Ytt = Yxx in the semi-infinite domain with initial conditions

    y(x,0) = f(x) =
    \begin{cases}
    1 & \text{if } 0≤x≤1 \\
    0 & otherwise
    \end{cases}

    Yt (x,0) = 0 and boundary condition of y(0,t) = 0

    so I changed f(x) into D(x) where

    D(x) =
    \begin{cases}
    1 & \text{if } 0≤x≤1 \\
    -1 & \text{if } -1≤x≤0 \\
    0 & otherwise
    \end{cases}

    2. Relevant equations
    Y(x,t) = Φa(x,t) +Φb(x,t)

    where Φa(x,t) = 1/2 D(x+t)
    and Φb(x,t) = 1/2 D(x-t)

    3. The attempt at a solution
    These are the sketches. Can somebody please tell me if they are all correct for this question!

    upload_2015-1-7_17-53-8.png

    upload_2015-1-7_18-18-31.png
     
  2. jcsd
  3. Jan 7, 2015 #2

    Orodruin

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    The decomposition into φa and φb looks correct and they seem to travel in the correct direction. However, the bottom result for y is not correct. You can see this easily as it does not fulfil the boundary condition, i.e., it has y(0) = 0.5.
     
  4. Jan 7, 2015 #3
    Both of the y seems to have problem...do you have any idea how can I fix this? Because If I want to make y 0 whenever x is 0 then my two parts diagram will be affected right? I thought both a and b parts are correct in the diagram so I don't want to change them...
     
  5. Jan 7, 2015 #4

    LCKurtz

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    Your graphs would be much easier to interpret if you would plot both waves and their sum in the same picture. Also show your x axis passing through y = 0 instead of under the graph, and finally, use a symmetric range like x = -4 to 4.
     
  6. Jan 7, 2015 #5
    At t = 0.25, the red and the blue plots are correct. They just don't add up to the black curve as you have drawn it.

    Chet
     
  7. Jan 7, 2015 #6
    how does it add up then? I mean I just went straight up at x = -0.25 cause the blue part starts there...or do I just have to make y =0 at x =0 then continue as normal? Also can you confirm if t =0 is correct?
     
  8. Jan 7, 2015 #7
    No. You added the curves up incorrectly between -0.25 and +0.25. Otherwise, the rest is OK
    Confirmed.

    Chet
     
  9. Jan 7, 2015 #8
    So the boundary condition is satisfied in t = 0? But how though when x is 0 isn't the y 1? shouldn't it be 0?
     
  10. Jan 7, 2015 #9
    Also for this question when t = 1 is this correct?

    upload_2015-1-7_22-21-37.png
     
  11. Jan 7, 2015 #10
    I don't understand this question grammatically.

    Chet
     
  12. Jan 7, 2015 #11
    I mean when t=0 how is the boundary condition being satisfied?

    Also can you confirm if sketch for t=1 is correct
     
  13. Jan 7, 2015 #12
    At each of the jumps, the value exactly at the jump is the average of the values on either side.
    No. The middle sketch is not correct. Anyway, you should only be looking at the region x > 0 (for the combined solution).

    Chet
     
  14. Jan 7, 2015 #13
    For t=0.25 is this sketch for y correct then

    upload_2015-1-7_23-36-52.png

    and for t = 1 by middle sketch you mean the Φb is not correct and other two are correct? and as for considering x>0 for combined solution, that's what I have done for the y sketch of t = 1....I have ignored the x<0 region of the Φb sketch...but added the x>0 region of Φa and Φb
     
  15. Jan 7, 2015 #14
    No. y should be zero between x = -0.25 and x = + 0.25.
    No. Both the middle sketch and the bottom sketch are wrong.

    Then just show that region so we're not confused. But, if you do show the region x < 0, then you should show the correct sketches there.

    Chet
     
  16. Jan 7, 2015 #15
    What am I doing wrong for the middle and the bottom sketches? I mean for the middle part specially I am considering the range D(x) = -1 if -1<x<0 so as t = 1 i am adding -1+1<x<0+1

    so my range becomes 0<x<1 for Φb = -0.5

    and when considering D(x) = 1 if 0<x<1 for t = 1 I am adding again so 0+1<x<1+1 so my range becomes 1<x<2 for Φb = 0.5

    What's wrong in here....I don't understand
     
  17. Jan 7, 2015 #16
    Like I said, if you are going to show the region x <0 on the sketch, it should be shown correctly. In the region between x = -1 and x = 0, the middle sketch should be -0.5.

    Chet
     
  18. Jan 7, 2015 #17
    Is this correct for t =0.25 y sketch

    upload_2015-1-8_0-19-58.png
     
  19. Jan 7, 2015 #18
    And for t=1 are the middle and bottom part correct now?

    upload_2015-1-8_0-31-32.png
     
  20. Jan 7, 2015 #19
    No. Both are wrong. On the middle sketch, it should be -0.5 from x = 0 to x = 1, and 0 from x = -1 to x = 0.

    Chet
     
  21. Jan 8, 2015 #20
    In that case my first middle sketch for t=1 is correct then if you have a look. I do have -0.5 for 0 to 1 and 0 for -1 to 0. Why did you say it was wrong initially then?
     
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