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String gets deformed and released - derive equation

  1. Sep 7, 2015 #1
    1. The problem statement, all variables and given/known data

    This graph shows an infinite string on the x-axis. The middle part is deformed as shown. The string is released at time t=0
    - give the analytical solution of the problem, i.e. find upload_2015-9-7_14-51-42.png for t>0
    - sketch the string displacement \xi(x) for a sequence of times t. (I'd like to know what some good times would be, but I guess that would be clear once I get the equation.
    - sketch the characteristics diagram (x,t diagram) for the problem. (I know I could just plug in points to the equation, but is there an organized way to do this?)


    upload_2015-9-7_14-43-14.png
    2. Relevant equations

    (see part 3)



    3. The attempt at a solution
    I know the graph above is expressed as follows:
    tex?%5CLARGE%20%5Cxi%28x%2C0%29%3D%5Csin%7Bx%7D%20%5Ccdot%20rect%7B%5Cfrac%7Bx%7D%7B2%5Cpi%7D%7D.gif
    I need to get the derivative of this in order to find upload_2015-9-7_14-56-41.png it looks like it'll be the chain rule with cosine as the derivative of sine, and the derivative of the rect will be impulses at -pi and +pi, but how do I express that mathematically? I think it's Dirac delta or something, but could someone please remind me how to write it? It's been a while since I had to do this.
    On the other hand, there are no t's in the equation so does this mean the derivative would be 0?
     
  2. jcsd
  3. Sep 8, 2015 #2

    Orodruin

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    The fact that they say that the string is "released" indicates that the initial condition on the velocity is zero. Once you know this, you can simply apply d'Alembert's solution to the wave equation and you are done.
     
  4. Sep 8, 2015 #3
    Thanks for your response. But I have to graph it too, what would be some good times to plug in? And how do I make the characteristics diagram?
     
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