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Can somebody help me understand this BVP question?

  1. Jan 18, 2017 #1
    1. The problem statement, all variables and given/known data
    So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show.

    2. Relevant equations
    utt = c2uxx

    3. The attempt at a solution
    The problem set is in the attachment
     

    Attached Files:

    Last edited: Jan 18, 2017
  2. jcsd
  3. Jan 18, 2017 #2

    DrClaude

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    Staff: Mentor

    No, it means starting from the generic wave equation given in the preamble of the question and showing that the specific equations given in problem 1 actually correspond to the specific physical situation.
     
  4. Jan 18, 2017 #3
    I feel like I'm missing something obvious; I don't see how I can "show" that. The problem describes the physical process and then proceeds to write down the boundary conditions that correspond to that process. I don't get what I'm supposed to do. Am I just supposed to explain in words why the boundary conditions are valid?
     
  5. Jan 18, 2017 #4

    DrClaude

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    I would go about it as if the equations were not given. Start with the wave equation and discuss the specific physical problem, setting up the parameter ##a^2## and the initial conditions, arriving at the equations given.
     
  6. Jan 18, 2017 #5
    So If I understand you correctly, this is how I would go about answering the question.

    Suppose that a 1 foot flexible piece of wire is stretched between the points (0,0) and (1,0). The tension in the wire is 10 Ib and the weight of the wire is 0.032 Ib. The parameter "a" in the wave equation is defined as a2 = Tension/density; therefore since

    mg = 0.032 Ib and δ(linear mass density) = m/L, a2 = (10 Ib) (32 ft/s2)(1 ft)/(0.032 Ib) = 104 (ft/s)2

    The wave equation then becomes ytt(x,t) = 104 yxx(x,t)

    At t = 0 the string lies completely on the x-axis but has a velocity of 1 ft/s in the positive y - direction. The wire is under no external forces.

    Since the wire is being stretched between the endpoints, that implies that the endpoints are fixed, therefore
    y(0,t) = y(L,t) = 0 for t ≥ 0

    it was said that the wire lies completely on the x-axis at t = 0, therefore
    y(x,0) = 0 for 0 ≤ x ≤ 1
    Since the velocity of the wire at t = 0 was 1 ft/s
    yt(x,0) = 1 for 0 < x < 1

    So in short, the physical process described above can be summarized in the following way

    ytt(x,t) = 104 yxx(x,t) for 0 < x < 1 for t > 0
    y(0,t) = y(L,t) = 0 for t ≥ 0
    y(x,0) = 0 for 0 ≤ x ≤ 1
    yt(x,0) = 1 for 0 < x < 1
     
  7. Jan 18, 2017 #6

    Ray Vickson

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    Good. You have done exactly what the question asked you to do.
     
  8. Jan 18, 2017 #7
    The question just seems weird to me, at least the wording does. It feels circular. Same thing with question 2. In this class we haven't gone over how to solve these PDE's yet, so I'm thinking that for question 2 when its asking me to show that the string hangs in the fashion described by the parabolic function, I should just take the appropriate derivatives and substitute back into the wave equation and confirm that the equality holds, correct?
     
  9. Jan 18, 2017 #8

    Ray Vickson

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    How can we tell? We do not know the approach taken by your textbook or course notes, so we don't know whether the course (initially, at least) emphasizes things like physical derivations of PDEs for some phenomena, or whether it essentially starts with a PDE and then discusses boundary conditions and the like. I have seen different books on the subject take very different approaches to these questions.
     
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