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Wave equation ( partial differential equations)

  1. Sep 1, 2011 #1
    Consider a string of length 5 which is fixed at its ends at x = 0 and x = 5. The speed of waves along the string is v = 2 and the displacement of points on a string is defined by the function f(x,t). At the initial time the string is pulled into the shape of a triangle, defined by

    f(x,0) = x for 0 <= x < 1
    f(x,0) = 5/4 - x/4 for 1 <= x <= 5

    and then released from rest.

    a. What are the 2 initial conditions for this problem?
    b. What are the 2 boundary conditions for this problem?
    c. Use the method of separation of variables and Fourier methods to find an equation for how the shape of the string changes with time.

    How do I do this question, please somebody help me i'm terribly lost its depressing :(
     
  2. jcsd
  3. Sep 1, 2011 #2
    I assume that your textbook (or resource or whatever) has the actual wave equation, and the procedure for solving i listed somewhere. I suggest you start there.

    Now, as to some helpful hints if you don't happen to have that text available.
    -In the problem statement, you noted that the string was clamped down at two ends. Further you mentioned a speed for the string. These are your initial and boundary conditions.
    -I suggest you quickly go through this:
    http://en.wikipedia.org/wiki/Wave_equation
    Skip to the section called 'One Space Dimension"

    Cheers,
    -Malus
     
  4. Sep 1, 2011 #3

    I think the boundary conditions are:
    f(0,t) = 0 and f(5,t)=0 but im not sure about the initial conditions can you please tell me what they would be? would one of them possibly be df(x,t)/dt = 2?
     
  5. Sep 1, 2011 #4

    vela

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    That's correct.
    That's not correct. Where did you get df(x,t)/dt = 2 from?

    What does the word initial imply?
     
  6. Sep 1, 2011 #5
    isnt the velocity at all point on the string always going to be 2 do then partialf(x,t)/partial(t) = 2?? what would at least one of them be?
     
  7. Sep 1, 2011 #6

    vela

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    Oh, OK. I see what you're thinking.

    If you have a horizontal string, the wave moves down the string, that is, in the horizontal direction, and v=2 (units?) is the speed at which it propagates down the string.

    If you look at a single point on the string, it moves vertically. The velocity of each point is called the transverse velocity, and it's given by ∂f/∂t, which, like f, is a function of x and t.

    The one of the initial conditions has to do with ∂f/∂t. It has nothing to do with the speed at which the wave moves down the string.
     
  8. Sep 1, 2011 #7

    vela

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    Here's a Flash animation of a wave on a string.

    http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html

    Set the damping to 0, choose no end for the right end of the string, and set it so you can send a pulse down the string.

    When you hit the pulse button, you'll the pulse move down the string. The speed at which it moves down the string corresponds to v=2 in this problem.

    If you watch one of the green dots, you'll see it only moves vertically. Its velocity is the transverse velocity ∂f/∂t.
     
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