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Homework Help: PDE Wave Equation/boundary condition question

  1. Sep 13, 2009 #1
    1. The problem statement, all variables and given/known data
    I need to visualize the wave equation with the following initial conditions:
    u(x,0) = -4 + x 4<= x <= 5
    6 - x 5 <= x <= 6
    0 elsewhere
    du/dt(x,0) = 0

    subject to the following boundary conditions:
    u|x=0 = 0

    2. Relevant equations
    I'm not sure I understand the boundary conditions as they written can someone help me?

    3. The attempt at a solution
    Not sure where to start with the boundary conditions.
  2. jcsd
  3. Sep 13, 2009 #2


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    This represents a semi-infinite string tied down at the origin and given a triangular displacement between 4 and 6. So at time t = 0 the string looks like this:

    It is on the x axis from (0,0) to (4,0) then goes on a slope 1 straight line to (5,1), then on a slope -1 back down to (6,0), then out the x axis.

    The du/dt(x,0) = 0 statement says the string is released from rest, not given an initial velocity.

    Th boundary condition u|x=0 = 0 would be better written as u(0,t) = 0 for t >0 which says the string is tied down at the end.

    Have you studied D'Alembert's solution and how to handle semi-infinite strings with it? That's where you need to look.
  4. Sep 18, 2009 #3
    im sorry if im doing wrong by posting in a another thread,but i have a similar problem...
    its an infinite string which is straight ( U(x,0)= 0) and has input velocity defined by some arbitrary function g(x)...
    from this conditions i have to derive

    (sorry,dont know Latex)

    where c is (obviously) its velocity...
    im having problem with boundary conditions...well,everything tends to cancel out :biggrin:
    Last edited by a moderator: Apr 24, 2017
  5. Sep 18, 2009 #4


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    If you start with the the 1-dimensional wave equation for the vibrating string:

    [tex] u_{tt} = c^2 u_{xx}[/tex]

    have you studied the D'Alembert substitution:

    [tex] r = x + ct,\, s = x - ct[/tex]

    yet? This leads to the equation:

    [tex]u_{rs} = 0 [/tex]

    which leads to the solution you are seeking. Perhaps if you show what you have done so far I would know where to help you.
    Last edited by a moderator: Apr 24, 2017
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