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Standing wave with specific initial/boundary conditions

  1. Apr 30, 2015 #1
    1. The problem statement, all variables and given/known data

    "Solve the wave equation with the following initial conditions and boundary conditions."

    2Y/∂x2 = 1/v2 * ∂2Y/∂t2

    Boundary conditions:
    ∂Y/∂x(x=0, t)=0 and Y(x=L,t)=0

    Initial Conditions:
    ∂Y/∂t(x, t=0) = 0
    Y(x,t=0) = δ(x-L/2)

    2. Relevant equations

    Using separation of variables:

    Y(x,t) = X(x)*T(t)

    X(x) = C*cos(kx/L) + D*sin(kx/L)
    T(t) = E*cos(wt) + F*sin(wt)

    3. The attempt at a solution

    So, I'm relatively familiar with this process but I have a couple of hangups relating to these specific initial conditions. First, the requirement that the partial of y with respect to x=0 at all times t (first boundary condition) seems, to me, to make a wave impossible. It gives D=0 AND C=0 OR k=0, but of course all of these are unsatisfactory solutions.

    More analytically, it seems that requiring the slope to always be 0 at x=0 is really restricting that piece of the wave from moving. I'm just not sure how to handle it.

    Moving past that though (I assumed y(x=0,t)=0 as my first boundary condition giving C=0), I'm also not sure how to handle the initial condition. The initial condition equation, x-L/2, has an integral of 0 when integrated from 0 to L. This makes sense since the expression has equal area above and below y=0, but I still don't know how to handle it in terms of getting an actual solution.

    Just thinking out loud as I type this, does the delta in the expression indicate that I should be treating it as an absolute value equation so that all of it is forced to be above y=0?
     
    Last edited: Apr 30, 2015
  2. jcsd
  3. Apr 30, 2015 #2

    SammyS

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    The Boundary condition, ∂Y/∂x(x=0, t)=0, only makes D = 0 with no restriction on C.
    The other Boundary condition gives you k .

    It's a standing wave. Right?

    What are the conditions at nodes and at anti-nodes?
     
    Last edited: Apr 30, 2015
  4. Apr 30, 2015 #3
    Hey, thanks, that was exactly what I needed! I realized the first boundary condition only set D=0, but I erroneously was thinking that the second boundary condition gave C=0 when, in fact, you're right: it gives k.

    I then applied the second initial condition in terms of absolute value, which I think is the correct way to go about it, and everything worked out.

    Thanks again!

    edit: and yes, it's a standing wave.
     
  5. Apr 30, 2015 #4

    SammyS

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    You're very welcome.
     
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