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Partial derivatives with Wave Function

  1. Oct 1, 2008 #1
    1. The problem statement, all variables and given/known data
    Knowing: y(x,t) = Acos(kx-ωt)
    Find the partial derivatives of:
    1) dy/dt
    2) dy/dx
    3) d^2y/dt^2
    4) d^2y/dx^2

    2. Relevant equations

    3. The attempt at a solution
    These are the answers the actual answers:
    1) dy/dt = ωAsin(kx-ωt) = v(x,t) of a particle
    2) dy/dx = -kAsin(kx-ωt)
    3) d^2y/dt^2 = -(ω^2)Acos(kx-ωt) = a(x,t) of a particle
    4) d^2y/dx^2 = -(k^2)Acos(ks-wt)

    now here are my questions:
    1) how come when I do the partial derivative of y with respect to t, kx becomes the constant and vice versa with dy/dx, how come ωt becomes the constant? is it because of implicit differentiation?
    2) What does it give me to find: dy/dx? the slope? also what about d^2y/dx^2?

    Thanks for the Help!
  2. jcsd
  3. Oct 2, 2008 #2
    \dfrac{\partial ^{2}y}{\partial x^{2}}=\dfrac {1}{c^2}\dfrac{\partial ^{2}y}{\partial t^{2}}
  4. Oct 2, 2008 #3
    Ooooh.. Interesting!

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