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Seismic displacement ofr a plane wave

  1. Nov 3, 2008 #1
    1. The problem statement, all variables and given/known data
    phi(x,y,z,t)= cos(wt- the dot product of K and x)
    k= kx i hat + ky j hat + kz k hat units are m^-1
    x= x i hat + y j hat + z k hat units are m

    t is time and w is the angular velocity w/ units of sec^-1

    show that abs. value of k = w/alpha by plugging the displacement potential into the P-wave equation:

    laplacian(phi) = 1/alpha^2 times d^2phi/dt^2


    2. Relevant equations
    laplacian= d^2phi/dx^2 + d^2phi/dy^2 + d^2phi/dz^2
    only interested in d^2phi/dx^2



    3. The attempt at a solution
    I have carried out the dot product of k dot x:
    x, y, z= sub x, y, z
    X,Y,Z= X, Y, and Z
    cos(wt-kxX + kyY + KzZ)

    now I am stuck and don't know where to go/what to do...... any tips/suggestions would be awesome!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 3, 2008 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Well, you have: [itex]\varphi(\vec{x},t)=Cos(\omega t- k_x x-k_y y -k_z z)[/itex] so why not find [itex]\frac{\partial \varphi}{\partial t}[/itex] and [itex]\nabla ^2 \varphi[/itex] and see what [itex]\alpha[/itex] must be if [itex]\varphi[/itex] satisfies your given wave equation?
     
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