Seismic displacement ofr a plane wave

[Briguy21]

Homework Statement

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

Homework Equations

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

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!

 

[gabbagabbahey]

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