Electric field for a plane wave in free space

[jb646]

This isn't really a homework problem, I just need to know how to do a problem similar to this one for the final and I don't want to fail, so I posted it here.

The problem is:
Given the electric field for a plane wave in free space: E(r,t)=E₁cos(wt-ky)k
a)what is the statement for w and ka in the wave equation and how are they related to each other?
b)what is the direction of wave propagation?
c)write an expression for the accompanying magnetic field B
d)show that the electric field given above is is satisfied with the wave equation

relevant equations: since this is more of an explanation problem I worked the equations into the attempt at a solution

a)do I have to rearrange the equations to solve for w and k or do I just integrate to solve for them?

b)im pretty sure it is in the x-direction, that makes sense to me

c)∇xE=-(∂B/∂t)
so do I just calculate the cross products:
x y z
d/dx d/dy d/dz
E1cos(wt-ky) 0 E2cos(wt-ky)

and set that equal to -(∂B/∂t), if so, how to I un-partialize it [can you tell I'm not really a physics major, just taking a required class...sorry for the lack of terminology and general knowledge]

 

[Mindscrape]

a) I think part a just wants you to say what omega and k are, i.e. k is the wave propagator and omega is angular frequency. Though by all means, rearrange so that you have it in terms of wavelength and displacement.

b) What physical reason do you have to believe it's propagating in the x-direction?

c) For a plane wave, remember that the magnetic field is orthogonal to the electric field with a factor of c.

I think this wikipedia entry will greatly help you with this problem.

http://en.wikipedia.org/wiki/Electromagnetic_radiation