Plane Wave Equation Propagation and Oscillation Directions

[DrPapper]

Homework Statement

Write down the equation for a plane wave traveling in perpendicular to the plane x+y+z=constant traveling in the direction of increasing x, y, and z.

Homework Equations

From the given information how do I determine the unit vector that goes next to E(0)? How do I determine the r vector in the exp?

The Attempt at a Solution

My present solution is:

E=E(0)(X/sqrt3+Y/sqrt3+Z/sqrt3)exp^i[nk((x+y+z)/sqrt3)-wt] Sorry I can't seem to get latex to work so this isn't such a mess to look at.

The n is the index of refraction.

The k is the wave number in a medium with index n.

My reasoning is since the wave is traveling perpendicular to the x y z plane all values are positive for the direction the field oscillates in. The wave proceeds in the positive directions so the exponent has all positive values. I'll enclose a picture of my solution to make it easier. It's the Problem 2. Part 3. I'm uncertain of all my answers, but knowing how to do three will definitely tell me how to do 2.2 Thanks in advance. :D

 

[Simon Bridge]

You put double-dollar signs either side of display math to get latex to render (and double hash for inline math)
Thus $\LaTeX2e$

Start by writing the general form for a plane wave with general wave vector $\vec k$

From what you've written, the wave iscillates in the E dimension... it may take positive or negative values in that direction... about some equilibrium value usually taken as E=0.

 

[DrPapper]

Thank you for your response Simon. I believe what you're telling is is to first write

$$E=E_0 e^i(\vec k \cdot \vec r - \omega t)$$

Where $$\vec k = (x_1\vec x + x_2\vec y + x_3\vec z) $$ is the Euclidean direction vector with coefficients of $$x_n$$ to the unit vectors.

I don't understand what is meant when you say "From what you've written, the wave oscillates in the E dimension". I'm thinking you're saying I've written it such that the wave oscillates in a dimension that does not exist? Or is there some E dimension I'm missing? I'm not being sarcastic, I'm a bit lost in this course so it very well could be there's something called the E dimension. Would you please expand on that a bit more?

Also, are you referring to what I've written in the photo or in typeset? The typed part is pretty wonky, and looking at it again it is not what I'd actually written as my response, would you please check the picture (if you've not already). Or I can re-do it in latex if that's preferable - just takes me some time to type it all out since I'm not very experienced with latex.

Again, thank you for your time and help.

 

[Simon Bridge]

I try to avoid pics.

You have written a wave in E, in post #1, ... what physical quantity does this letter represent (you didn't say)?
$\vec E$ is usually the electric field vector... this does usually have a direction in space which you will have to deduce from the physics of electric fields but not all waves oscillate in a physical direction... i.e. pressure or particle density.

The equations in post #3 are, indeed, what I had in mind.
Considering your problem statement, what is $\vec k$ going to be in this case?

Note: latex a^{b(c+d)} gets you $a^{b(c+d)}$
... in general put curley brackets around everything you want to get grouped together.

 

[DrPapper]

I was trying to write that my attempt at the solution is:

$$E=E_0(\frac{\hat x + \hat y + \hat z }{\sqrt 3})e^i(nK_0(\frac{x+y+z}{ \sqrt 3})-\omega t)$$

where

$$E_0$$ is just the amplitude of the electric field

n is the index of refraction for a material

$$K_0$$ is the wave number of when in the material having an index of n

LOL I should have just started with latex.

 

[DrPapper]

Darn it, I did it wrong again:

$$E=E_0(\frac{\hat x + \hat y + \hat z}{ \sqrt 3})e^{i(nK_0(\frac{x+y+z}{ \sqrt 3})-\omega t)}$$