Phasor representation of plane wave propagation

[Defennder]

Homework Statement

I was looking through my notes when I saw the following expression of a plane wave represented as a phasor $$A_{0}e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$

Now I can certainly understand a plane wave propagating along a given coordinate axis say, x or z, and the phasor representation of that would simply be $$A_{0}e^{i(kx-wt)}$$.

But somehow when I'm forced to consider the wave traveling in an arbitrary direction wrt to some fixed coordinate system, I don't understand how that reduces to the equation above (the first latex expression). I suppose it's due to the fact that k and r are both vectors and that k in the 2nd latex expression is a scalar given by $$\frac{2\pi}{\lambda}$$.

I don't remember ever learning about k being treated as a vector and if so what direction is it oriented in? I searched the internet and found:

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

but apparently the page doesn't explain why it's possible to substitute the expression k.r into the plane wave phasor itself. And why the dot product? I'm guessing that it's because we would then obtain three separate phasors for each x,y,z component, but I still don't see why exactly is it used.

Could someone point me to a textbook or an online resource for clearer explanation?

Homework Equations

The Attempt at a Solution

 

[Kurdt]

The scalar version is commonly called the wavenumber. The wave vector has a magnitude of the wavenumber and just points perpendicular to the wave fronts (i.e. the direction the wave is travelling). Theres not much more to it than that.

 

[Defennder]

So it's a matter of definition that k points along the direction of the wave and has magnitude of 2pi/lamda ? Then why is the dot product used?

 

[Kurdt]

Well the dot product is just the projection of one vector onto another. If the wave is not traveling in the direction of r then we need to use only a component of it.

 

[Defennder]

I think I'll post the context in which the question is asked:

http://img166.imageshack.us/img166/4619/crystaldiffractiongc4.th.jpg

My question is, I originally thought r was the vector along which the wave was propagating, but now as you can see from the picture, r is the position vector to a specified point on the crystal. What bothers me most is how the dot product can be justified when I can relocate the origin O somewhere else and you can see from the picture, r will change and so will k.r, which is kind of weird. After all, when crystal diffraction is concerned, we aren't quite concerned where O is located only the Bragg angle for which diffraction occurs and this is dependent only the relative position of the wave source and the crystal and not on O.

 

[Kurdt]

Ok scattering isn't really my thing so I'm a bit stumped at the minute but I will attempt to find out or else I'll post it in the HH forum.

 

[malawi_glenn]

I don't understand your question really, but as far as i have learned from scattering theory is that the plane wave is: $$e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$
where $$\vec{r}$$ is the postion vector from an arbitrary point (O) to the scattering point.

But anyway, when you determine the scattering amplitude, you do at least one integration over the crystal volume, then your coice for O didn't matter.

See for example Kittel chapter 2 about reciprocal space and scattering

 

[Kurdt]

A. P. French briefly covers 3d waves that explains a bit more in vibrations and waves, which luckily has quite a good website here:

http://physics.nmt.edu/~raymond/classes/ph13xbook/node20.html

I think the course is following French's textbook more than anything else but its still good. Like malawi glenn said it doesn't really matter the choice of the origin as long as you're consistent. Because its periodic one might have to add in something for the phase.

 

[Defennder]

Actually that's what bothers me. The fact that O is entirely arbitrary and $$\vec{r}$$ is entirely dependent on O as well as the fact that $$e^{i(\vec{k}\cdot\vec{r}-\omega t)}$$ depends on $$\vec{r}$$. I've looked through a book which I browsed through in the library concerning this, and there was something about reciprocal lattice or some sort which I didn't learn in the course. I don't have the book with me now though. Is knowing this necessary to understand why the choice of O doesn't matter? I asked my prof about how $$\vec{r}$$ is apparently
arbitrary and he said that $$\vec{r}$$ is actually defined wrt a lattice point.

Sorry for the late reply. But the HW forum for physics moves so fast that the thread was buried a few pages behind so I thought no one would reply. I didn't notice when it resurfaced again.

Could you provide the titles of the textbook you made passing references to? I'm not too familiar with physics texts.

 

[Kurdt]

I believe malawi glenn is referring to Charles Kittel's "Introduction to Solid State Physics" which is onto its 8th edition now I think. The book I suggested only has a page on the subject and isn't really useful. The website I linked to however goes into a bit more depth.

 

[Defennder]

Kurdt, I'm really beginning to think that that question I'm asking should be framed in the context of wave scattering and that the thread is mistitled, what with the choice of O being arbitrary and malawi_glenn saying that the integral (whatever type of integral it is) performed over the lattice structure unit cell will always give the same answer.

If you could recommend some texts regarding this I'll be grateful to check them out myself. Please state the full title though.

 

[Kurdt]

I think Kittel's book is a good place to start. The chapter is about reciprocal lattices but starts off with Bragg diffraction and talks of scattered wave amplitudes so they're obviously related. I asked in the homework help forums for others to have a look since solid state isn't really my thing and I quickly realized once you posted that picture that I wouldn't be much use. If malawi glenn comes back to this thread he may be better equipped to make more suggestions.

 

[malawi_glenn]

Defennnder, I don't know, but it seems to me that you just is not so familar with plane wave representation in 3D. $$\vec{k}$$ is the direcetion where the wave is going, and $$\vec{r}$$ is a point in space.

So it doesent matter where you put the origin, since the crystal is symmetric with repsect to translation and that you integrate over the whole volume.

What kind of course did this question come to you? Maybe you could visit your library and check out some books.