K-vector: fn of space?

1. Apr 27, 2007

Swapnil

Is it possible for the k-vector to be a function of space (in the context of EM waves)? What would it imply if this was the case?

2. Apr 27, 2007

arunma

Well I know that the k-vector can be related to a wave's frequency (this is called a dispersion relation). Is that what you were asking about?

3. Apr 27, 2007

Swapnil

No... I was just curious about the spatial dependence of the k-vector (if such a thing is possible).

4. Apr 28, 2007

Meir Achuz

If the n of the medium varied in space, then so would k.
k=nw/c

5. Apr 28, 2007

Staff: Mentor

In a non-planar wave (e.g. a spherical wave radiating from a pointlike source), the direction of $\vec k$ obviously depends on location.

6. Apr 29, 2007

christianjb

The equation for a spherical wave is
$e^\left(ik|\mathbf{r-r}_0|\right)$

k doesn't depend on direction

7. Apr 29, 2007

Staff: Mentor

That equation contains only the magnitude of the vector $\vec k$, whose direction is always away from the source (located at ${\vec r}_0$):

$$\vec k = k \frac{\vec r - {\vec r_0}}{|\vec r - {\vec r_0}|} = \left( \frac{2\pi}{\lambda} \right) \frac{\vec r - {\vec r_0}}{|\vec r - {\vec r_0}|}$$

8. Apr 29, 2007

christianjb

I see what you're saying, but it's easier to treat k as a scalar in this case, where k has no dependence on direction.

9. Apr 29, 2007

robphy

The wave vector can probably best thought of as
"the gradient of the phase of the wave". Thus, one can visualize it as fields of vectors perpendicular to the wavefronts.

(The physical quantity described by the "k-vector" is actually more naturally thought of as a "covector" (or "one-form"), but that's another story.)