Converting between wavenumber and wavelength

[aliens123]

Homework Statement

Converting between wavenumber and wavelength

Relevant Equations

None

By dimensional analysis, we have that the wavenumber: $$k = \frac{\text{radians}}{\text{distance}}$$
And the wavelength:
$$\lambda = \frac{\text{distance}}{1 \text{wave}}$$

Then:
$$\lambda k = \frac{\text{radians}}{\text{distance}}\frac{\text{distance}}{1 \text{wave}} = \frac{\text{radians}}{1 \text{wave}}$$
Now:
$$2\pi \text{radians} = 1 \text{wave} $$
$$\frac{\text{radians}}{1 \text{wave}} = \frac{1}{2\pi} $$
So
$$\lambda k = \frac{1}{2\pi} $$
But this contradicts the "well known"
$$\lambda k = 2\pi $$
So where did I go wrong?

 

[kuruman]

You were not specific enough with your definition of $k$. Instead of $k=\dfrac{\mathrm{radians}}{\mathrm{distance}}$, you should have said $k=\dfrac{\mathrm{radians~in~1~wave}}{\mathrm{distance~of~1~wave}}$. Then you get the definition for the wavenumber $k$ with "radians in 1 wave"= $2\pi$ and "distance of 1 wave"= $\lambda$. Dimensional analysis is not a good heuristic tool for figuring out where constants go, if they belong anywhere.

 

[TSny]

I agree with @kuruman . Symbols such as $\frac{\text{radians}}{\text{wave}}$ can be ambiguous.

For example suppose I write $\frac{\text{in}}{\text{ft}}$. What does this mean? If it's interpreted to mean the number of inches per foot, then it equals 12. But if it means the ratio of an inch to a foot, it equals 1/12.

You wrote

$$\lambda k = \frac{\text{radians}}{\text{distance}}\frac{\text{distance}}{1 \text{wave}} = \frac{\text{radians}}{1 \text{wave}}$$

Here, the meaning of $\frac{\text{radians}}{1 \text{wave}}$ is the number of radians of phase in one wavelength. So it equals $2\pi$.

Then you wrote

Now:
$$2\pi \, \text{radians} = 1 \text{wave} $$
$$\frac{\text{radians}}{1 \text{wave}} = \frac{1}{2\pi} $$

Here, the meaning of the first equation $2\pi \, \text{radians} = 1 \text{wave}$ is to say that moving along a wave such that the phase increases by $2\pi \, \text{radians}$ is the same as moving 1 wavelength. You could rearrange this as $\frac{2\pi \, \text{radians}}{1 \text{wave}} = 1$. The ratio on the left equals 1 in the sense that the numerator and the denominator represent the same amount of movement along the wave. Dividing both sides by $2 \pi$ then gives the second equation. But note that now the meaning of $\frac{\text{radians}}{1 \text{wave}}$ is the ratio of how much you need to move along a wave to change the phase by 1 radian to moving along a wave by one wavelength. This ratio is $\frac{1}{2 \pi}$. That is, changing the phase by 1 radian only takes you along the wave by $\frac{1}{2 \pi}$ of a wavelength.

So, here the meaning of the symbol $\frac{\text{radians}}{1 \text{wave}}$ is different than the meaning of the same symbol when you used it in your expression for $k \lambda$.