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Converting between wavenumber and wavelength

  • Thread starter aliens123
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Homework Statement
Converting between wavenumber and wavelength
Homework Equations
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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

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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

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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##.
 
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