Comparing K and k in Hecht's Optics Book

[manimaran1605]

What is the difference between propagation number(K) and wave number(k) described in Optics book written by hecht

He defined K=2π/λ and k=1/λ and both of them have the same units (i.e, meter inverse)
What does wave number of a Harmonic function tells about?

 

[olivermsun]

The units look the same, but they aren't quite the same. It's a somewhat confusing notation in my opinion (especially when you use 'K' and 'k'!)

K=2π/λ is in units of radians/m and k=1/λ has units cycles/m, where "radians" and "cycles" are usually omitted. You just have to remember which kind of units you're using, so you know whether there needs to be a factor of 2π inside the sine/cosine/exponential.

For example, a wave of "propagation number(K)" would be sin(Kx), while a wave with "wave number(k)" would be sin(2πkx).

 

[jtbell]

[Hecht] defined [propagation number] K=2π/λ and [wave number] k=1/λ

You have it backwards. He defines the propagation number as $k = 2 \pi / \lambda$ (lower-case Latin letter "k") and the wave number as $\kappa = 1 / \lambda$ (lower-case Greek letter "kappa", not upper-case Latin letter "K"). Those different symbols ($k$, $\kappa$, and $K$ in LaTeX; or k, κ, and K in PF's default font) tend to confuse people. Look carefully!

As olivermsun noted, the units are different. $k$ is much more commonly used. $\kappa$ is mainly used by spectroscopists. I don't know if they have a practical reason for it, or if it's just a historical convention.

 

[olivermsun]

The $k = 2 \pi / \lambda$ version is often convenient for working with waves when you also use the angular frequency $\omega = 2 \pi / T$ (where $T$ is the wave period).

That way, you can write things like $e^{i(kx - \omega t)}$ without having $2\pi$s all over the place.

 

[jtbell]

That way, you can write things like $e^{i(kx - \omega t)}$ without having $2\pi$s all over the place.

Exactly. The OP (manimaran1605) should compare equation (2.24) in Hecht with the other equations in that group on page 16, which are different ways of writing the same wave equation using different combinations of constants.