An intuitive meaning to the phase constant k?

[rem1618]

k being the one from the harmonic wave ψ(x,t) = Asin(kx - ωt) where k = 2π/λ

The way I see it right now, k is just defined this way to get the period of sin(x) to be λ by using sin(kx), so I wondered if there's something more to it. (Though I know it can be used as an vector to declare the direction of the wave.)

I played with the trig stuff a bit. I know that a harmonic wave can be traced by a point in a circular motion. sin(x) traces the circle with a period of 2π so I take that to be a circle with 2π circumference and radius 1. Doing the same with sin(kx) gets a circle with λ circumference and radius 1/k.

So k seems to be related to 1/r or the curvature? I found it interesting too that if I directly substitute k with 1/r, the ω in ω = v/r from mechanics can be found with the ω in ω = kv from waves.

Is there some insight to take away from this?

 

[SteamKing]

k is also known as the 'wave number'.

See: http://en.wikipedia.org/wiki/Wavenumber

 

[ModusPwnd]

From that wiki, I like the phrase "spatial frequency". Thats how I internalize it. Its the spatial version of time's "omega" (angular frequency).

 

[Philip Wood]

The radius and circumference of the 'generating' circle for sinusoids aren't related to either $\omega$, or its spatial analogue, k. The circle radius gives the amplitude of the sinusoid.

I don't think you're going to get a much better intuitive feel for k than what I've just mentioned: it's the spatial analogue of $\omega$. But there's rather a neat extension to the idea when you deal with waves propagating in 3 dimensions. We then define a vector, $\vec{k}$, having a magnitude, k, equal to $\frac{2\pi}{\lambda}$ and direction that of the direction of wave propagation. This enables neat mathematical handling of wave propagation equations. For example, the displacement, y, at any point in the path of a plane sinusoidal wave may be written as
$$y =y_0 sin [\omega t - \vec{k}.\vec{r} + \epsilon].$$

 

[lightarrow]

For the OP.
To understand the meaning of k: fix the time t, as if you made a photo of the experiment and see what happens mathematically varying the point x of the space (you can fix t = 0).
To understand the meaning of ω: fix a point x of the space (you can choose x = 0) and see what happens varying the time.

You then understand why, as someone has already written, k can be called "spatial frequency" and ω can be called "temporal frequency".

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lightarrow