mangoes8061
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#1
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I do not understand where the 2π/λ comes from in y(x,t)=Asin(2π/λx- ωt) ? Why do we have to multiply the x by this number? I've seen it called the angular wavenumber but that confused me even more.
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Callicious
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A good amount of understanding for this can be gleamed from Fourier Analysis- Lambda is the "spatial" period of the wave- how often it "repeats" in space.

Consider angular frequency, which is the natural Fourier equivalent of time:
\omega = \frac{2\pi}{T}
for a regular old' sine wave- where T is the time period. Angular frequency is a measure of the frequency of the wave in time. Similarly, you have
k = \frac{2\pi}{\lambda}
which is a measure of the "spatial frequency" of the wave. Like angular frequency, it's angular: if k\delta{x} is 2\pi then you have had one full wave over \delta{x}.

That's the way I understood at it A-level without having to worry about the specifics- it's just natural/how it works. Lord knows I didn't know what Fourier was, but being able to draw a parallel between time/space and just treating them as dimensions (rather than as explicit concepts of time and space separately) made the understanding clear of why we would use a k.

We often take kx and \omega{t} to represent periods in an angular form, i.e. 2\pi for either is one full period. Most of this stems from the fact that we're likely to run into sines and cosines, and it's just very convenient to use this system (which often falls out naturally from the math.)

Some examples where k falls out naturally:
- total energy of a particle (quantum) has \omega for wavefunction
- kinetic term in schrodinger has k for wavefunction
- electric fields leverage wavevectors for direction of light propagation
- basically anything to do with sound propagation (and not just through air.)

Anyway, here's an example of where it naturally falls out. Consider the wave equation in 1D

d_t^2f(x,t) = c^2d_x^2f(x,t)

To solve, propose an ansatz

Ae^{i(kx-\omega{t})}

Which does (verify it yourself) satisfy the problem if you let

c = \frac{\omega}{k}

Note that c is the speed of the wave in the medium. Anyhow, in this case, having k and \omega make the ansatz fairly obvious. That's just how we've got our angular system set up- \omega = 2\pi{f} and so having k similarly defined makes it fairly easy to do things.
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