This is a typical wavefunction, which is a function of both position (x) and time (t). If you’re a good JC student in Singapore taking Mathematics at the H2 level, then you should be able to figure out that this function travels in the direction of positive x (if you can’t tell you can come ask me, heh) and it also travels in this direction with increasing time.
Now, we see that there’s an amplitude (A), and there’s an angular frequency (ω), and there’s a wavenumber (k). These quantities should, hopefully, be familiar to you, or at least I hope you’ve heard of them! Haha. These are all A Level knowledge, so hopefully you have!
I’d like to say that although what I’m about to say applies to all waves, not all waves have such a nice looking function like the one above! I just chose the easiest one of all so that it’s easy to digest what I’m going to say.
Going on, I’d like you to see that:
What I’ve done here is to take the second partial derivative, meaning I differentiate the wavefunction with respect to x only, twice, each time keeping t constant.
Aright, now, let’s try to take the partial derivative again, now with respect to t:
Cool! To sum it all up so that you follow me better:
It should be obvious to you then, that:
Does anyone want to fashion a guess what this means? Heh. Well, for waves we know that (from basic JC level Physics):
And therefore, we can say that:
But hey! We learnt in Secondary School that the speed v of any wave is related to the frequency f and wavelength λ by:
And therefore:
And thus our previous equation shows that:
That is, if we differentiate the wavefunction with respect to position twice, we obtain the derivative of the wavefunction with respect to time divided by the speed of the wave squared! What an interesting inherent symmetry! And this holds for all waves!
I’d really like to go on, but well, I don’t think most of you are acquainted with electromagnetic waves, are you? For electromagnetic waves, the equation turns out such that:
If you don’t know, ε is the permittivity of vacuum, and μ is the permeability of vacuum. Interesting enough, we note that:
Since all quantities on the left are constants, then we conclude that the speed of light (or rather, electromagnetic waves) must also be a constant regardless of anything, and therefore we now have:
Well well, I hope this is good enough! Because really, I can’t explain why light has a constant speed unless one deals with more advanced electromagnetic theory and wave mechanics. But haha, this is good enough I think, for all of you to chew upon! :p
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