# Parent page: Waves

How do we describe a wave? Well, a transverse wave (in 1D) might like to be described at a particular point on whatever transmission medium it’s *waving* through (or w/e).

So we’re looking at the displacement of a point from equilibrium in the medium (say, in the y-axis). This depends where you’re looking and when you’re looking. In 1D, we’ll talk about where as “x” and when as “t”.

So the displacement from equilibrium as a function of x and t …

Each point on a “string” oscillates like as in **SHM** with the same amplitude and frequency – but, in general, the points on the string are * out of phase *with one another. Only points at

*of the string are “in phase” – be it spatially in phase, or temporally in phase.*

**specific parts**Spatially, we say that points at integer multiples of the wavelength are in phase and temporally, we can see that a point is in phase with integer multiples of the wave having traveled a wavelength, so (P.S. we’re getting that from **T** the time it takes to travel one wavelength (i.e. the period) )

needs to be described by some function – either **Sine** or **Cosine** will work – but let’s just go with cosine for now, so that at x=0 and t=0 we start at a maximum, i.e. the amplitude, i.e. and that’ll get us started.

The function will be something like for some theta governed by x and t. The points on the string move like SHM – but hey we haven’t learned about that yet (if’n you’re learning physics solely from *this guy* that is). So let’s think. How should we think about that theta?

Well, it’s units should be nil (no unit pls, I’m just a simple scalar) so that’s something to keep in mind. Uhhhh… Oh, it’d be smart to put in a factor of so that when certain conditions are met, the whole thing gets put on repeat – and what’d we say those conditions were again?

Well, when you move any integer multiple of the wavelength down the line, you’ll find a point acting the same as the one you left off of, so if we’re going down the line for an instant in time** t**, then for every wavelength traveled, we should find ourselves wondering “oh jeez, is this the Twilight zone?” Yes, it is.

So for we’ll have a repeating offender.

Do a little magic, and where and

Now we’ve got something like

We also know that the wave at time **t=t′** should be the same as the wave at an earlier time so substitute that in there, and find

And there we have the wave equation!

Keep in mind: !!!

Of course there are other forms involving partial differential equations, but hey, I’ll recommend a **Walter Lewin **video, just Google it.

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