Wave Equation – A derivation

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 y(x,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 specific parts of the string are “in phase” – be it spatially in phase, or temporally in phase.

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

y(x,t) 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. y(x=0,t=0)=+A and that’ll get us started.

The function will be something like y(x,t)=Acos({\theta}) 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 2\pi 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 {\theta}=\frac{2{\pi}}{\lambda}x we’ll have a repeating offender.

Do a little magic, and {\theta}=\frac{2{\pi}}{\lambda}x = \frac{2{\pi}f}{v}x = {\omega}t where \omega = 2{\pi}f and t=\frac{x}{v}

Now we’ve got something like y(t)

We also know that the wave at time t=t′ should be the same as the wave at an earlier time t = t'-\frac{x}{v} so substitute that in there, and find \footnotesize {y(x,t)=Acos({\omega}{(t-{\frac{x}{v}})}) = Acos({\omega}{({\frac{x}{v}}-t)})=Acos({\omega}{({\frac{x}{{\lambda}f}-t})}) = Acos(kx-{\omega}t)}

And there we have the wave equation!

Keep in mind: cos(\theta)=cos(-{\theta})!!!

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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One thought on “Wave Equation – A derivation

  1. Pingback: Waves – Delblg

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