Waves

Mathematically simple to describe yet capable of really making your brain hurt nonetheless, waves are everywhere.

It’s in the drink you’re drinking ,the air you’re breathing, the thoughts you’re thinking, your heart that’s (hopefully) beating – and pretty much in all the matter you’re made of.

They’re everywhere, trust me. It’s behind you right now I bet. Don’t look now, it’s there, so rest assured.

Jokes aside, this is all due to the fact that waves really are simple shit. And so too are the things that waves describe, such as Simple Harmonic Motion (SHM).

I’ll outline a few common mathematical descriptions of waves in other posts – there’s the Wave Function and Fourier Series – but for now, I’ll just try and answer the question, “What even are these so-called ‘waves’ and why should my tax-payer’s money go towards funding them?!”

Well, it isn’t something your taxes fund, but rather a consequence of how the Universe works.

Wikipedia says that a wave is an “oscillation accompanied by a transfer of Energy that travels through a medium (Space or Time)… Wave motion transfers energy from one point to another, which displaces particles of the transmission medium…” Yadda, yadda, yadda.

It’s accurate, but how about a picture?

wave-types
Two kinds of waves

There we can see two different types of waves: transverse and longitudinal waves. The difference between the two is that the direction of energy transfer lies perpendicular to the direction the wave travels in (for transverse waves) and for longitudinal waves, the energy transfer is in the same direction as the direction of travel.

The same mathematics describes the two, so it’s k.

A simple equation that describes the behavior of most waves (periodic waves) is the following: v = {\lambda}f where the variables are the velocity of the wave (a constant value, dependent on the medium), the wavelength (distance separating two “identical” points on the wave), and the frequency (the amount of wiggles per second (usually it’s per second)). The units of velocity are (in SI units) meters per second, so wavelength is obviously gonna be in meters, leaving frequency with the units “per second” denoted s^{-1}

This equation is pretty simple but nice and intuitive, which is why I like it. Think about it for a second – if the wave’s velocity is constant, let’s say for now it’s traveling at 1 m/s, and we say it’s wavelength is 1 meter long, then the time it takes for it to travel one wavelength is therefore 1 second. If you were to keep the velocity the same, but halve the wavelength (now it’s 0.5 meters long), then the time it takes for the wave to travel by one wavelength also halves, or equivalently, we get two wavelengths per second, ergo the frequency doubles. Therefore it makes sense to say f \propto \frac{1}{\lambda} . The two quantities are inversely proportional.

The mathematical functions that physicists (and other normal people) use to describe waves are the Sine and Cosine functions – when you plot em, they even look like waves, so it’s a match made in heaven really. Briefly here, y(x,t)=Asin(kx-{\omega}t) which describes a wave pretty much entirely (go look at the derivation and/or The Wave Function if you’re keen to really understand that expression) – but that’s in an ideal situation, really. In real life, we’ve got Damping effects, Friction, all kinds of bullshit like that, so it’s complicated.

Life’s complicated. Deal with it.

(Just remember to pay your bills and taxes, oh and support your community by voting in local and national elections too)

In any case, let’s just focus all our brain cells onto one variable at a time.

The capital “a” (famously denoted A) is the Amplitude of the wave (famously regarded as the largest displacement from the equilibrium position).

The lowercase “k” is something called the wave numberit’s nothing for you to worry about, it’s just there to make this equation look pretty. k=\frac{2\pi}{\lambda} just remember that.

Lowercase “x” is the displacement along the x-axis (this equation describes 1D waves – but hey, that’s pretty good, you decide what axis you care about, point and shoot, and blam-o, you’re in business (if you make money off this, I deserve a 10% cut at least)).

That weird looking “w” is actually the Greek letter “omega”, and omega, like k, is fluff-stuff: \omega=2{\pi}f where f is the frequency.

Finally, the lowercase “t” is time. What’s it doing there? Idk, check out a derivation of the wave equation, and maybe you’ll get it. Maybe. 😉

Rewrite that without the fluff, and you get: y(x,t)=Asin(\frac{2{\pi}x}{\lambda}-2{\pi}ft) = Asin(2{\pi}{(\frac{x}{\lambda}-ft)}) = Asin(2{\pi}f{(\frac{x}{v}-t)}) as per our previous equation v={\lambda}f

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