The simplest and the purest of them all, the SIN wave. To understand how "waves" work in general (ie osc, lfo, etc.) I think it's essential to understand the basic components of the SIN function. In mathematics and physics alike, it is usually written like this.
y = Amplitude * [ sin (2*pi*frequency*x)]
Well, actually that's a lot of jargon so let's sort it out.
First of all, the big picture is that the right side of the equation vibrates, and makes y (the speaker diaphragm) vibrate. x is the time domain. The more the x, the more to the right you are in the waveform editor. I think most digital musicians are quite comfortable with that.
Take notice that x is in the sin parenthesis, and Amplitude is outside of the it. So no matter how the Amplitude changes, it will never affect the x value.. it only scales the wave in the y direction, making the speaker vibrate more. Quite obvious.. but not quite so obvious when written in math.
So now we go into the contents of the sin. This is where most of our phase cancellation headaches and whatnots occur.
For your viewing ease, here's the equation again:
y = Amplitude * [ sin (2*pi*frequency*x)]
First, frequency = w/2pi and you can write down w/2pi instead of frequency. The important thing is that you see a "2pi" in the equation. And in the field of trigonometry, for some very geeky reasons, "pi" means 180 degrees and "2pi" means 360 degrees. Screw the logic behind this, but just remember that it's from this very reason that effects such as phasers use the unit degrees for how much phase shift you want. 180 degrees is a complete phase shift, meaning the wave is turned completely upside down, hills becoming valleys and vice versa. So when you have a whole pie, or 180 degrees, you're completely out of phase. Beyond 180 degrees, it's the same as going from 180-0 so that's why it's usually left out. The guy who built the first phaser must have been a math geek.
Here's a sin wave:
and here's a sin wave that's 90 degrees, or 1/2pi out of phase, which happens to be called a cosine wave:
Well, now that we've got 2 waves, let's talk about mixing them together.. which naturally comes to a musician's mind. The act that we usually called "mixing", in mathematical term is an act of addition. We'll add 2 sines of the same frequency(5hz):
How uninteresting.. the volume got louder, obviously.
Let's try one with 2 sines at a slightly different frequency. (80hz and 75hz)
Yowsers! There's a pattern in volume!!!!! Actually, this pattern can be calculated by frequency of one sine wave - the frequency of the other sine. And yes, the resulting difference comes in units of hertz, which means when this number gets high enough, it will become audible. So a 140hz sine and 60hz sine will cause a difference of 80hz, fast enough to be audible AND makes quite a bad harmony. Therefore, you get a "beating" of 60 hertz and that sucks in a mix. But when this beating becomes fast enough, you get a strange effect.... part of what make up the notorious AM. (it adds AND subtracts the two waveforms) For a beating to occur, you need 2 waves. One of which you feed the AM. The other wave is generated by the AM module and then you get that ringing-metalico-ziing-zang tone. But that's if the beating is fast. When it's low, it's just phase cancellation and it messes up your bass.
We've done addition/subtraction of the waves.. how about multiplication? Yep, we're all familiar with this. It's FM. This produces a funny lookin' wave.. a sin wave made of a sin wave.. ain't that kind of neat? Anyway, the point of FM is to impose a waveform onto another. So the modulation signal takes the overall shape of the carrier. And since you get more bumps than you had before, you get more overtones, or brighter harmonics. That's the basic 2OP (operator) FM.
Heey, now that we've mentioned harmonics, here's a fun fact. When you FM, you get more lobes and you get more harmonics. Think of lobes as "change", and when you FM, the rate of change increases and so does the harmonic content.
See how many harmonics an FMed sin wave has as opposed to a sin wave:
Here's the whimpy sin:
Now check out a square wave.. it's just FULL of harmonics.. so full, that a jpg this small can't even depict them all!
It's obvious, but a sin wave is smooth. He's a chill-out, curvy, round type of fella that has NO harmonics. And the square wave?? Nooo waaaay. He's tough, rigid, has 90 degree corners and has a bunch of harmonics. See the difference? Harmonics has to do with the rate of change. Sin waves have a smoooth rate of change, and a square wave is very sudden... it's theoretically impossible to do a 90 degree corner, (because speakers need time to move) but it's pretty sudden nonetheless. Now think about synced waves... These guys are even more rougher, tougher, meaner sounding. Check this out:
See all the points and sudden jerks? These are what cuases the high, cutting edges of sync basses and leads. The edges are MUCH more pointier than 90 degrees. People don't call them "edgy" for no reason! Edges cut... RIGHT through the mix. The harmonics are just outrageous, all over the spectrum, and also kind of un-uniform, which gives it that distorted quality.
And when sounds cut RIGHT through the mix, they're precieved as loud, or punchy, or OUCHY. So let's talk about the energy of a sound. Everyone is probably comfortable with decibels but when it comes to defining just how much a difference a 3 dB cut will make is pretty tough. This is because the way sound is measured, in SL (sound intensity level) has a lot to do with the distance, and decibels are usually derived from a relative comparison of the SL.
The calculations are much, much, uhhh... MUCH too complex to write here.. Here's the theory behind it. Since energy means the "amount of work" in physics, it's measured in watts. People have decided to use a certain intensity of sound to use as a 0 dB guideline, and then measure sounds starting from there. Basically, it's something that's been "decided" on.
In a more practical sense:
"If a sound is not reflected or interrupted, the intensity drops 6 dB ( i.e. 0.25 of its value) every time we double the distance. Thus, if the SL is 90 dB at 2 metres from the source, it will be 84 dB at 4 metres and 78 dB at 8 metres."
http://www.avatar.com.au/courses/PPofM/ ... sity3.html
So now the sound has shot out of your speakers! Let's see what happens as it travels through the air in the Acoustics section. Thanx for taking your time to read all this.
By the way, feel free to use the images if they're of any use. They're always a drag to make!
<font size=-1>[ This Message was edited by: kensuguro on 2002-05-03 04:24 ]</font>
I never expected to cross over such an extremely complex topic like this one, when talking about sound. It really goes far away into the deepest scientific speculations of modern time. Acoustics is related to so many important things that it’s amazing: health, printer technology, avionics, atmospheric perfume sprays, communications, Digital TV, some modern heating apparatus, it’s just too much. I’ll keep exploring cos it’s opening my mind to many things, and… of course, to the understanding of music itself. See you later…
