Let's talk about beats, beats are phenomenon that happens it's like interference in time and it happens when you have two sounds don't have to be sounds but that's usually what we're going to look at, two sounds waves that have almost the same frequency but not quite. Alright now in order to understand the phenomenon of the beats it's very, very useful to have a trigonometric identity that you probably aren't familiar with because it's been cut out of a lot of pre-calculus circular unfortunately but whatever. So let's look at these 2 waves, I've cosine of 2 pi times 10 times t plus cosine of 2 pi times 12 times t. Now you maybe wondering why did I get this 2 pi there, why did I chose the 10, what's the significance? Well let's go ahead and look at what this means, as you know the period of a sine or a cosine is 2 pi over the coefficient of t. So the coefficient of t is 2 pi times 10 but 2 pi's cancel and that means that my period is one tenth and that means that my frequency is amazingly 10 hertz. So the frequency here is 12 hertz, so these are almost the same but they're not exactly the same, so we'll expect this phenomenon of beats.
Alright so what is this trigonometric identity that I was mentioning? Well if I add 2 cosines together, what I always end up with is 2 times the cosine of the difference of these 2 things divided by 2, well jeez 12 minus 10 is 2 I divide by 2 I get 1 so this is 2 pi t times the cosine of the sum of these 2 things divided by 2 alright well what's that? Well that's 11, well 2 pi times 11 t. Alright so who cares, why is that useful? Well let's go ahead and look over here at a graph of this function now I don't want to just give you the graph for free I want to show you how you get. Alright so I've got 2 times the cosine of 2 pi t times the cosine 2 pi times 11 t. Alright now we know that the frequency of this guy is 1 hertz and the frequency of this guy is 11 hertz. This guy is oscillating much faster than this guy is. So this guy is just kind of taking his time he's going to go like that and this one is going real fast inside right? So that actually helps us make the graph, in fact it basically gives us the graph because what we're going to say is that during these oscillations this guy is essentially constant because he's going so slow. So the way we're going to graph this is we're going to pretend that this is a constant.
Now we know how to graph 3 times the cosine of something or 5 times the cosine of something right we just make little squiggly lines between negative 3 and positive 3 or negative 5 and positive 5. So here in order to graph this function we're going to make little squiggly lines between what? Well we're going to make them between 2 times the cosine of 2 pi t and negative 2 times the cosine of 2 pi t. So let's graph 2 times the cosine of 2 pi t, well that looks like that. It's a cosine graph what do you want, and then let's graph negative 2 times the cosine of 2 pi t, well it's going to be kind of the opposite alright we can do that not even hard. Alright so now we need to graph the cosine of 2 pi times 11 t between these 2 green dashed lines. So let's do it, here we go, we don't get to escape the green lines, so this is the sum of those two sound waves. Notice what we have, we have something that's oscillating at a frequency of 11 hertz but look what happens. It's amplitude is big and then small and then big and then small and then big and then small and you can see what happens after that it just keeps on going back and forth. So the frequency here of this envelope the screen is called the envelope alright the frequency was 1 hertz but within this time period we got loud twice. This counts as half of getting loud, this is a whole getting loud and this is a half of a getting loud.
Alright so that means that we're going to get loud 2 times per second or at twice the frequency of the envelope. It's 2 times because you're going positive and negative and positive and negative, so it kind of washes out it doesn't matter whether it's a peak or a trough in this wave alright so for that reason the beat frequency, these are called beats, the beat frequency is always equal to the difference of the 2 frequencies. So here I had 12 and 10 my beat frequency is the absolute value of 12 minus 10 hertz or 2 hertz I'll expect it to get loud 2 times per second. Alright so let's see how a problem would be solved using this technique, so this is a sample problem that I've seen on you know some of these state tests a tuning fork with a frequency of 440 hertz that's middle a, that's the proper frequency for middle a is played along with the middle a string of a violin. And 2 beats per second are heard what are the possible frequencies of the a string on the violin? Well if 2 beats per second are heard then that means that the difference between the violin strings' frequency and the tuning fork frequency must be 2 hertz.
Now it doesn't tell us whether it's big or small it just tells us that the difference is 2, so there's 2 answers 438 hertz or 442 hertz. Not a difficult problem at all but you do have to know what to do, this whole idea of beats is extremely important in musical instruments I've often asked classes you know "have you ever heard this before?" And I've had lots of students "Oh yeah I've heard it before what does it mean?" It means you're out of tune that's how you tune a musical instrument. You play your note along with the proper note and you listen for beats. The thing is that it's much easier to hear beats loud, soft, loud, soft than it is to tell the difference between 438 hertz and 440 hertz if you just hear them separately. Alright you got to have perfect pitch to tell the difference between those two things, but if you hear them together you hear beats or you don't pretty obvious. Another important phenomenon that uses this and maybe it's not a phenomenon maybe it's kind of a unfortunate state of affairs is radar guns. They use exactly this, so they'll shoot a laser beam off your car and then it comes back and because you're moving the frequency that comes back is different from the frequency that they sent out. It's not very much different but it's a little bit different and so then they look at the beats they use that to measure how fast you're going and they pull you over. Alright that's beats.