Okay, so we have talked a lot about Helmholtz resonators, and when I say we've talked a lot about them, I mean like you have been doing a lot of the math to figure out how to solve for the resonant frequency of a Helmholtz resonator. But it's possible that you're thinking, what? What the heck is a Helmholtz resonator and why is it important for speech? Like why is this one of the first things we learn when we're going through to talk about how we actually speak? First we start with spring and mass resonator, then we go to Helmholtz resonator.
None of this really seems to make any sense. I'm going to tell you why it makes sense. So really quickly, just to go back through, what is a Helmholtz resonator?
I mean this is in the other videos, but we just kind of like blow over it really quickly. So this is a Helmholtz resonator right here. It is like a hard balloon, basically. It's a system with only one resonant frequency. It has an opening, it has a neck, it has a circular bowl.
You know that because you have had to calculate the resonant frequency by using the opening, the area of the opening, the length of the neck, and the volume. And so this is just... This is one.
This picture here is one. It's basically a bottle. I mean, like, you could fill it up with stuff, and you could consider it like the weirdest hydro flask that anybody would have.
Definitely not double-walled, so, like, your water wouldn't stay very chilled, but whatever. Also, I've been drinking, not today. but yesterday. And so while I was drinking this, I thought, you know what? That's a Helmholtz resonator, bro.
And you know the old trick? Everybody knows this. But you can like blow in a bottle.
You can change the pitch of it by having more water in it or not in it, and that changes the volume. that's available like the volume for air that's available in this this is sort of like a Helmholtz resonator uh and it does only have one resonant frequency that resonant frequency right there then again you can change it you can put water into it um I thought about pouring water into it but uh I'm not gonna do that as it it seems like a bad idea you It seems like I would spill on everything. So it's just going to be empty.
Anyway, this is not a great Helmholtz resonator. I mean, it's obviously not like a cylinder. The neck is a little too long. This is not, again, it is a cylinder.
It's not a sphere. So there are some problems with this as a Helmholtz resonator, but it's basically the same thing. So why is it important to know about a Helmholtz resonator? We can think about this bottle like a Helmholtz resonator. You can think about this if for some reason you don't want to think about the bottle.
But this whole video hinges on the bottle. So you should just get over and think about the bottle. Anyway, what happens is air moves over this opening. Remember the Bernoulli principle from 3.10.
What happens with the Bernoulli principle? I have, if you took it with me. You saw that I have a straw that's in a mug and then I'm going to blow through a second straw and the air moving over the top of the second straw decreases the air pressure enough that the liquid inside the mug is drawn up the straw and out into the airstream. So then just by blowing through a straw you can essentially suck liquid up out of another straw and then shoot it out in front of you.
So that's the Bernoulli principle. What's going on there is that air moving across the top of something creates a lower air pressure in that area where the air is moving over. So the air blows over the top of the bottle and the air that's in the bottle comes up into that lower air pressure here and gets blown out. Well what happens, there's nowhere else for this bottle to fill up with air. So eventually it becomes a vacuum inside the bottle.
You can't continue taking air out. So eventually what happens is now inside the bottle is a low pressure zone. So air that's blowing over the top of the bottle instead of sucking the other air out is now going to go into the bottle.
Once the air goes in the bottle, now it's a high pressure zone and it's going to come back out because of the Bernoulli principle. Then this is going to turn back into a vacuum and air is going to go into the bottle again. The cycle, like the rate at which that cycles from high pressure to low pressure, is determined by the size of the bottle.
That's the resonant frequency. So again, the rate at which you go from high pressure to low pressure and back and forth, that is a cycle of a wave. Like we've been talking about waves this whole time, waves through the air, waves through the water, waves, waves, waves. This is sort of like a wave, high pressure, low pressure.
So high pressure is your compression, low pressure is your rarefaction. And so you've got all the components of a wave right here. And what happens is that that cycle time, the cycles per second of high pressure and low pressure, is the resonant frequency of the bottle.
So the energy resonates at that frequency and it eventually creates a tone, which is this. Why is that important for speech? Well, because the pharynx, the mouth, and the nasal passages all have resonant frequencies.
And all those different resonant frequencies are what gives speech its characteristic sound, specifically the formats, which you probably know from, well, we talk about it in 310. We talk about it in 205. So if you had those with me, you know about that. We talk about it a little bit in 367. And also, you definitely have talked about it in phonetics. And we're going to talk about it a ton in this class. After this point.
So all of those resonant frequencies from all of those different areas give you formants. Those are zones of high resonant energy. So all of these different areas, the pharynx, the mouth, the nasal passages, those are like little Helmholtz resonators giving off a characteristic frequency.
So. I make scientific equipment at home by drinking beer. And what I was thinking, like I told you, sit around drinking this beer, eating a hamburger, and thinking that this is like a Helmholtz resonator. And so I thought I could probably figure out the resonant frequency of the bottle mathematically and then use audacity to see if those matched up.
So... Got a tape measure here and what we're going to do is we're going to take some measurements and calculate the resonant frequency of this bottle with the same formula that you already know. So first thing we need to know is the area of the opening and the way that we do that is this opening is pretty much an inch in diameter.
So the formula for area finding the area of the opening is pi r squared. But what I know is the diameter. So what I have to do is cut the diameter in half.
So that's half an inch. But we also want to use millimeters, not inches. So we have to convert 0.5 inches to millimeters, which is 12.7 millimeters. If you put that into the area formula, it's pi r squared. So r is 12.7, 12.7 squared times pi equals...
This number over... wait, I wonder... I don't...
This number either here or here. I don't know which direction the video... it doesn't matter.
I'll check. Here. Oh, it's over here!
This number here. This number here, 506.71 square millimeters. That's the area of the opening right here.
Next we need a length. The length of the neck of the bottle is three and a half inches. So you just convert that in Google to 88.9 millimeters. And then the last thing is we have to figure out the volume of the entire thing.
And I was trying to, you know, think about like, well, what you do is you take the area here. So like you do the diameter. thing again to get the radius then convert the radius and then you figure out the area and then you multiply it by the height and that gives you the thing well no you don't have to do that you know the volume it's a 12 ounce beer There you go, 12 ounces. So you can take 12 fluid ounces and you can convert that to cubic millimeters, and 12 ounces is 354,882 cubic millimeters.
So now we've got all the components that we need to figure out the resonant frequency of a Helmholtz resonator. We've got the area, 506.71. We've got the length, 88.9.
We've got the volume, 340. 54,882. So we just feed it into the formula that you guys know so well at this point. Like you can't even stand to look at it anymore probably.
But here it is. Over here. Somewhere.
Resonant frequency, the formula we've got is C over 2 pi times the square root of A over VL. C, of course, is the speed of sound, which we're just going to estimate at 340,000. It's a little different than that.
It's like 340,372 or something. That's fine. We'll just round it to 340,000 over 2 pi.
multiplied by AVL. We're just going to move over all the things that we had in the previous slide, this one right here. So 506.71 divided by 354,882 times 88.9. So going through this, um, I was, uh, when I was making the answer key for the exam, I did not. follow the order of operations.
You have to do what's under the division sign first, otherwise it doesn't work. I was just like multiplying and then dividing. I had it written out long ways, not up and down. And so I got the wrong answer.
This may have happened to some of you. And if it did, you're not going to lose too many points because it's such an easy thing. to mess up on.
But essentially you need to do the bottom of this calculation first. So you end up with 3,154,000. Oh, sorry, 31,549,009.8 under 506.71.
That's still within the square root. We can reduce that more to 0.000016061, which is cool because it's a palindrome. That's also the square root.
So you take the square root of it. It's basically 0.004. And you can also reduce the 340,000 over 2 pi to 54,113, basically.
And so you multiply those together and you get 216.9 hertz, or 217 hertz. 217 hertz. That's what we should be getting out of this. Mathematically, that's what we should be getting out of this.
It sounds... Pretty close, but of course we can test it in Audacity. You now know this as well as anybody else. So what I did was I played this sound in Audacity and I analyzed it.
Here's the spectrum. You can see that it's a really low frequency down here. Really low.
So. in the realm of like 200 hertz. So this is, it's looking good. And then I did the analyze plot spectrum tool. And this right here is 199 hertz.
So basically 200. And the math showed that we should get 217 hertz. Remember, this isn't a perfect Helmholtz resonator. The math is going to be a little bit...
different but 217 is what we get mathematically 200 is what we get using Fourier transform it's pretty close it's pretty close so the math actually does work out now you can see it like in action it's not just some abstract thing that you have to calculate and you don't really understand like well how does that relate to anything it relates to this exactly to this so it's not exact Exactly right. It's possible the volume in this is wrong. So 12 ounces is how much beer is actually in this.
But the bottle, I mean, so I'm sure that most of you had beer. Beer goes up to about here, just a tiny ways up the neck. And there's all this volume here that isn't being measured. There's 12 ounces of beer, but then there's also this area here that we have to account for because air can get into it. So it's Probably about 13 ounces, maybe it's 13 and a half ounce, the whole bottle itself.
And if you recalculate everything using 13 ounces instead of 12 ounces, the frequency you get is 208.36, basically 208. And remember what we got out of here was 200, so we're getting a little bit closer. So probably the volume... It's just a little bit wrong in the calculations and then also it's not a perfect Helmholtz resonator.
So it's there's going to be some differences but it's pretty close. And that's our Helmholtz resonator right here. Again why is this important? Because you've got a pharynx right here. It's not exactly this shape but you know sort of.
you're going to have a resonant frequency there because air is flowing through that area. You're going to have high pressure and low pressure zones. You're going to have a resonant frequency of that cavity. Same for your mouth, your nasal passages.
They're all going to have resonant frequencies, and those create formats. So what you're hearing here, this is basically a format. It's a resonant frequency, but it's also a format when you think about it in terms of speech. It's like a format. It's a resonant frequency because there's only one for a Helmholtz resonator but it could also be a format if we think about it in terms of a voice.
My guitar that you saw in one of the first videos. Guitars have resonant frequencies too. Not really electric guitars but my acoustic which you might...
oh no that's the electric. um acoustic is in the closet but the acoustic guitar has a resonant frequency all acoustic instruments have resonant frequencies in fact one of the ways that people tell if a violin is a Stradivarius violin is by looking at the resonant signature of the violin to look for the formats so violins have formats and Stradivarius violins have specific formats you Um, okay. That's the Helmholtz resonator.
I will, I always, I always end before I can actually end because I have to be back at this screen to click stop. Anyway, I'll see you soon.