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So, let's switch gears a few minutes, and I want to talk about something called

resonance. And, resonance will have it has, an

important impact in, in many aspects of our life.

you know, probably one of the simple things that you can think about In terms

of resonance, for those of you who ride in an automobile, there's a suspension

system. If you were to look under the car,

there's a spring that mounts is part of the suspension system mounting the wheels

to the automobile. And, if you drive down the road, if you

hit a bump, you'll feel the car kind of oscillate a little bit.

And then as the road flattens out, it, it It will, you know, so will your response

to the automobile, well, that's a resonance, okay?

And, but resonances happen mechanically, they happen actually, you know, we can

think about a resonance of electrical circuits, and we'll talk about that a

little bit later. And Mark will cover that I'm sure.

And we can also talk about resonance in acoustic space.

And the, the example that I wanted to discuss is the Helmholtz Resonator.

Named after Helmholtz so you can look up under Wikipedia to learn about his

contributions to, acoustics. But the Helmholtz resonator we have to

make a few assumptions before we, we talk about it.

But it, basically what we're interested in is, is a bottle here and this bottle

has a, a, a neck on it Right here, and basically the, the mass

of the air, that's in this neck of the bottle here, moves as a unit.

it's as if it's a plug, moving up and down in the neck of the bottle.

All right, and then we have a volume of the bottle here

And that volume of the bottle ends up in the cavity, in this open bottle, creates

or provides a stiffness. we also have a surface area of the neck

of the bottle, so, the, the're going to have a length associated with this neck

of the bottle. We're going to have a surface area of the

opening, so S of the, of the port here of the bottle, and then the volume.

Alright? And, there are a few assumptions that are

important in thinking about the bottle, one is the wavelength of sound.

for the Helmholtz's resonator effect to apply, the wavelength of the sound must

be much greater than, L the length, the fluid in the neck.

And that way the fluid in the neck moves in a unit mass, like I was describing.

the wavelength also had to be much greater than the, third root of the

volume. and when that's the case, the acoustic

pressure in the cavity provides an actual stiffness.

And then, of course, if the wavelength is much greater than the square root of the

surface area. The opening of the bottle then radiates

like a simple sound source. Alright?

And the analogy is a mass-spring system. Okay?

And just like the string that I showed earlier, if I were to displace this mass

by some fixed amount of distance, we would get oscillation.

And the mass would vibrate up and down against the spring.

That's basically what happens in the Helmholtz resonator.

Now before we go on to talk about the mathematics simple mathematics relatively

behind the Helmholtz resonator, I thought I would would demonstrate the effect for

you. Since we've been talking about the

Helmholtz resonator, a little bit, I thought it might be fun to demonstrate

it. And I thought maybe before I demonstrate

it, I would tell you a story, my father, when he was little.

Basically had to he had to always take the milk bottle back and have it refilled

at the local store. at that point in time you didn't buy

milk, you know, plastic containers they were all glass bottles.

And so you returned the glass bottle and, and you'd get a new one or have it

refilled. he took the bottle back to the store one

night, and he was little and it was kind of dark outside.

And he noticed as he was walking along, there was something howling, and he

became scared, so he started running. And he told me, the faster he ran, the

louder it howled. what he was really running from was a

Helmholtz resonator, he thought he was running from a ghost.

but I thought it was a funny story and I wanted you to hear the sound of a bottle

in resonance. And we, you know, we're talking about

this a little bit, you end up having a neck the neck of the, the, the air in

the, the, in the volume here, actually will vibrate like a mass.

And it's actually the reason why we can think about using a port in a ported

loudspeaker design, because we can tune that port so that it actually radiates.

The volume of air that sits in the bottle itself inside of here, and I have water

in here as well right now, but the volume of air that sits in the bottle actually

serves as the spring. And so if I blow across the top of the

bottle, [SOUND] you can hear, [SOUND] a particular frequency.

And that frequency is associated with the resonance of this mass column of air

that's moving against this spring that's in the bottle.

Now, [SOUND] let's see if we can change [COUGH] change the frequency.

You hear a lower frequency. You hear a lower frequency because now

the volume of air in the bottle is much larger now.

And because of that, the spring stiffness changes, and it's actually more

compliant. So, the mass of the, in the, of the air

moving in the neck hasn't changed, what we've done is, basically, made a softer

spring, and so that lowers the frequency, [SOUND].

[SOUND] Hear the changes. That's the basics of the Helmholtz

resonator, and it's incredibly applicable to speaker design, both in thinking about

how the spring stiffness is defined by the volume.

The volume of the box you build will actually serve as a spring.

If you use a port, it's kind of like the neck of the bottle.

So there's a lot of great relationships between the Helmholtz resonator concept

and speaker design. Okay, with that demonstration in place,

let's, let's talk a little bit about how we could calculate the resonant frequency

for a given bottle. So first, the mass the mass of the volume

of the air moving is represented as shown here.

It's the density of the air, the surface area, and the length.

So. S times L and [INAUDIBLE] L, this L prime

is in a, is what we call an effective length of the neck.

And a is the radius of the opening, so the effective length is related to the

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So, our force here is our spring constant times the displacement.

So this basically has to be our spring constant K, this is our displacement and

that's how we get this result right here, which is a significant result.

So, given a mass and a stiffness, we can compute the resonance of the system.

And that is going to correspond to a frequency at which it naturally

oscillates. Now, there are some details in vibrations

that I won't cover here, you know, natural frequencies corresponds to

systems that have no mechanism for dissipating energy.

So, you know, in your cars, I just talked about earlier, you typically have a shock

absorber. and basically what that does is, it

dissipates energy in the system. The natural frequency corresponds to the

system that's described without any kind, any mechanism for dissipating energy,

But let's, let's continue with that just a bit, as you'll see here, our frequency

which our system would naturally oscillate.

Can be defined by omega n, so it's the square root of k over m so the stiffness

divided by the mass. Alright.

And if we substitute from the expressions that we had earlier we can put the

stiffness associated with the bottom, Helmholtz resonator in the equation.

And we can insert the mass that we computed, this is one over the mass, the

mass is right here, obviously, but it needs to be in the denominator, as it is

here, all right? So, it's the stiffness K divided by the

mass and with some simplification you see that we have an expression that relates

the speed of sound in air. To the square root of the surface area

here associated with that plug of moving air all divided by the volume of the, of

the enclosed, the volume of the bottle. And the effective length of that moving

mass. Okay?

And remember, we, circular frequency is related to frequency by 2 pi.

So, if we want to compute the natural frequency in hertz or 1 over seconds we

basically have to divide and make [UNKNOWN] by 2 pi.

And if we do that, then we get an expression for the natural frequency

associated with that oscillation. Now, this will show up at a frequency,

you know, can show up, obviously, at a frequency in our audible range.

and you heard that earlier when I blew across the bottle.

And of course, as I drank some of the water out of the bottle and changed the

volume I didn't change this, I didn't change this.

But when I changed the volume itself, you heard the frequency change.

so we demonstrated the, result of this equation, by simply removing water from

the [INAUDIBLE], from the bottle. And then blowing across the bottom again

to hear a change in the, in the frequency.