In this course students learn the basic concepts of acoustics and electronics and how they can applied to understand musical sound and make music with electronic instruments. Topics include: sound waves, musical sound, basic electronics, and applications of these basic principles in amplifiers and speaker design.
Acoustic Resonance: Helmholtz Resonators
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Do curso por University of Rochester
Princípios da Engenharia de Som e Produção Musical: Parte 1 - Som Musical e Eletrônico
299 classificações
University of Rochester
299 classificações
Na lição
Week 1 - Introduction RC & Introduction to Electrical Circuits MB
Introduction to oscillations and sound waves, simple oscillating systems, sound pressure, sound waves, the speed of sound, wavelength, frequency and pitch, sound pressure level, loudness, making sound, properties of musical sound versus “noise”. Electronics fundamentals - charge, current, voltage, power, resistance, Ohm’s law, DC circuits, finding currents and voltages in simple circuits.
Conheça os instrutores
Robert ClarkProvost and Senior Vice President for Research and Professor
Mechanical Engineering
Mark BockoDistinguished Professor and Chair
Electrical and Computer Engineering
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
length plus 1.5 times the radius. And I'm not going to go into the
derivation of that you can take my word for it at this point in time.
But the reason is is that, there is an effective mass loading of that, if you
will, plug of air in the neck due to the surrounding air.
So, in some way, some, in, some distance proximate to that opening of the bottle.
Some of the air loads or, or, or, or adds mass to the plug of the air in the in the
neck, itself. but basically what you see is, is you
have an area, S, a length, L, that gives you a volume cubed.
And since density is kilograms per meter cubed, the units on that in the end are
kilograms, okay? And that gives us a mass.
Then we have a change in pressure, in the bottle, the mass, the mass itself in the
bottle is fixed, and the volume changes with the oscillation of the mass in the
neck of the bottle. So, you can think about that as a, as a
piston that's, that's, that's in the bottle if we, you know, I'll try a sketch
for just a second. But if I think about my neck of the
bottle here, I'll make it simple and the volume of the bottle here.
Then, if I, were to imagine this as a solid, mass oscillating up and down, you
can see as I, move down into the bottle, it's going to effectively reduce the
volume of the bottle. And as I, expand up in the direction, it
will increase the volume slightly. so the mass is fixed, and the volume
changes with the oscillation, which means the density changes, since, mass is,
density is the mass divided by the volume.
And if we explore that a little bit further, we can basically express the
change in density normalized by, the median density to.
a change of volume of the bottle versus the reference volume.
And that change, if we identify the displacement by this Greek symbol here,
theta, and multiply that by the surface area that gives us our delta, our small
change in volume. Normalized by the volume, so this is,
this is our displacement of the piston, if you will, that, that plug of mass of
air in the neck of the bottle that's moving.
And if we do some substitutions, we can actually compute the pressure in the, in
the bottle. And through the expression shown here.
Which is our density, our speed of sound squared and air, the surface area of the
opening of the neck. The volume of the bottle and then of
course the displacement. Alright?
Alright, so let's let's let F here represent a force and we know force is
the product of pressure over any area, so, if this is pressure, this is surface
area. Remember, pressure is force divided by
area so if we multiply it by area we're left with units of force.
that would be in newtons for the metric system, so, if I were to, apply a force,
a fixed force to this mass, the mass would move down a little bit, right?
And then we would compress our spring a little bit,
And, if it were a static force, we could compute static equilibrium here, based
upon this displacement that we impose by the force.
And that leads us to something known as a spring constant.
Which is the, is defines the stiffness of the spring here and the spring constant
is related to the force and the displacement.
But remember we could actually express the force in terms of the pressure and
the surface, area, and so, we can develop the expressions that we see here.
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.
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