Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 6 Helmholtz Resonators and...

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Unit 1Unit 1

Session 6 Session 6

Helmholtz ResonatorsHelmholtz Resonators

and Vibration Modesand Vibration Modes

Unit 1Unit 1

Session 6 Session 6

Helmholtz ResonatorsHelmholtz Resonators

and Vibration Modesand Vibration Modes

Physics 1251 Physics 1251 Unit 1 Session 6Unit 1 Session 6Helmholtz Resonators and Vibration ModesHelmholtz Resonators and Vibration Modes

Foolscap Quiz: Foolscap Quiz: What is the frequency of a simple What is the frequency of a simple harmonic oscillator that has a harmonic oscillator that has a spring constant of k = 50.0 N/m spring constant of k = 50.0 N/m and a mass m of 1.00 kg?and a mass m of 1.00 kg?

Frequency = Frequency = f = 1/(2f = 1/(2π)π)√(K/m)√(K/m)

f = 0.1592f = 0.1592√(50.0/1.00)√(50.0/1.00)

f = 0.1592f = 0.1592√(50.0) = 1.13 Hz√(50.0) = 1.13 Hz

P =1/f = 1/1.13 Hz = 0.89 secP =1/f = 1/1.13 Hz = 0.89 sec

Physics 1251 Physics 1251 Unit 1 Session 6 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Helmholtz Resonators and Vibration Modes

Put seat number on the Foolscap. Put seat number on the Foolscap.

Do you wish to sit here “permanently?”Do you wish to sit here “permanently?”

Joe College

1/14/02 Session #1

Seat #123

Physics 1251 Physics 1251 Unit 1 Unit 1 Session 6 Unit 1 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Helmholtz Resonators and Vibration Modes

11′ Lecture:′ Lecture:• A A Helmholtz resonatorHelmholtz resonator is a simple is a simple

harmonic oscillator where the mass harmonic oscillator where the mass is provided by the air in a narrow is provided by the air in a narrow neck while the spring is provided by neck while the spring is provided by a volume of trapped air.a volume of trapped air.

• The natural frequency of a Helmholtz The natural frequency of a Helmholtz Resonator is given by the formula:Resonator is given by the formula:

f = [v/(2f = [v/(2π)]π)]√[A/ (V L)]√[A/ (V L)]

A: area of neckA: area of neck v: velocity of sound in air v: velocity of sound in air

V: volume of Bottle V: volume of Bottle L: length of neckL: length of neck


11′ Lecture (cont’d.):′ Lecture (cont’d.):• When an object has n masses and n When an object has n masses and n

springs, there are springs, there are n degrees of n degrees of freedomfreedom and and n modes of oscillationn modes of oscillation. .

Often each mode has a different Often each mode has a different frequency; occasionally some frequency; occasionally some frequencies are the same.frequencies are the same.


Does Air have mass and weight?Does Air have mass and weight?

How much?How much?

Density = Density = ρ = ρ = mass/volumemass/volume


Density of AirDensity of Air

• Density = Density = ρ =ρ =

Mass/VolumeMass/Volume

• ρ = 1. 2 kg/ ρ = 1. 2 kg/ mm33


The “Bulk Modulus” B is the The “Bulk Modulus” B is the springiness of a gas.springiness of a gas.

B is equal to the change in pressure B is equal to the change in pressure (in Pa) for a fractional change in (in Pa) for a fractional change in volume.volume.

B = B = Δp / (ΔV/V)Δp / (ΔV/V)

What is the increase in pressure if I What is the increase in pressure if I decrease the volume of trapped gas by decrease the volume of trapped gas by 50%? B = 1.41 x 1050%? B = 1.41 x 1055 Pa. Pa.

ΔpΔp = = (ΔV/V) B = 0.50 (1.41 x10(ΔV/V) B = 0.50 (1.41 x1055 ) = ) = 70 kPa .70 kPa .


Air has Air has “Springiness”“Springiness”

00 0.330.33 0.500.5020. N20. N 30. N30. N00

ΔV/V: ΔV/V: Force:Force:

ΔΔVV ΔΔVVVV

F = A B ( F = A B ( ΔΔV/V) = - (AV/V) = - (A22 B/V) x B/V) x


Largest Largest VolumeVolume

Lowest Frequency

Highest Frequency

Smallest Smallest VolumeVolume

k k ∝ 1/V ∝ 1/V so so f f ∝ 1/√V∝ 1/√V


Simple Harmonic Simple Harmonic Motion of Air Motion of Air

Air “spring” Air “spring” →→

Air “mass” Air “mass” →→↕↕

Oscillation of Oscillation of

air massair mass

TurbulenceTurbulence


Two 500 ml FlasksTwo 500 ml Flasks

• Same VolumeSame Volume

• Same Length of Same Length of neckneck

• Different Different diameterdiameter

Same frequency?Same frequency?Larger Larger →→diametediameterr

←←SmalleSmallerrdiametediameterr

f = 1/(2f = 1/(2π)π)√[k/m]√[k/m]f = 1/(2f = 1/(2π)π)√[(A√[(A22B/V) / (ALB/V) / (ALρρ)])]v= √ B/v= √ B/ρρ

f = v/(2f = v/(2π)π)√[A/ (V L)]√[A/ (V L)]


Helmholtz ResonatorHelmholtz Resonator

• OcarinaOcarina

Open holes increase area of Open holes increase area of “neck.”“neck.”

http://www.ocarina.co.uk/


Application of Helmholtz Resonator:Application of Helmholtz Resonator:

Ported Speaker CabinetPorted Speaker Cabinet

Air “Spring”Air “Spring”

Air “mass”Air “mass”


Normal or Natural Modes of OscillationNormal or Natural Modes of Oscillation


Spring Spring ——————→→

Mass Mass ————→————→

Two Masses on Two Coupled Two Masses on Two Coupled SpringsSprings

Spring Spring ————→————→

Mass Mass ————→————→

Mode 1Mode 1 Mode 2Mode 2


80/2080/20A Simple Harmonic Oscillator has only A Simple Harmonic Oscillator has only one Normal or Natural Mode of Oscillation one Normal or Natural Mode of Oscillation

and only one frequency of oscillation.and only one frequency of oscillation.


80/2080/20The number of Normal or Natural The number of Normal or Natural Modes of Oscillation is equal to the Modes of Oscillation is equal to the

number of simple harmonic oscillators number of simple harmonic oscillators that are coupled together.that are coupled together.


80/2080/20Two Normal or Natural Modes of Two Normal or Natural Modes of Oscillation are called Oscillation are called ““degeneratedegenerate”” if if

they have the same frequency.they have the same frequency.

Physics 1251 Physics 1251 Unit 1 Session 6 Unit 1 Session 6 Helmholtz Resonators and Vibration Mode Helmholtz Resonators and Vibration Mode

Summary:Summary:• A Helmholtz Oscillator is a SHO comprised A Helmholtz Oscillator is a SHO comprised

of an enclosed air volume and a narrow of an enclosed air volume and a narrow neck and has a single frequency.neck and has a single frequency.

• A normal or natural mode of vibration or A normal or natural mode of vibration or oscillation is one of the fundamental oscillation is one of the fundamental ways that a device can move.ways that a device can move.

• The number of modes is equal to the The number of modes is equal to the number of simple harmonic oscillators in number of simple harmonic oscillators in the system.the system.

• Degeneracy means two or more normal Degeneracy means two or more normal modes have the same frequency.modes have the same frequency.

Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 6 Helmholtz Resonators and...

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