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

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Foolscap Quiz: What is the frequency of a simple harmonic oscillator that has a spring constant of k = 50.0 N/m and a mass m of 1.00 kg? Frequency = f = 1/(2π)√(K/m) Frequency = f = 1/(2π)√(K/m) f = √(50.0/1.00) f = √(50.0) = 1.13 Hz P =1/f = 1/1.13 Hz = 0.89 sec

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Put seat number on the Foolscap. Do you wish to sit here “permanently?” Joe College 1/14/02 Session #1 Seat #123

Physics 1251 Unit 1 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes 1′ Lecture: A Helmholtz resonator is a simple harmonic oscillator where the mass is provided by the air in a narrow neck while the spring is provided by a volume of trapped air. A Helmholtz resonator is a simple harmonic oscillator where the mass is provided by the air in a narrow neck while the spring is provided by a volume of trapped air. The natural frequency of a Helmholtz Resonator is given by the formula: The natural frequency of a Helmholtz Resonator is given by the formula: f = [v/(2π)]√[A/ (V L)] A: area of neckv: velocity of sound in air V: volume of Bottle L: length of neck

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes 1′ Lecture (cont’d.): When an object has n masses and n springs, there are n degrees of freedom and n modes of oscillation. Often each mode has a different frequency; occasionally some frequencies are the same. When an object has n masses and n springs, there are n degrees of freedom and n modes of oscillation. Often each mode has a different frequency; occasionally some frequencies are the same.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Does Air have mass and weight? How much? Density = ρ = mass/volume

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Density of Air Density = ρ = Mass/Volume Density = ρ = Mass/Volume ρ = 1. 2 kg/ m 3 ρ = 1. 2 kg/ m 3

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes The “Bulk Modulus” B is the springiness of a gas. B is equal to the change in pressure (in Pa) for a fractional change in volume. B = Δp / (ΔV/V) What is the increase in pressure if I decrease the volume of trapped gas by 50%? B = 1.41 x 10 5 Pa. Δp = (ΔV/V) B = 0.50 (1.41 x10 5 ) = 70 kPa. Δp = (ΔV/V) B = 0.50 (1.41 x10 5 ) = 70 kPa.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Air has “Springiness” N 30. N 0ΔV/V:Force: ΔVΔVΔVΔV ΔVΔVΔVΔVV F = A B ( ΔV/V) = - (A 2 B/V) x

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Largest Volume Lowest Frequency Highest Frequency Smallest Volume k ∝ 1/V so f ∝ 1/√V k ∝ 1/V so f ∝ 1/√V

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Simple Harmonic Motion of Air Air “spring” → Air “mass” → ↕ ⃕ ⃔ ⃕ ⃔ ⃔ ⃕ ⃔ ⃕ Oscillation of air mass Turbulence

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Two 500 ml Flasks Same Volume Same Volume Same Length of neck Same Length of neck Different diameter Different diameter Same frequency? Larger → diameter ←Smaller diameter f = 1/(2π)√[k/m] f = 1/(2π)√[(A 2 B/V) / (ALρ)] v= √ B/ρ f = v/(2π)√[A/ (V L)]

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Helmholtz Resonator Ocarina Ocarina Open holes increase area of “neck.”

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Application of Helmholtz Resonator: Ported Speaker Cabinet Air “Spring” Air “mass”

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Normal or Natural Modes of Oscillation

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Spring ———→ Mass ————→ Two Masses on Two Coupled Springs Spring ————→ Mass ————→ Mode 1 Mode 2

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes 80/20 A Simple Harmonic Oscillator has only one Normal or Natural Mode of Oscillation and only one frequency of oscillation.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes 80/20 The number of Normal or Natural Modes of Oscillation is equal to the number of simple harmonic oscillators that are coupled together.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes 80/20 Two Normal or Natural Modes of Oscillation are called “degenerate” if they have the same frequency.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Mode Summary: A Helmholtz Oscillator is a SHO comprised of an enclosed air volume and a narrow neck and has a single frequency. A Helmholtz Oscillator is a SHO comprised of an enclosed air volume and a narrow neck and has a single frequency. A normal or natural mode of vibration or oscillation is one of the fundamental ways that a device can move. A normal or natural mode of vibration or oscillation is one of the fundamental ways that a device can move. The number of modes is equal to the number of simple harmonic oscillators in the system. The number of modes is equal to the number of simple harmonic oscillators in the system. Degeneracy means two or more normal modes have the same frequency. Degeneracy means two or more normal modes have the same frequency.