Physics 1251 The Science and Technology of Musical

Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 6 Helmholtz Resonators 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) f = 0. 1592√(50. 0/1. 00) f = 0. 1592√(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 Seat #123 Session #1

Physics 1251 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. The natural frequency of a Helmholtz Resonator is given by the formula: f = [v/(2π)]√[A/ (V L)] A: area of neck v: 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.

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 • 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 105 Pa. Δp = (ΔV/V) B = 0. 50 (1. 41 x 105 ) = 70 k. Pa.

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Air has “Springiness” ΔV V ΔV/V: Force: F = A B ( ΔV/V) = - (A 2 B/V) x 0 0 ΔV 0. 33 20. N 0. 50 30. N

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

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

Physics 1251 Unit 1 Session 6 Helmholtz Resonators and Vibration Modes Two 500 ml Flasks • Same Volume • Same Length of neck • Different diameter ←Smaller Larger → diameter Same frequency? 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 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 Two Masses on Two Coupled Springs Spring ———→ Mass ————→ 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 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. • Degeneracy means two or more normal modes have the same frequency.