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Resonance in a Closed Tube Constant Frequency, Changing Length Movement of Air • Net displacement of molecules is zero. • Amplitude at resonance points is a relative maximum, because the sound is loudest. • Constant frequency, b/c the pitch doesn’t change. – Constant velocity, b/c constant T. – Constant wavelength, b/c v = f λ Molecular Movement Represented: Molecular Movement Represented: Molecular Movement Represented: Distance between resonance points: Constant for same frequency. Decreases as f increases. Node to node. Δx = ½ λ Molecular Movement Represented: First resonance point: ≈ Half of difference ( ½ Δx). Decreases as f increases. (Duh!) Antinode to node. xinitial ≈ ¼ λ … End correction! Tube Length vs. Wavelength: L≈¼λ L≈¾λ L ≈ 5/4 λ Tube Length vs. Wavelength: L = 1/4 λ, 3/4 λ, 5/4 λ, …odd/4 λ L≈¼λ L≈¾λ L ≈ 5/4 λ Calculating Wavelength: L≈¼λ λ≈ 4L Δx = ½ λ λ = 2 Δx λ ≈ 4/3 L λ ≈ 4/5 L Frequency, Wavelength, and Speed of ANY wave, including Sound: v=fλ Know two, find the third! Wavelength measured: λ = 2 Δx Speed calculated: v = 331.5 + 0.607 T Frequency: Measured or calculated Open Tube Using the analysis of a closed tube as a guide, determine the lengths of an open tube that will resonate. Answer: last page. Example Data: Closed Tube Resonant Lengths (cm): • 466.1 Hz: 23.0, 60.2, 95.8, 132.9 • 500.0 Hz: 19.0, 53.1, 85.8, 121.0, 154.7 • 1000. Hz: 8.7, 23.9, 42.2, 58.7, 74.6, 92.9, 109.5, 126.3, 143.1, 158 Example Data: Calculations Average separation of resonant points: • 466.1 Hz: 36.6 cm • 500.0 Hz: 33.9 cm • 1000. Hz: 16.6 cm Observations / Conclusions There appears to be an inverse relationship between frequency and separation distance. f α 1 / Δx , “f is inversely proportional to Δx.” f = k / Δx , where k is the proportionality constant [slope is the proportionality constant in a a direct relationship]. To prove inverse relationship: •Calculate k for each frequency. •Are the values of k equal? Open Tube: Open Tube: Resonant Lengths Open Tube: Resonant Lengths L = ½ λ, 2/2 λ, 3/2 λ L = 2/4 λ, 4/4 λ, 6/4 λ, …even/4 λ