Resonance in acoustic tubes 1. wavelength 2. plain wave propagation 3. reflection 4. phase matching Two ways to measure the period of a sine wave. Time Frequency = 1 / Time Sound: pressure fluctuation that travels through space. Speed of sound = 35,000 cm/s Space Wavelength = spatial period Wavelength = speed of sound * period duration λ=c*T λ=c/f because f = 1 / T Space Wavelength = spatial period 1. wavelength. Sine wave has a spatial period, peaks and valleys located in space. Space sound propagates from source in a sphere. However, sound in a tube propagates in a plane – effectively, no curvature However, sound in a tube propagates in a plane – effectively, no curvature 2. Plane wave propagation 3. Reflection Sound reflects off of surfaces - more reflection off of hard surfaces - less reflection off of soft surfaces - scattered reflection off of uneven surfaces hard soft uneven Sound traveling in a closed tube reflects off the ends of the tube. Sound traveling in an open tube also reflects off the ends of the tube. Reflection off of a soft surface The vocal tract is a tube that is open at one end and closed at the other. has two kinds of reflection: 1. hard surface at closed end 2. soft surface at open end Sound waves traveling though space interfere with each other. A direction direction B Destructive interference: A + B = 0 A direction direction B A+B Constructive interference: A + B = AB A direction direction B A+B Constructive interference: A + B = AB 1 A direction 1 direction B 2 A+B Constructive interference: A + B = AB A direction -1 direction B -1 A+B -2 Reflected waves in a tube interfere with each other. constructive interference = resonance destructive interference = nonresonance Q: What frequencies will resonate in a tube? = Q: What sine waves will show constructive interference? two factors wavelength and tube length key: wave must “fit” in tube fit = reflect in phase An example of reflecting “in phase” - a sine wave that “fits” in a closed tube wavelength = tube length An example of reflecting “in phase” - a sine wave that “fits” in a closed tube wavelength = tube length the reflected wave is in phase Δ constructive interference An example of reflecting “in phase” - a sine wave that “fits” in a closed tube wavelength = tube length the reflected wave is in phase Frequency of this resonance: Δ constructive interference f = c/λ Another example of reflecting “in phase” - a sine wave that “fits” in a closed tube wavelength = ½ * tube length A general formula for calculating the resonant frequencies of sine waves that will resonate in a tube closed at both ends: Fn = nc/2L n = resonant frequency number (1,2,3, ...) c = speed of sound (35,000 cm/s) L = tube length (in cm) Now consider a tube that is open at one end, and closed at the other. Now consider a tube that is open at one end, and closed at the other. Reflection from the open end is different. Phase shift! A sine wave that “fits” in a tube that is open at one end, and closed at the other. A sine wave that “fits” in a tube that is open at one end, and closed at the other. phase shift at open end A sine wave that “fits” in a tube that is open at one end, and closed at the other. λ phase shift at open end A sine wave that “fits” in a tube that is open at one end, and closed at the other. λ phase shift at open end resonant frequency is: f = c/(4/5*L) Another sine wave that “fits” this tube. resonant frequency: f = c/(4/3L) λ A general formula for resonant frequencies of tubes open at one end and closed at the other: fn = (2n-1)c/4L n = resonance number (1,2,3...) c = speed of sound (35,000 cm/s) L = tube length (in cm) the vowel schwa [ә]: a tube open at one end (lips) and closed at the other (glottis) Vocal tract length: ~ 17.5 cm F1 = c/4L = 35,000/70 = 500 Hz F2 = 3c/4L = 1500 Hz F3 = 5c/4L = 2500 Hz Peter Ladefoged saying [ ә ]: 2500 Hz 1250 Hz 400 Hz

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