Resonance in acoustic tubes1. wavelength2. plain wave propagation3. reflection4. phase matchingTwo ways to measure the period of a sine wave.Time Frequency = 1 / TimeSound: pressure fluctuation that travels through space.Speed of sound = 35,000 cm/sSpaceWavelength = spatial periodWavelength = speed of sound * period durationλ = c * Tλ = c / f because f = 1 / TSpaceWavelength = spatial period1. wavelength. Sine wave has a spatial period, peaks and valleys located in space.Spacesound propagates from source in a sphere.However, sound in a tube propagates in a plane – effectively, no curvatureHowever, sound in a tube propagates in a plane – effectively, no curvature2. Plane wave propagation3. ReflectionSound reflects off of surfaces - more reflection off of hard surfaces - less reflection off of soft surfaces - scattered reflection off of uneven surfaceshard soft unevenSound 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 surfaceThe vocal tract is a tube that is openat one end and closed at the other.has two kinds of reflection:1. hard surface at closed end2. soft surface at open endSound waves traveling though spaceinterfere with each other.ABdirectiondirectionDestructive interference: A + B = 0ABdirectiondirectionA+BConstructive interference: A + B = ABABdirectiondirectionA+BConstructive interference: A + B = ABABdirectiondirectionA+B112Constructive interference: A + B = ABABdirectiondirectionA+B-1-1-2Reflected waves in a tube interfere with each other.constructive interference = resonancedestructive interference = nonresonanceQ: What frequencies will resonate in a tube?= Q: What sine waves will show constructive interference?two factors - wavelength and tube lengthkey: wave must “fit” in tubefit = reflect in phaseAn example of reflecting “in phase”- a sine wave that “fits” in a closed tubewavelength = tube lengthAn example of reflecting “in phase”- a sine wave that “fits” in a closed tubewavelength = tube lengththe reflectedwave is in phaseΔ constructive interferenceAn example of reflecting “in phase”- a sine wave that “fits” in a closed tubewavelength = tube lengththe reflectedwave is in phaseΔ constructive interferenceFrequency of this resonance:f = c/λAnother example of reflecting “in phase”- a sine wave that “fits” in a closed tubewavelength = ½ * tube lengthA general formula for calculating the resonant frequencies of sine waves that will resonate in a tube closed at both ends:Fn = nc/2Ln = 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.phase shift atopen endA 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 atopen endλA sine wave that “fits” in a tube that is open at one end, and closed at the other.phase shift atopen endλresonant frequency is: f = c/(4/5*L)Another sine wave that “fits” this tube.λresonant frequency: f = c/(4/3L)A general formula for resonant frequenciesof tubes open at one end and closed at the other:fn = (2n-1)c/4Ln = 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 cmF1 = c/4L = 35,000/70 = 500 HzF2 = 3c/4L = 1500 HzF3 = 5c/4L = 2500 HzPeter Ladefoged saying [ ә ]:2500 Hz1250 Hz 400
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