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 c T 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 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 constructive interference f c Another example of reflecting in phase a sine wave that fits in a closed tube wavelength 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 phase shift at open end A sine wave that fits in a tube that is open at one end and closed at the other 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 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 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 2500 Hz 1250 Hz 400 Hz