Since last midterm: 1. the decibel scale 2. resonances 3. normal vibration modes (standing waves) strings strings tubes tubes 3. human hearing

The decibel scale (a way of measuring loudness) Intensity (I) ~ (pressure difference) 2 W/m 2 = J/m 2 s minimum audible sound

Logarithms Powers of 10 The only problem are the …

log ab = log a + log b log 1/a = -log a

NO CALCULATORS DURING THE MIDTERM

The point of using the decibel scale is the Weber-Fetchner “Law” A doubling of volume feels like the same increase, regardless of how much increase in intensity actually occurred.

Thresholds for hearing and pain

Weakest sound heardWeakest sound heard 0dB0dB Normal conversation (3-5')Normal conversation (3-5') 60-70dB60-70dB Telephone dial toneTelephone dial tone 80dB80dB City Traffic (inside car)City Traffic (inside car) 85dB85dB Train whistle at 500'Train whistle at 500' 90dB90dB Subway train at 200'Subway train at 200' 95dB95dB Level at which sustained exposure may result in hearing lossLevel at which sustained exposure may result in hearing loss dB dB Power mowerPower mower 107dB107dB Power sawPower saw 110dB110dB Pain beginsPain begins 125dB125dB Pneumatic riveter at 4'Pneumatic riveter at 4' 125dB125dB Jet engine at 100'Jet engine at 100' 140dB140dB Death of hearing tissueDeath of hearing tissue 180dB180dB Loudest sound possibleLoudest sound possible 194dB194dB Environmental Noise Weakest sound heard 0dB Normal conversation (3-5') 60-70dB Normal piano 60-70dB Telephone dial tone 80dB City Traffic (inside car) 85dB Walkman on 5/10 94dB Subway train at 200‘ 95dB Level at which sustained exposure may result in hearing loss dB Power mower 107dB Symphonic music peak dB Pain begins 125dB Jet engine at 100‘ 140dB Rock concert peak 150dB Death of hearing tissue 180dB

Normal modes of vibration, standing waves If you bang on an object, it will vibrate in a complicated way. But this complicated motion is a superposition of NORMAL MODES (just like a complicated sound can be decomposed into simple sine waves).

fundamental 2 nd harmonic 3 rd harmonic 4 th harmonic Normal modes of strings (standing waves) Animation courtesy of Dr. Dan Russell, Kettering University

Standing waves are a superposition of two counter moving waves Animation courtesy of Dr. Dan Russell, Kettering University

l 1 = 2 L f 1 = v/ l 1 = v/(2L) l 2 = L f 2 = v/ l 2 = v/L=2 f 1 … L Frequencies of standing modes of a string

How the velocity depends on the string: Mersenne’s laws fundamental frequency tension mass per length length

Other objects have their normal modes too: Square membrane:

Circular membrane:

Bottle of beer:

This is not a string now, it’s the graph of the pressure x distance Standing sound waves in air tubes

v string v sound nodes at the ends nodes or antinodes at the ends air tubes x strings

closed end open end pressure displacement

l /4

Voice

Anatomy

The sound wave produces by the vocal chords contains many frequencies that may or may not be enhanced by the resonances (formants) of the vocal tract 6dB/octave

Formants stay fixed as pitch changes

1. vocal chords vibrate with a given frequency 2. formants enhance some of the overtones (harmonics) 3. different formats = different vowels 4. consonants are formed with non-steady changes in lips, tongue, …

Hearing 1. Physiology 2. Place Theory 3. Psychophysics of hearing Fundamental tracking Fundamental tracking Aural harmonics Aural harmonics Sheppard tones and pitch perception Sheppard tones and pitch perception 4. Sound localization interaural level difference interaural level difference interaural time differenceinteraural time difference head-transfer function head-transfer function

55 mm mm 2

Uncoiled cochlea (schematic) stiffer limber

Cross section of cochlea

Two frequencies f and 2f (one octave) 3.5 mm “same” interval corresponds to the same frequency ratio (fixed distance along the cochlea)

distance along the basilar membrane excited hair cells sharpening The amount of sharpening determined the just noticeable difference in frequencies

frequency up and down by = 0.1% frequency up and down by = 0.5%

note D note D minus fundamental note D minus fundamental and 2 nd harmonic Fundamental tracking: the absence of the fundamental does not change the perceived pitch

Aural harmonics sin(2 p 50 t) sin(2 p 50 t)+ 0.2 sin(2 p 100 t) +0.1 sin(2 p 150 t) +… extra frequencies “aural harmonics” 400Hz, 400Hz+802Hz, 400Hz+1202Hz

Shepard tones

Sound localization How do we know where the sound is coming from ? interaural level differences (ILD) interaural level differences (ILD) interaural time differences (ITD) interaural time differences (ITD) head-related transfer function (HRTF) head-related transfer function (HRTF)

Interaural level difference: one ear will be on the shadow cast by the head diffraction makes it ineffective at low frequencies we can detect even 0.5 dB in ILD

Interaural time difference: peaks and through will arrive at ears at different times t ~ L/v ~ (0.15 m)/(340m/s) ~ s difference in arrival time distance between ears much shorter than synaptic delays !

Phase ambiguity:  =10 cm, f=340 m/s /0.2 m = 1700 Hz distance between ears

300 Hz: 2000 Hz:

Head-related transfer function: includes the reflection, refraction and diffraction from ears, chest, head, …