Introduction to Acoustics

Sound • Vibration of air molecules. • Sound is a vibration of air molecules that can cause the ear drum to vibrate, producing an auditory sensation. • To analyze sound, we need to characterize patterns of vibration. • Most basic type of vibration is simple harmonic motion (SHM) • Period and frequency • Amplitude I The frequency here is 1 cycle per second per second (Hz) I Vibration that slow would not cause a sensation of sound

Frequency and Amplitude

Frequency: How many times per second the ear drum vibrates in • and out, and is measured in Hz (number of round trips per second). Period is the amount of time required for each round trip, and for speech is usually measured in milliseconds (1/1000 of a second). • Amplitude: The extent to which the ear goes in and out, i.e. how far in Visualizingand how far sound: out, Waveform is the amplitude or energy of vibration. Usually measured in decibels in speech (db).

Eardrum

B A C

Cycle C

A (from rest) Distance

Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 4 / 21

Tuesday, January 21, 14 A more realistic example: 100 Hz

Visualizing sound: Waveform • 100 cycles in one second • How many seconds does each cycle take? I A more realistic example: 100 Hz I •100 cycles1 sec in= one1000 second milliseconds I How many seconds does each cycle take? How many milliseconds does each cycle take? I •1 sec = 1000 milliseconds

I How many milliseconds does each cycle take?

Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 6 / 21 VisualizingVisualizing sound: Waveform sound: Waveform

I 500 Hz = 500 cycles in one second I How many milliseconds in each cycle?

Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 7 / 21 Period and amplitude are unrelated in SHM • Period • determined by the physical properties of the vibrating system • length of pendulum bob • stiffness of spring • size, stiffness of tuning fork tine (hair cells in ear) • Amplitude • depends on initial conditions: how the object was set into motion • how far from equilibrium position it it displaced • how hard it is pushed Tuning forks • Period (frequency) is determined by the size and stiffness of the fork. • Frequency is perceived as pitch. • Example • Amplitude is determined by how hard you strike it to set it into motion. • Amplitude perceived as loudness. • Example Complex Vibrations • Very few of the sounds in the world exhibit SHM. These are pure tones . • How do we characterize other patterns? • Fundamental Frequency (f0) • 1/period of repetition of complex pattern • example of complex waves with different f0 • Fundamental frequency in speech: Each fundamental period corresponds to one open and closing of the vocal folds • The fundamental frequency is a property of the sound source. • Fourier Analysis (spectrum) Fourier Analysis (spectrum) • Any pattern of vibration can be analyzed as the sum of SHMs each with its characteristic amplitude and frequency. • For a periodic sound (with an observable fundamental frequency), simple harmonic (pure tone) components always occur at integer multiples of the fundamental frequency, and are referred to as harmonics . • 2nd harmonic frequency = 2f0. 3rd harmonic frequency = 3f0. • example of complex waves with same f0, different harmonic amplitudes • Harmonic amplitudes are a properties of the sound filtering produced by the vocal tract. Complex Sounds: Spectrum

• Complex periodic sounds can be represented as the sum of a number of pure tone components.

• The pure tone components are found at integer multiples of the fundamental frequency of the sound and are called harmonics.

• The harmonic content of a complex sound can be represented in a graph called a spectrum. • The horizontal axis of a spectrum corresponds to frequency. • The vertical axis corresponds to amplitude of the harmonic component in the complex wave. Auditory system: performs Fourier Analysis on Basilar membrane Two simultaneous vibrations: addition Two simultaneous vibrations: waveform

10 ms

1

300 Hz 0

-0.99981 0 0.013 Time (s)

2100 Hz 0

-0.9894 10 0.013 Time (s)

300 Hz + 0 2100 Hz

-0.9946 0 0.013 Time (s) Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 13 / 21 Two simultaneous vibrations: spectrum Two simultaneous vibrations: spectrum

100

90

80

70

60

50

Amplitude in db 40

30

20

10

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency in Hz

Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 14 / 21 Two simultaneous vibrations: spectrogram Spectrogram of 300 Hz + 2100 Hz

Linguistics 285 (USC Linguistics) Lecture 4: More Acoustics September 4, 2015 4 / 19 amp(2100) = amp (300) amp(2100) > amp (300)

1 1

0 0

-0.9946 -0.9911 0 0.013 0 0.013 Time (s) Time (s)

100 140

90

120 80

70 100

60

50 80

Amplitude in db 40 Amplitude in db

60 30

20 40

10

0 20 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency in Hz Frequency in Hz Formant Frequencies and Vowels • Vowels can be distinguished by the regions of the spectrum that have high amplitude. • These regions are called formants, and they are lawfully related to the shape of the supralaryngeal vocal tract and reflect its filtering action.

/i/ /ɛ/ 315 2080 3275 Hz 500 1800 2500 Hz

F0=100 VisualizationsVisualizations of sound of sound

Type time frequency amplitude Example pos Waveform x-axis y-axis 2 4 6 8 10 ms

Spectrum x-axis y-axis

5000

4000 ) z H (

y c n

darkness, e u q

e 2000 r Spectrogram x-axis y-axis F color Frequency(Hz) 500

0 0 0.7 0 250 Time (s) 500 ms

Louis Goldstein (USC Linguistics) Ling 285: Lecture 3: Intro to Acoustics September 1, 2015 9 / 21