Music 171: Sinusoids. Tamara Smyth, Department of Music, University of California, San Diego (UCSD) January 10, 2019

Transcription

1 Music 7: Sinusoids Tamara Smyth, Department of Music, University of California, San Diego (UCSD) January 0, 209

2 What is Sound? The word sound is used to describe both:. an auditory sensation in the ear 2. the disturbance in a medium that causes an auditory sensation Nearly all objects will vibrate when disturbed. Sound is the result of a wave, created by vibrating objects: alternating regions of low and high pressure that propagate through a medium from one location to another. Music 7: Sinusoids 2

3 Sound Wave Sound is a mechanical wave: it requires a medium in which to propagate. Sound is a traveling wave: it propagates through a medium from one location to another (with negligible change in shape). In fluids (e.g. air, water), sound is a longitudinal compression wave: medium particle displacement is parallel to the direction of wave propagation. The waveform illustrates the variation (either temporal or spatial) of the wave. Sound waveforms show change/variations in pressure. Music 7: Sinusoids 3

4 Waveform: wavelength The wavelength is the length of one cycle of the spatial waveform: distance between crests or between troughs Waves are often characterized by their length: infrasonic: lower than lowest audible frequency. ultrasonic: higher than highers audible frequency. Music 7: Sinusoids 4

5 Waveform: period Waves may also be characterized by their period. The period is the time to complete one cycle of the temporal wavdeform:.5 Period Time (s) If we know wave velocity, we can determine the period from the wavelength: The frequency is: period = wavelength velocity the number of cycles per second (Hz) inverse of the period frequency = period = velocity wavelength Music 7: Sinusoids 5

6 Waveform The waveform of the sound shows the time evolution of the pressure variations, illustrating: amplitude: maximum particle displacement from rest position (Pa or N/m 2 ). period: time to complete one cycle (s). frequency: number of cycles per second (Hz). wavelength: length of one complete cylce (m) (need wave speed to compute). Music 7: Sinusoids 6

7 Properties of Sound Waves Speed of sound : in air: 340 m/s in water: 480 m/s Amplitude range of hearing (humans) Threshold of audibility: N/m 2 Threshold of feeling (or pain!): 200 N/m 2 Frequency range of hearing humans: Hz dogs: Hz beluga whale: Hz Period of lowest and highest audible frequencies /20 Hz = 0.05 s / Hz = 0.05 ms Shortest human audible wave (air) 340/20000=.7cm Longest human audible wave (air) 340/20=7m Quantity depends on temperature: For air, the speed of sound is c = 20. T, where T is the absolute temperature found by adding 273 to the temperature on the Celsius scale. Music 7: Sinusoids 7

8 Why Sinusoids are Important Sinusoids are fundamental in physics: many systems oscillate in a quasi-sinusoidal motion known as simple harmonic motion: repetitive movment through an equilibrium position maximum displacement on either side of equilibriums is equal (if no losses). Bandlimited sound (e.g. digital audio) made be viewed as the sum of a finite number of sinusoids having a different amplitudes, frequencies, and phases. Music 7: Sinusoids 8

9 Sinusoids Sinusoids is a collective term referring to both sine and cosine functions. A sinusoid is a function having the following form: x(t) = Asin(ωt+φ) or x(t) = Acos(ωt+φ), where x is the quantity which varies over time and A peak amplitude ω radian frequency (rad/sec) = 2πf f frequency (Hz) t time (seconds) φ initial phase (radians) ωt+φ instantaneous phase (radians) 2 Amplitude Time (s) Figure : Sinusoid where A = 2, ω = 2π5, and φ = π/4. Music 7: Sinusoids 9

10 Cycle/Phase/Period One cycle of a sinusoid is 2π radians. The phase refers to the position within the cycle. π/2 π 3π/2 2π The initial phase is the phase at time = 0 and is dependent on whether the sinusoid is expressed with a sin or cos function. Since sinusoids are periodic with period 2π, φ = φ±2π Thus, the range of φ may be restricted to 2π: or π < φ < π, 0 < φ < 2π. The period T of a sinusoid is the time (in seconds) it takes to complete one cycle (2π radians). Music 7: Sinusoids 0

11 Initial Phase The initial phase φ (also called phase offset or phase shift), given in radians, tells us the position of the waveform cycle at t = 0. Sine function with φ = 0 (sin(ω 0 t+0)): 2 Amplitude Time (s) Sine function with φ = π/2 (sin(ω 0 t+π/2)): 2 Amplitude Time (s)... or equivalently cos(ω 0 t+0). Music 7: Sinusoids

12 Frequency (Hz vs. Radians/second The frequency f of the waveform is given in cycles per second or Hertz (Hz). Frequency is equivalent to the inverse of the period T of the waveform, f = /T Hz. The radian frequency ω, given in radians per second, is equivalent to the frequency in Hertz scaled by 2π, ω = 2πf (rad/sec). What is the frequency in Hz of the following sinusoid? amplitude time (s) What is the frequency in rad/sec? Music 7: Sinusoids 2

13 Sine and Cosine Functions The sine and cosine function are very closely related and can be made equivalent simply by adjusting their initial phase: sinθ = cos(θ π 2 ) or cosθ = sin(θ+ π 2 ). Amplitude Time (s) Figure 2: Phase relationship between cosine (solid blue line) and sine (broked green line) functions. Music 7: Sinusoids 3

14 Amplitude Envelopes In his work, On the Sensations of Tone, Hermann von Helmholtz characterized tones by the way in which their amplitudes evolved over time, that is, by their amplitude envelope. He described the envelope as having three parts:. the attack: the time it takes the sound to rise to its peak 2. the sustain: the steady state portion of the sound (where the amplitude has negligiable chane) 3. the decay: the time it take for the sound to decay or fade out. Music 7: Sinusoids 4

15 Amplitude Amplitude Amplitude Time (s) Attack Steady State Decay Time (s) Time (s) Figure 3: A sinusoid with an amplitude envelope. Music 7: Sinusoids 5

16 ADSR Envelope The duration of the attack and decay greatly influence the quality of a tone: wind instruments tend to have long attacks, while percussion instruments tend to have short attacks. Another envelope, called ADSR, has a fourth segment inserted between the attack and the sustain. This envelope attempts to mimic the envelopes found in acoustic instruments Figure 4: An ADSR envelope. Music 7: Sinusoids 6

17 Power and Intensity The sound waveform illustrates the time evolution of pressure variations. Related to pressure is:. sound power emitted by the source: a fixed quantity (measured in watts, W), analogous to the wattage rating of a light bulb 2. sound intensity a distance from the source: power per unit area carried by wave (W/m 2 ), influenced by interference from environment and other sources, akin to brightness of a light at a point in room. Music 7: Sinusoids 7

18 Sound Intensity Sound intensity is a measure of the power of a sound that actually contacts an area, such as the eardrum. The range of human hearing is threshold of audibility (I 0 ): 0 2 W/m 2 threshold of feeling: W/m 2 Intensity is related to pressure squared. Music 7: Sinusoids 8

19 Linear vs. Logarithmic Scales On a linear scale, a change between two values is perceived on the basis of their difference e.g., a change from to 2 would be perceived as having the same increase as from 4 to 5. On a logarithmic scale, a change between two values is perceived on the basis of their ratio e.g., a change from to 2 (ratio of :2) would be perceived as having the same increase as a change from 4 to 8 (also a ratio of :2). Linear Logarithmic ,000 Figure 5: Moving one unit to the right increments by on the linear scale and multiplies by a factor of 0 on the logarithmic scale. Music 7: Sinusoids 9

20 Decibels (db) Human hearing: better measured on a logarithmic scale amplitude is often expressed in decibels (db). The Bel is a logarithmic scale, named after telecommunications pioneer, Alexander Graham Bell. used to compare two quantities: the difference between A and B is defined as The decibel is log 0 (A/B) Bels. more convenient for expressing relative amplitudes of sound (or, for instance, the power gain of an amplifier). defined as one tenth of a Bel: B = 0 db. to convert from Bel to db, multiply by 0. Music 7: Sinusoids 20

21 Intensity Difference in db The difference/change between two signals with intensities I and I 2 may be expressed in decibels as: L = 0logI 2 /I db. When decibels are used as absolute measurements, there is an implied fixed reference, e.g. the threshold of audibility: where I 0 = 0 2 W/m 2. L I = 0logI/I 0 db, Music 7: Sinusoids 2

22 Sound pressure Level Intensity is proportional to sound pressure amplitude squared. The sound pressure level L p (SPL) is equivalent to sound intensity level and is expressed in db by: L p = 0logI/I 0 = 0log ( p/(2 0 5 ) ) 2 = 20logp/(2 0 5 ) = 20logp/p 0. where p 0 = is the threshold of hearing for amplitude of pressure variations. Music 7: Sinusoids 22