Lecture 2: Acoustics

Transcription

1 ELEN E4896 MUSIC SIGNAL PROCESSING Lecture 2: Acoustics 1. Acoustics, Sound & the Wave Equation 2. Musical Oscillations 3. The Digital Waveguide Dan Ellis Dept. Electrical Engineering, Columbia University E4896 Music Signal Processing (Dan Ellis) /16

2 1. Acoustics & Sound Acoustics is the study of physical waves Waves transfer energy without permanent displacement of matter Common math for different media gas, liquid, solid, EM Intuition: Pulse going down a rope E4896 Music Signal Processing (Dan Ellis) /16

3 The Wave Equation For 1-D medium with displacement y(x, t) : curvature c 2 2 y x 2 = 2 y t 2 simple to derive from freshman physics... acceleration y y(0,t) = m(t) x y + (x,t) y(l,t) = 0 E4896 Music Signal Processing (Dan Ellis) /16

4 Solution: The Wave Equation y(x, t) =y + (x sum of leftward-moving and rightward-moving traveling waves shape does not change (set by initial conditions) y c 2 c 2 y x 2 = c ct)+y (x + ct) y + E4896 Music Signal Processing (Dan Ellis) /16 2 y t 2 x y + y - y(x,t) = y + + y - y

5 Terminations & Reflections Boundary conditions include fixed points e.g. held ends of string Superposition of traveling waves must match constraints hence reflections y(x,t) = y + + y Any impedance change results in some reflection y + x = L y energy loss... E4896 Music Signal Processing (Dan Ellis) /16

6 1-D Waveguides Plucked/struck string guitar, piano... tension S mass/length ε Acoustic tube e.g. clarinet or trumpet vocal tract L ω 2 = π2 S L 2 ε pressure Solid bar xylophone,... x E4896 Music Signal Processing (Dan Ellis) /16

7 2. Musical Oscillations (Pseudo) periodic oscillation is central to musical pitch 4000 Frequency Time Musical instruments create pitch in different ways... E4896 Music Signal Processing (Dan Ellis) /16

8 Simple Harmonic Motion Basic 2nd order mechanical oscillation ẍ = 2 x x = A cos( t + ) resonance Spring + mass + damper e.g. tuning fork F = kx ζ m x ω 2 = k m Not great as a musical instrument only fundamental low amplitude E4896 Music Signal Processing (Dan Ellis) /16

9 Relaxation Oscillator Alternating states closed state collects energy from steady source open state releases energy, then reverts to closed e.g. Vocal folds p u Oscillation period depends on tension easy to adjust hard to keep stable E4896 Music Signal Processing (Dan Ellis) /16

10 Strings Canonical wave equation example tension S mass/length ε L ω 2 = π2 S L 2 ε guitar (plucked) piano (struck) violin (bowed...) Control of period vary length L (frets) vary tension S and/or length L (piano) Initial conditions = excitation E4896 Music Signal Processing (Dan Ellis) /16

11 Air Column Wave equation in air U 0 e jωt kx = π x = λ / 2 pressure = 0 (node) vol.veloc. = max (antinode) pressure waves traveling in tube resonance of tube depends on length coupled energy input Clarinet, oboe, organ, flute finger holes disrupt waveguide (scattering) first reflection determines oscillation period E4896 Music Signal Processing (Dan Ellis) /16

12 3. Digital Waveguides 1-D waveguide is easily discretized spatial sampling time sampling Julius Smith, energy nonlinear element acoustic waveguide scattering junction (tonehole) ω = π c 2 L (quarter wavelength) waveguide delay line scattering low-pass feedback final load instrument body resonances nonlinear function for energy input E4896 Music Signal Processing (Dan Ellis) /16

13 Digital Waveguides Direct physical model + simplifications Nut -1 String Waveguide Bridge dispersion + radiation load y + (x,t) Initialize with pluck shape -1 Delay Lines hreflec s(t) y + (0,t) = y - (0,t) y - (x,t) y - (L,t) = hreflec(t) * y + (L,t) Karplus- Strong Delay z-l Initialize with random values L = SR/f0 s(t) h LP E4896 Music Signal Processing (Dan Ellis) /16

14 Waveguide Simulation Karplus-Strong Plucked String algorithm Initialize with random values Delay z-l Lowpass L = SR/f 0 Direct implementation of traveling waves E4896 Music Signal Processing (Dan Ellis) /16

15 Karplus-Strong in Pd Pd s delwrite~ / delread~ implement the delay line E4896 Music Signal Processing (Dan Ellis) /16

16 Summary Wave equation: Simple physics leads to oscillations Musical acoustics: Different ways to control steady tones Digital waveguides: Natural, efficient simulation of tones E4896 Music Signal Processing (Dan Ellis) /16