Lecture 2: Acoustics. Acoustics & sound

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1 EE E680: Speech & Audio Processing & Recognition Lecture : Acoustics The wave equation Acoustic tubes: reflections & resonance Oscillations & musical acoustics Spherical waves & room acoustics Dan Ellis <dpwe@ee.columbia.edu> Columbia University Dept. of Electrical Engineering Spring 006 E680 SAPR - Dan Ellis L0 - Acoustics Acoustics & sound Acoustics is the study of physical waves (Acoustic) waves transmit energy without permanently displacing matter (e.g. ocean waves) Same math recurs in many domains Intuition: pulse going down a rope E680 SAPR - Dan Ellis L0 - Acoustics

2 The wave equation Consider a small section of the rope: y S ε φ φ 1 S displacement y(), tension S, mass ε d lateral force is F y S sin( φ ) S sin( φ 1 ) y S d... E680 SAPR - Dan Ellis L0 - Acoustics Newton s law: Wave equation () F ma y S d y εd t Call c S Ú ε (tension to mass-per-length) hence: the Wave Equation: c y y.. partial DE relating curvature and acceleration t E680 SAPR - Dan Ellis L0 - Acoustics

3 Solution to the wave equation If (any f( )) then y y y(, t) f ( ct) f '( ct) f ''( ct) also works for Hence, general solution: y t y t yt (, ) f( ct) c f '( ct) c f ''( ct) c y y t yt (, ) y ( ct) y ( ct) E680 SAPR - Dan Ellis L0 - Acoustics Solution to the wave equation () y ( ct) y ( ct) and are travelling waves - shape stays constant but changes position: y y time 0: y - c T time T: y y - c is travelling wave velocity ( / t ) y moves right, y moves left resultant y() is sum of the two waves E680 SAPR - Dan Ellis L0 - Acoustics

4 Wave equation solutions (3) What is the form of y, y? - any doubly-differentiable function will satisfy wave equation Actual waveshapes dictated by boundary conditions - e.g. y() at t 0 - plus constraints on y at particular s e.g. input motion y(0, t) m(t) rigid termination y(l, t) 0 y y(0,t) m(t) y (,t) y(l,t) 0 E680 SAPR - Dan Ellis L0 - Acoustics Terminations and reflections System constraints: - initial y(, 0) 0 (flat rope) - input y(0, t) m(t) (at agent s hand) ( y ) - termination y(l, t) 0 (fied end) - wave equation y(,t) y ( - ct) y ( ct) At termination: y(l, t) 0 y (L - ct) y (L ct) i.e. y and y are mirrored in time and amplitude around L inverted reflection at termination y y(,t) y y L y simulation [travel1.m] E680 SAPR - Dan Ellis L0 - Acoustics

5 Acoustic tubes Sound waves travel down acoustic tubes: pressure - 1-dimensional; very similar to strings Common situation: - wind instrument bores - ear canal - vocal tract E680 SAPR - Dan Ellis L0 - Acoustics Pressure and velocity ξ(, t) Consider air particle displacement : ξ() Particle velocity v(, t) hence volume velocity ξ t u(, t) A v(, t) (Relative) air pressure p(, t) 1 ξ -- κ E680 SAPR - Dan Ellis L0 - Acoustics

6 Wave equation for a tube Consider elemental volume: Area da Force p da Volume da d Mass ρ da d Force (p p / d) da Newton s law: F p d da ma ρdad p v ρ t v t Hence c ξ ξ c t ρκ E680 SAPR - Dan Ellis L0 - Acoustics Acoustic tube traveling waves Traveling waves in particle displacement: ξ(, t) ξ ( ct) ξ - ( ct) Call u ( α) ca ξ ( α) α ρc Z A Then volume velocity: ξ u(, t) A u ( ct) u - ( ct) t And pressure: p(, t) 1 ξ -- Z κ 0 [ u ( ct) u - ( ct) ] (Scaled) sum & diff. of traveling waves E680 SAPR - Dan Ellis L0 - Acoustics

7 Acoustic tube traveling waves () Different resultants for pressure and volume velocity: c c Acoustic tube u u - u(,t) u - u - Volume velocity p(,t) Z 0 [u u - Pressure ] E680 SAPR - Dan Ellis L0 - Acoustics Terminations in tubes Equivalent of fied point for tubes? Solid wall forces u(,t) 0 hence u u - u 0 (t) (Volume velocity input) Open end forces p(,t) 0 hence u -u - Open end is like fied point for rope: reflects wave back inverted Unlike fied point, solid wall reflects traveling wave without inversion E680 SAPR - Dan Ellis L0 - Acoustics

8 Standing waves Consider (comple) sinusoidal input: u 0 () t U 0 e jωt Pressure/volume must have form Ke j( ωt φ) Hence traveling waves: ( ωt φ A) u ( ct) Ae j k ( ) u - ( ct) Be jk ωt φ B k ω Ú c λ cú f πc Ú ω where (spatial frequency, rad/m) (wavelength ) Pressure, vol. veloc. resultants show stationary pattern: standing waves - even when A B simulation [sintwavemov.m] E680 SAPR - Dan Ellis L0 - Acoustics Standing waves () U 0 e jωt k π λ / pressure 0 (node) vol.veloc. ma (antinode) For lossless termination ( u u - ), have true nodes & antinodes Pressure and vol. veloc. are phase shifted - in space and in time E680 SAPR - Dan Ellis L0 - Acoustics

9 Transfer function u 0 () t U 0 e jωt Consider tube ecited by : - sinusoidal traveling waves must satisfy termination boundary conditions - satisfied by comple constants A and B in ut (, ) u ( ct) u - ( ct) Ae j( k ωt) Be jk ( ωt ) e jωt ( Ae jk Be jk ) - standing wave pattern will scale with input magnitude - point of ecitation makes a big difference... E680 SAPR - Dan Ellis L0 - Acoustics Transfer function () For open-ended tube of length L ecited at 0 U 0 e jωt by : jωt kl ut (, ) U 0 e cos ( ) k coskl (matches at 0, maimum at L) i.e. standing wave pattern e.g. varying L for a given ω (and hence k): ω --- c U 0 e jωt U 0 U L U 0 e jωt U 0 U L magnitude of U L depends on L (and ω)... E680 SAPR - Dan Ellis L0 - Acoustics

10 Transfer function (3) Varying ω for a given L, i.e. at L : U L ult (, ) U 0 u( 0, t) coskl u(l) u(0) cos( ωl Ú c) L u(l) at ωl/c (r1)π/, r 0,1,... u(0) Output vol. veloc. always larger than input Unbounded for L ( r 1) πc ( r 1) λ ω -- 4 i.e. resonance (amplitude grows w/o bound) E680 SAPR - Dan Ellis L0 - Acoustics Resonant modes For lossless tube with L m λ -- 4, m odd, λ wavelength, ul ( ) u( 0) is unbounded, meaning: - transfer function has pole on frequency ais - energy at that frequency sustains indefinitely L 3 λ 1/ 4 ω 1 3ω 0 - compare to time domain... L λ 0/ 4 e.g 17.5 cm vocal tract, c 350 m/s ω 0 π 500 Hz (then 1500, ) E680 SAPR - Dan Ellis L0 - Acoustics

11 u k u - k Area A k Area A k1 Scattering junctions u k1 u - k1 At abrupt change in area: pressure must be continuous p k (, t) p k1 (, t) vol. veloc. must be continuous u k (, t) u k1 (, t) traveling waves u k, u- k, u k1, u- k1 will be different u k u - k Solve e.g. for u - k and u k1 : (generalized term.) 1 - r 1 r r 1 r r 1 r - 1 r 1 u k1 u - k1 r A k1 A k Area ratio E680 SAPR - Dan Ellis L0 - Acoustics Concatenated tube model Vocal tract acts as a waveguide Lips L Lips u L (t) Glottis u 0 (t) 0 Glottis Discrete appro. as varying-diameter tube: A k, L k E680 SAPR - Dan Ellis L0 - Acoustics A k1, L k1

12 Concatenated tube resonances Concatenated tubes scattering junctions lattice filter u k e -jωτ 1 e -jωτ u - k e -jωτ 1 e -jωτ e -jωτ 1 e -jωτ Can solve for transfer function - all-pole 1 Imaginary part Approimate vowel synthesis from resonances sound eample: ah ee oo E680 SAPR - Dan Ellis L0 - Acoustics Oscillations & musical acoustics Pitch (periodicity) is essence of music: - why? why music? Different kinds of oscillators: - simple harmonic motion (tuning fork) - relaation oscillator (voice) - string traveling wave (plucked/struck/bowed) - air column (nonlinear energy element) E680 SAPR - Dan Ellis L0 - Acoustics

13 Simple harmonic motion Basic mechanical oscillation: ẋ ω Acos( ωt ϕ) Spring mass ( damper) F k m ζ e.g. tuning fork ω k m Not great for music: - fundamental (cosωt) only - relatively low energy E680 SAPR - Dan Ellis L0 - Acoustics Relaation oscillator Multi-state process: - one state builds up potential (e.g. pressure) - switch to second (release) state - revert to first state etc. e.g. vocal folds: p u ( ) Oscillation period depends on force (tension) - easy to change - hard to keep stable less used in music E680 SAPR - Dan Ellis L0 - Acoustics

14 Ringing string e.g. our original rope eample tension S mass/length ε L Many musical instruments - guitar (plucked) - piano (struck) - violin (bowed) Control period (pitch): - change length (fretting) - change tension (tuning piano) - change mass (piano strings) ω π S L ε Influence of ecitation... [pluck1a.m] E680 SAPR - Dan Ellis L0 - Acoustics Wind tube energy Resonant tube energy input nonlinear element acoustic waveguide scattering junction (tonehole) ω π c L (quarter wavelength) e.g. clarinet - lip pressure keeps reed closed - reflected pressure wave opens reed - reinforced pressure wave passes through Finger holes determine first reflection effective waveguide length E680 SAPR - Dan Ellis L0 - Acoustics

15 4 Room acoustics Sound in free air epands spherically: radius r Spherical wave equation: p r -- r p r c p t solved by prt (, ) P e r j ( ωt kr ) p Intensity falls as r E680 SAPR - Dan Ellis L0 - Acoustics Effect of rooms (1): Images Ideal reflections are like multiple sources: virtual (image) sources reflected path source listener Early echoes in room impulse response: direct path early echos h room (t) t - actual reflections may be h r (t), not δ(t) E680 SAPR - Dan Ellis L0 - Acoustics

16 Effect of rooms (): modes Regularly-spaced echoes behave like acoustic tubes: Real rooms have lots of modes! - dense, sustained echoes in impulse response - comple pattern of peaks in frequency response E680 SAPR - Dan Ellis L0 - Acoustics Reverberation Eponential decay of reflections: h room (t) ~e -t /T Frequency-dependent - greater absorption at high frequencies faster decay Size-dependent - larger rooms longer delays slower decay Sabine s equation: 0.049V RT Sα Time constant varies with size, absorption t E680 SAPR - Dan Ellis L0 - Acoustics

17 Summary Travelling waves Acoustic tubes & resonance Musical acoustics & periodicity Room acoustics & reverberation Parting Thought: Musical bottles E680 SAPR - Dan Ellis L0 - Acoustics

Lecture 3: Acoustics

CSC 83060: Speech & Audio Understanding Lecture 3: Acoustics Michael Mandel mim@sci.brooklyn.cuny.edu CUNY Graduate Center, Computer Science Program http://mr-pc.org/t/csc83060 With much content from Dan