EE1.el3 (EEE1023): Electronics III Acoustics lecture 19 Musical acoustics Dr Philip Jackson www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3 Musical acoustics Objectives: • identify the acoustic source and filter elements • relate instrument properties to acoustic theory • calculate resonance frequencies for instruments Topics: • string • percussion • woodwind • brass • voice S.1 Preparation for musical acoustics • For one instrument from each class, identify what cre- ates a sound source, and a resonator that enhances it? { voice { woodwind { brass { percussion { string S.2 Types of musical instrument • string & percussion • woodwind & brass • voice S.3 Acoustic model of musical instruments Source: • mechanical vibration Time domain: • coupled vibration s(n) = h(m) ∗ x(n) aero-acoustic • 1 = X h(m) x(n−m) Filter: m=0 • sound box Frequency domain: • pipe resonance • instrument's timbre S(!) = H(!) X(!) sound x(n) acoustical s(n) output generation filter, h(m) sound pressure S.4 Acoustic filters Mechanical resonance • String (guitar, violin, piano) • Bar (xylophone, triangle) • Membrane (tympani, drums) • Shell (bell, cymbal) Acoustic resonance • Open & closed pipes • Low-frequency resonance S.5 Low-frequency resonance When the wavelength is very large compared to the size of an object, acoustic variables are effectively uniform over it at any time instant: it acts as a lumped acoustic element. S x A low-frequency resonator δx can be formed from a chamber with a hole or VV+ δV a neck, e.g., a bottle. P0 P0 + δp The gas density varies inversely with the volume δρ δV Sδx = − = − (1) ρ0 V V inducing an adiabatic change in pressure 2 2δρ δp = c δρ = ρ0c (2) ρ0 which combine to give 2Sδx δp = −ρ0c (3) V S.6 The Helmholtz resonator The force from pressure difference across the air in the neck opposes an inertial force due to its acceleration: 2 2S x Fpressure = Sp = −ρ0c (4) V Finertia = mx¨ = ρ0SLx¨ (5) Ignoring any losses, we obtain the SHM equation of motion: c2S Lx¨ + x = 0 (6) V 2 2 with !H = c S=LV and Helmholtz resonance frequency: s c S f = (7) S H 2π LV L V S.7 Acoustic resonators Open pipes LO Half-closed pipes LS Low-frequencyS Helmholtz L V S.8 Acoustic filters Waveguide geometry can be used to enhance or attenuate given frequencies: Low pass (expansion) High pass (contraction) Band stop (anti-resonator) S.9 Acoustic sources Energy: Mechanism: • plucking • emission of vibration into sound • bowing • mechanical and acoustic resonance • striking • aeroacoustic flow oscillation • blowing S.10 Strings & percussion: source Mechanical vibration of strings, bars, membranes, shells T y + dy dx dx ds θ dy y dx T x x + dx From previous analysis, we obtained the 1D wave equation: @2y @2y − v2 = 0 (8) @t2 @x2 for vibration on a string, where the wave velocity is q v = T/ρL and general solutions have the form: x x y(t; x) = y(+) t − + y(−) t + (9) v v for any arbitrary waveforms, y(+)(·) and y(−)(·) S.11 Modes of a string We obtain (complex) solutions for sinusoidal waves x x (+) j! t− (−) j! t+ y(t; x) = y e ( v ) + y e ( v ) (10) Boundary conditions, y0 = yL = 0, give resonant mode shapes j!t nv y (x; t) = An sin (2πf(n)x) e , where f(n) = n 2L S.12 Modes of a membrane Solving boundary conditions for 2D standing waves gives: Modes of a shell S.13 Strings & percussion: filter Resonance of the sound box to provide instrument's timbre: guitar, violin, piano S.14 Mode shapes of a guitar's sound box S.15 Woodwind & brass: coupled source & filter Oscillation of the reed/lips matches a resonance frequency of the pipe, which determines the pitch of a note: clarinet, oboe, saxophone, trumpet, horn, trombone, tuba S.16 Aero-acoustic source Oscillation of the air matches a pipe resonance frequency: whistle, pipe, flute, organ S.17 Organ pipes S.18 Organ pipe's source and its boundary conditions Flow-coupled source Pulse reflections S.19 Resonances of organ pipe's filter n c f(n) = 2LO (2m − 1)c f(m) = 4LS S.20 Voice as an instrument Vocal source x(n) and filter h(m): s(n) = h(m) ∗ x(n) S(!) = H(!) X(!) voice output vocal−tract filter source S.21 Musical acoustics • Acoustical techniques for making music: { sound generation, resonance and filtering • Musical instruments: { string & percussion { woodwind & brass { voice • Musical sound sources: { mechanical vibration: string, bar, membrane, shell { coupled vibration: reed, lips { aero-acoustic: overblow, whistle • Musical use of resonance: { sound box amplification from cavity { pitch selection using pipe resonance { resonance to provide instrument's timbre S.22 Preparation for sound localisation • What acoustic factors give us the ability to localise sounds with our two ears? { identify and describe at least two acoustic cues that we use, based on the sound pressure signals arriving at the ears S.23 Appendix Trigonometric identities for mode shape derivation ejx = cos x + j sin x (11) ejx + e−jx cos x = = cosh jx (12) 2 ejx − e−jx sin x = = −j sinh jx (13) 2j S.24.