Research Methods in Acoustics Lecture 6: Musical Acoustics Jonas Braasch

Wave Equation for Pressure Explain x=

Solution for the Wave Equation We have three oscillating terms: forward propagating wave backward propagating wave oscillating function in time Theoretically we also have a second oscillating term in time, oscillating backwards! In our environment, we can exclude this solution.

Solution for the Wave Equation forward propagating wave backward propagating wave

Acoustic Impedance of a Plane Wave Acoustic Impedance Definition Acoustic Impedance of a plane wave Acoustic Impedance of an occluder

Wave Equation for Pressure forward propagating wave backward propagating wave L p0 v0 Z0= v0 p0 x x=0 L−x x=L

Velocity wave in organ pipe open pipe stopped pipe

Velocity wave in organ pipe open pipe l/2 stopped pipe Standing wave at l=l/2 Z0<<ZW

Impulse response of a cylindrical resonator t=0 ms l p+ t=0.5·l/c ms l t=1.0·l/c ms p+ r=p−/p+=−1 l t=1.0·l/c ms p−=−p+ p− l

Impulse response of a cylindrical resonator t=1.0·l/c ms p−=−p+ p− l t=1.5·l/c ms p− l t=2.0·l/c ms p− l p+ t=2.0·l/c ms p+=−p− l T=2·l/c T=2·l/c l=c·T=2·l w=c/2l f=c/4pl

Impulse Response at l=0 p t with T=2·l/c p t with T=2·l/c R=1 T 2·T T 2·T 3·T 4·T 5·T 6·T p with T=2·l/c R=0.9 t T 2·T 3·T 4·T 5·T 6·T with T=2·l/c

Velocity wave in organ pipe open pipe l/2 stopped pipe l/4

Impulse response of a closed resonator t=0 ms l p+ t=0.5·l/c ms l t=1.0·l/c ms p+ r=p−/p+=1 l p− t=1.0·l/c ms p−=p+ l

Impulse response of a cylindrical resonator t=1.0·l/c ms p− p−=p+ l t=1.5·l/c ms p− l t=2.0·l/c ms p− l t=2.0·l/c ms p+=−p− p+ l

Impulse Response at l=0 p R=0.9 t T 2·T 3·T 4·T 5·T 6·T with T=2·l/c

function y=OrganPipeOpenSound(l,att) l=0.5; % m c=343; % m/s Fs=48000; PipeTabs=round(Fs*l/c); l=PipeTabs/Fs*c; ForwardDelayLine=zeros(PipeTabs+1,1); BackwardDelayLine=zeros(PipeTabs+1,1); x=randn(Fs,1); PipeTabs=PipeTabs+1; for n=1:(length(x)) ForwardDelayLine(2:PipeTabs)=ForwardDelayLine(1:PipeTabs-1); ForwardDelayLine(1)=-1*BackwardDelayLine(1)+x(n); y(n)=ForwardDelayLine(1); BackwardDelayLine(1:PipeTabs-1)=BackwardDelayLine(2:PipeTabs); BackwardDelayLine(PipeTabs)=-att*y(n); end % of for wavwrite(y,Fs,['Organ_open_' int2str(l*10) '_' int2str(att*10) ]);

Closed pipe, 50 cm length, r=0.9

Closed pipe, 50 cm length, r=0.9

Closed pipe, 15 cm length, r=0.9

Closed pipe, 15 cm length, r=0.9

Closed pipe, 30 cm length, r=0.5

Closed pipe, 30 cm length, r=0.5

Closed pipe, 50 cm length, r=0.9

Closed pipe, 50 cm length, r=0.9

Open pipe, 3 m length, r=0.9

Open pipe, 3 m length, r=0.9

Open pipe, 1 m length, r=0.9

Open pipe, 1 m length, r=0.9

Open pipe, 30 cm length, r=0.9

Open pipe, 30 cm length, r=0.9

Convolution From our pipe experiment, we saw that our current sound at l=0 is a mixture from the exciting noise and the pressure wave that has been returned after T= 2l/c. Theoretically we can synthesize this sound by adding old copies of the exciting sound to the new ones: y(t)=x(t)+rx(t-T)+r2x(t-2T)+…

R=0.9 t T 2·T 3·T 4·T 5·T 6·T with T=2·l/c R=0.9 t

Convolution t t + t t d(t=0) d(t)=1 for t=0 =0 elsewhere rd(t − T) Impulse response of a cylindrical resonator

Convolution t t t t + t t t t x (t) d(t=0) rd(t − T) rx (t−T)

Convolution Now let us go into the digital domain with the time indices n and k. We can now generalize the last equation: Impulse Response Convolution impulse response 2. time signal With the help of the convolution, we can calculate the output y(k) for an input signal x(k) and the impulse response h(k).

Definition of Convolution A convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function . It therefore "blends" one function with another. http://mathworld.wolfram.com/Convolution.html In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. A convolution is a kind of very general moving average, as one can see by taking one of the functions to be an indicator function of an interval. http://en.wikipedia.org/wiki/Convolution

Convolution z(k)= x(k)*y(k)= S x(m) y(k-m) + k,m: samples in time inf z(k)= x(k)*y(k)= S x(m) y(k-m) m=-inf + k,m: samples in time x: signal y: impulse response z: convolved signal

Convolution [signal,Fs]=wavread('MarlabSoloMono2'); [ir,Fs]=wavread('gs201m.wav'); tic y=conv(signal,ir); toc y=0.8.*y./max(abs(y)); wavwrite(y,Fs,'ConvSignal.wav'); Elapsed time is 403.312000 seconds. Name Size Bytes Class Fs 1x1 8 double array ir 121466x1 971728 double array signal 326707x1 2613656 double array y 448172x1 3585376 double array

Pythagorean Tempering octave 2:1 fifth 3:2 fourth 4:3 Major 3rd 5:4 Minor 3rd 6:5 Major 6th 5:3 Minor 6th 8:5 We can determine music intervals through basic ratios between two frequencies In the Pythagorean Tempering we use fifth and octaves to establish a musical scale: C-G-D-A-E-B-F#-C#-G#-D#-A#-F Unfortunately, some keys really sound out of tune.

Equal Tempering In the equal tempered scale all semitones have the same semitone interval: This way all keys basically sound the same and are all acceptable. The standard reference tone is A4=440Hz, but reference tone up to 445 Hz are common today.

Matlab function for pitch determination function s=note(f) o=log2(f)-log2(440); n=round(12*o); cents = 100*(12*o-n); oct=floor((n-3)/12)+5; chroma=mod(n,12); chromalist = {'A'; 'A#'; 'B'; 'C'; 'C#'; 'D'; 'D#'; 'E'; 'F'; 'F#';'G'; 'G#'}; cents = sprintf('%+.0f',cents); s=[char(chromalist(chroma+1)),num2str(oct),' ',num2str(cents), ' cents'];

FREE-REED ORGAN PIPE Weber (1829)

The Flutter Echo Radiation of plane wave towards the wall loudspeaker placed in the center of the room speed of sound cs = 340 m/s distance from wall W/2 m (W=Room Width). W

The Flutter Echo the wave reaches the wall after: T1=W/(2cs) sec Example: Room Width= 5 m Tp =5/(2x340) s =0.00735 s =7.35 ms 5 m

The Flutter Echo the wave reaches the origin after: T1=W/cs sec Example: Room Width= 5 m Tp=5/(340) s =0.0147 s =14.7 ms 5 m

The Flutter Echo Now the pressure waves continues to the other side of the room ... 5 m

The Flutter Echo ... and after another 14.7 ms it is again back at the origin. ... since this event repeats itself periodially, in our case every 14.7 ms we speak about a flutter echo, or slap echo 5 m

The Flutter Echo If Tp > 50 ms, we hear each flutter echo as a separate auditory event. 5 m

The impulse response of a flutter echo amplitude Tp time

The absorption coefficient To what extent is the sound wave reflected? reflected wave transmitted wave Incoming wave xi=(1-a) xr+ axt Wall with a=absorption coefficient The absorption coeffient has values between 0 and 1, with 0 (total reflection) and 1 (full transmission).

The impulse response of a flutter echo amplitude Tp time Every slap echo is diminished by the absorption coefficient For the k-th echo we find: xk= x0(1-a)k

The impulse response of a flutter echo x(t)= x0(1-a)t/Tp ... which leads to a exponentional delay of the envelope: x(t)= x0 eln(1-a)t/Tp = x0 eln(1-a)t/Tp ≈ x0 e-at/Tp approximation for small a

Sound Intensity Sound intensity is defined as the sound power per unit area. The usual context is the measurement of sound intensity in the air at a listener's location. The basic units are watts/m2 or watts/cm2 . Many sound intensity measurements are made relative to a standard threshold of hearing intensity I0 :                                                                                                                             The most common approach to sound intensity measurement is to use the decibel scale:                                                                                                                       from http://hyperphysics.phy-astr.gsu.edu/hbase/sound/intens.html#c1

Inverse Square Law The sound intensity from a point source of sound will obey the inverse square law if there are no reflections or reverberation. A plot of this intensity drop shows that it drops off rapidly. from http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/invsqs.html

Sound Intensity Power Energy

MODEL STRUCTURE

BASIC EQUATIONS Adapted from Tarnopolsky et al. (2000)

SIMULATION OF THE RESONATOR Digital waveguide model (Smith, 1998) Reflectance at open end: solution of Levine and Schwinger (1948) Losses are considered at the beginning or end of the delay line

TYPES OF RESONATORS diameter diameter (reed) (bell) cylindrical 4.5 cm 4.5 cm conical I 3.2 cm 6 cm conical II 3.2 cm (12.9+l).0.064 cm

5. PIPE DIMENSIONS

PIPE FREQUENCIES cylindrical conical I conical II

PIPE FREQUENCIES cylindrical conical I conical II

PIPE FREQUENCIES cylindrical conical I conical II

PIPE FREQUENCIES cylindrical conical I conical II

SPECTROGRAMS [dB] po

SPECTROGRAMS [dB] po [dB] pi

SPECTROGRAMS [dB] po [dB] pi [dB] x

ATTACK TRANSIENTS 19 cm 34 cm 40 cm 44 cm

Acoustic Parameters p pascals sound pressure f hertz frequency ρ kg/m3 density of air c m/s speed of sound v particle velocity ω = 2 · π · f radians/s angular frequency ξ meters Particle displacement Z = c • ρ N·s/m³ acoustic impedance a m/s² Particle acceleration J W/m² sound intensity E W·s/mm³ sound energy density Pac watts sound power or acoustic power A m² Area

References T.D. Rossing: The Science of Sound, Addison Wesley; 1st edition (1982) ISBN: 0805385657 Jens Blauert, Script Communication Acoustics I (wave equation derivation), The script is currently translated by Ning into English Daniel A. Russell: Absorption Coefficients and Impedance http://www.kettering.edu/~drussell/GMI-Acoustics/Absorption.html http://www.phys.unsw.edu.au/~jw/Helmholtz.html

Ideal Gas Law Let us assume that our medium (e.g., air, water) has mass and elasticity (like air or water). Then the pressure p within the medium is dependant on the density r of the medium:

Ideal Gas Law Typically, the sound pressure p~ is much smaller than the static air pressure p=. We can write the total pressure as In a similar sense we can write the total density of the air as the static density of the air r= and the additional density that is introduced by the sound wave r~:

Ideal Gas Law Since the additional terms of the sound wave are small compared the static air pressure terms we can write: 1. Linearisation Since the additional terms of the sound wave are small compared the static air pressure terms we can replace the derivative with a constant K: Ideal Gas Law (I) Now, we would like to determine K

Adiabatic Law The adiabatic law provides a relationship between p and V Please note that V is proportional to the inverse of r

Adiabatic Law

Adiabatic Law