1. Research Methods in Acoustics Lecture 6: Musical Acoustics
Jonas Braasch

2. Wave Equation for Pressure
Explain x=

3. 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.

4. Solution for the Wave Equation
forward propagating wave
backward propagating wave

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

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

7. Velocity wave in organ pipe
open pipe
stopped pipe

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

9. 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

10. 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

11. Impulse Response at l=0 p t with T=2·l/c p t with T=2·l/c R=1 T 2·TT
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

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

13. 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

14. 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

15. 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

16. 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) ]);

17. Closed pipe, 50 cm length, r=0.9

18. Closed pipe, 50 cm length, r=0.9

19. Closed pipe, 15 cm length, r=0.9

20. Closed pipe, 15 cm length, r=0.9

21. Closed pipe, 30 cm length, r=0.5

22. Closed pipe, 30 cm length, r=0.5

23. Closed pipe, 50 cm length, r=0.9

24. Closed pipe, 50 cm length, r=0.9

25. Open pipe, 3 m length, r=0.9

26. Open pipe, 3 m length, r=0.9

27. Open pipe, 1 m length, r=0.9

28. Open pipe, 1 m length, r=0.9

29. Open pipe, 30 cm length, r=0.9

30. Open pipe, 30 cm length, r=0.9

31. 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)+…

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

33. 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

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

35. 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).

36. 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.
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.

37. 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

38. 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 seconds.
Name Size Bytes Class
Fs x double array
ir x double array
signal x double array
y x double array

39. 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.

40. 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.

41. 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'];

42. FREE-REED ORGAN PIPE
Weber (1829)

43. 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

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

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

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

47. 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

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

49. The impulse response of a flutter echo
amplitude Tp
time

50. 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).

51. 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

52. 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

53. 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

54. 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

55. Sound Intensity
Power
Energy

56. MODEL STRUCTURE

57. BASIC EQUATIONS
Adapted from Tarnopolsky et al. (2000)

58. 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

59. 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) cm

60. 5. PIPE DIMENSIONS

61. PIPE FREQUENCIES
cylindrical
conical I
conical II

62. PIPE FREQUENCIES
cylindrical
conical I
conical II

63. PIPE FREQUENCIES
cylindrical
conical I
conical II

64. PIPE FREQUENCIES
cylindrical
conical I
conical II

65. SPECTROGRAMS
[dB] po

66. SPECTROGRAMS
[dB] po
[dB] pi

67. SPECTROGRAMS
[dB] po
[dB] pi
[dB] x

68. ATTACK TRANSIENTS
19 cm
34 cm
40 cm
44 cm

69. 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

70. References
T.D. Rossing: The Science of Sound, Addison Wesley; 1st edition (1982) ISBN:
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