UEO n 168-EN-22

Master 1 IMA

Master 1 IMDEA

tutorial labs in Physical Acoustics

Laurianne Barguet, Louis Delebeque, Guillaume Penelet, LAUM, UMR CNRS 6613

annee universitaire 2014-2015

Organization of the tutorial labs

Formation of binomes

You are asked to form a binome with one of your colleague. Trinomes are not allowed. Attribute

yourself a binome number, and then proceed to the different tutorial labs (1 tutorial lab per 4h session)

according to the table below.

binome session 1 session 2 session 3 session 4

1 TP1 TP2 TP3 TP4

2 TP2 TP3 TP4 TP1

3 TP3 TP4 TP1 TP2

4 TP4 TP1 TP2 TP3

5 TP1 TP2 TP3 TP4

6 TP2 TP3 TP4 TP1

7 TP3 TP4 TP1 TP2

8 TP4 TP1 TP2 TP3

Preparation of the tutorial labs, reports.

Each of the tutorial lab requires preliminary, theoretical works. These preliminary works should

be treated before the lab session. Some of the theoretical derivations you are asked to perform have

not been yet treated in the lecture course (or wont be treated during the course), so that you may

need to consult textbooks such as A.D. Pierce (Acoustics : an introduction to its physical principles

and applications), M. Bruneau (Manuel dAcoustique Fondamentale), or C. Potel and M. Bruneau

(Acoustique Generale).

You are asked to write a report for each of the tutorial lab. This report can be handwriten or

(preferently) typewriten. It must be given to your advisor one week (at the latest) after the

tutorial lab, either by E-mail (pdf file) if it is typewritten, or in the personal box of your advisor (2nd

floor of the acoustics building).

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1.- Radiation impedance of a duct

Equipment

a duct

a bae

an impedance sensor and its devoted software (capteurZ)

Objectives

to measure the radiation impedance at the end of the duct

to compare experimental data with theory (sound emission from a piston in an infinite wall)

To do list

Preliminary questions (not to be treated during the tutorial labs)

1. Give the approximate expression of the radiation impedance Zrad., in the low frequency range,

of a duct terminated with an infinite bafle (this has been treated as an exercise during the first

semester).

2. Show that the expression of the input impedance at the entrance of a duct (length L) terminated

by a load impedance Zload is given by :

Zinput = 0c0j tan kL+ Zload0c0

1 + j Zload0c0 tan kL(1)

3. Now assume that Zload = Zrad., and assume that |Zrad.|

4where a stand for the radius of the duct, and where J1 and S1 stand for the Bessel function of

the first kind (and order 1) and the Struve function, respectively. In order to see how does the

radiation impedance vary with the frequency (more precisely with the parameter ka), you can

download and execute the matlab programs radiation impedance.m and struve.m at the following

url : www.univ-lemans.fr/~gpenelet/TPM1

Experiments

1. Proceed to the impedance sensor calibration (ask your advisor if you do not know how to do).

2. Measure the input impedance of the duct when it is terminated by a rigid wall.

3. From the obtained results, calculate the sound speed for each resonant frequency (phase velocity

of the wave, which accounts for energy dissipation due to viscous and thermal losses along the

inner walls of the duct). To that purpose, you should treat the problem of the propagation of

plane waves through a duct with dissipation, the dissipation being treated as a complex wave

number

k =

c0[1 + (1 i)] ,

with 1 to be determined ( may vary with frequency . . .). The wave velocity is then givenby c = /

2.- Monopole, dipole, quadrupole

Equipment

One small loudspeaker in an enclosure.

Two small loudspeakers

A microphone

Audio power amplifier

Appropriate software and data acquisition card for measuring Frequency Response Functions

A phase shifter

Objectives

The objective of this tutorial lab is to characterize the radiation of sound by elementary sources.

To that purpose, a single or several loudspeakers will be used in order to tune the directivity of sound

radiation. More precisely, the objective is to realize a monopole, a dipole, a lateral quadrupole and a

linear quadrupole.

To do list

Preliminary questions (not to be treated during the tutorial labs)

1. Assume you have a source which is made by four small pulsating spheres driven in phase and

placed at the corner of a (virtual) square, each source being separated by a distance d. Accounting

for the distance d between each elementary source, what is the maximum frequency above which

the assumption of a ponctual source (i.e. the 4 sources considered as a unique one) is no longer

valid ?

2. Accounting for the size and the distance between the sources, what is distance from the sources

above which the far-field approximation may be reasonably retained ?

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6Figure 1 monopole

Experiments

Monopole

1. Build a monopole (just take the loudspeaker enclosure) and choose an appropriate distance from

the source to the microphone.

2. Proceed to the measurement of the frequency responses functions of the acoustic source from

50 Hz to 500 Hz and for different values of the azimuthal angle (an appropriate choice of

the angular discretization should be made in order to be able to trace directivity patterns).

From the obtained data, trace the directivity pattern of the source at a frequency to be chosen

appropriately.

3. Calculate the volume velocity radiated by the source (consider this source to be equivalent to a

monopole, and use the theoretical formula for the radiation of a monopole)

Dipole

Figure 2 Dipole

1. Build a dipole (take a loudspeaker without an enclosure)

2. Proceed to the measurement of the frequency responses functions of the acoustic source from

50 Hz to 500 Hz and and for different values of the azimuthal angle . From the obtained data,

trace the directivity pattern of the source at a frequency to be chosen appropriately.

73. Is the observed directivity in agreement with theory ?

4. Proceed to the ratio between the frequency responses of the dipole and the one(s) of the

monopole. Is the obtained result in agreement with theory (in both terms of frequency response

and directivity pattern) ?

5. Determine from your measurements the distance separating the two (virtual) elementary sources

of the dipole (once again, use the theoretical formula for the radiation of an acoustic dipole).

Lateral quadrupole

Figure 3 lateral quadrupole

1. Build a lateral quadrupole

2. Proceed to the measurement of the frequency responses functions of the acoustic source from

50 Hz to 500 Hz and and for different values of the azimuthal angle . From the obtained data,

trace the directivity pattern of the source at a frequency to be chosen appropriately.

3. Proceed to the ratio between the frequency response of the lateral quadrupole and the one of the

monopole. Is the obtained result in agreement with theory (in both terms of frequency response

and directivity pattern) ?

Linear quadrupole

Figure 4 Oblique quadrupole

81. Build a linear quadrupole

2. Proceed to the measurement of the frequency responses functions of the acoustic source from 50

Hz to 500 Hz and and for different values of the azimuthal angle . From the obtained data, trace

the directivity pattern of the source at a frequency to be chosen appropriately. Proceed to the

ratio between the frequency response of the lateral quadrupole and the one of the monopole. Is

the obtained result in agreement with theory (in both terms of frequency response and directivity

pattern) ?

Bonus

If time is remaining, build once again an linear quadrupole, but apply a phase shift (e.g. of about

pi/3) to one of the two loudspeakers. Proceed to the measurement of the directivity pattern. Discuss

the obtained results, and compare with a model (cf. infra)

Theoretical analysis

This part should be treated after (or possibly before) the tutorial lab, and it is aimed at getting

a deeper insight about the differences between the experimental results and the analytical results for

the radiation of a monopole, a dipole, and a quadrupole.

1. Consider that each of the loudspeakers can be modeled as two pulsating sphere (of appropriate

radius, and separted by some distance d). Compute the radiation of a set of two loudspea-

kers in the different configurations mentioned above (i.e. monopole, dipole, linear and lateral

quadrupoles), and calculate the resulting acoustic pressure at some point corresponding to the

observation point in experiments.

2. Compare the resulting directivity pattern with the one obtained in experiments and with the

one obtained theoretically (assuming both the radius of the sources and the distance between

the sources are small compared with the wavelength).

3.- Guided waves in a cylindrical duct

Equipment

Duct with four microphones arranged on the four quadrants of a section at one end, and equipped

with two loudspeakers at the other end

A data acquisition system (with four channels) and four microphones

Audio power amplifier

Objectives

To measure the amplitudes of the modes 10 and 20 below and above their associated cut-off

frequencies

To do list

Preliminary questions (not to be treated during the tutorial labs)

1. From the knowledge of the inner diameter of the duct, determine the cut-off frequencies of the

modes 10, and 20.

2. Assuming that only the plane wave mode and the mode 10 and 20 are existing within the duct,

give the expression of the acoustic pressure measured by each microphone. Show that the linear

combinations of the measured acoustic pressure should allow to determine the amplitudes of each

mode (00, 10, 20), and explain how we can anticipate the positions of the nodal lines associated

with the modes 10 and 20.

Experiments

Calibration of the transducers

1. Proceed to a careful relative calibration of the microphones. A way to proceed is to excite the

device at low frequency (for which only the plane wave mode is present) and to compare the

signals delivered by each microphone relative to one of the four microphone (seen as a reference

microphone). Check that the microphone have the same sensitivities in both terms of amplitude

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and phase. If this is not the case, you should use these measurements to define a correction of

the frequency response functions you will be asked to measure in the following.

Measurements of cut-off frequencies

For the measurements described below, you are asked to use a 4 channel data acquisition card, and

an adequate data acquisition software, as well as Matlab for the post-processing of data. Moreover,

only one of the two loudspeakers should be excited at first.

1. Proceed to the measurement of the frequency response function Hij =pi()pj()

= Aij()eiij(),

where pi and pj are the complex amplitudes measured by microphones i and j, and where these

two microphones should be chosen appropriately to determine the cut-off frequency of the mode

10 from the phase ij() of Hij . Compare the obtained value of the cut-off frequency f10 with

a theoretical estimate.

2. Try to do a similar experiment in order to determine f20.

Measurements of the amplitudes of each mode

1. Only one of the two loudspeakers is used to excite acoustic waves in the duct. Proceed to

the measurement of the frequency response functions Hi1 =pi()p1()

for each microphone, where

i = 2, 3, 4 and where j = 1 is to refer to the reference microphone you chose.

2. From the results above, determine the amplitudes of the plane wave mode as well as those of the

modes 10 and 20, at the axial position x0 under consideration (the position along x of the four

microphones). To that purpose, having answered to the preliminary questions is necessary . . .

3. Now, the two loudspeakers are used to excite acoustic waves in the duct, and both loudspeakers

are driven out of phase. Proceed to the measurement of the frequency response functions

Hi1 =pi()p1()

for each microphone, where i = 2, 3, 4.

4. From the results above, determine the amplitudes of the plane wave mode as well as those of

the modes 10, and 20, at the axial position x0 under consideration.

5. Proceed to the same measurements when the two loudspeakers are driven in-phase.

6. Discuss your results and conclude.

4.- Sound scattering by a rigid cylinder

Equipment

a PVC duct

a loudspeaker (tweeter) and an audio power amplifier

a microphone

A data acquisition system and appropriate software

three matlab program to calculate sound scattering of a plane wave by a rigid cylinder, that can

be found at the url www.univ-lemans.fr/~gpenelet/TPM1

Objectives

to measure the directivity pattern associated to sound scattering of an incident plane wave by a

rigid cylinder

to measure the frequency response at a given observation point near the cylinder

To do list

Preliminary questions (not to be treated during the tutorial labs)

1. From what you have learn during your lecture course in the first semester, and with the help of

an appropriate textbook (e.g. in [C. Potel and M. Bruneau, Acoustique Generale, ed. Ellipse,

Paris 2006]), show that if one consider the problem of a plane progressive wave which arrives

perpendicular to the axis of an infinite and rigid duct (see. Fig. 5) of radius a, then the acoustic

pressure field is the sum of the incident wave (peak amplitude P0)

r a, pi(r, , t) = P0eikr cos eit, (3)

= P0J0 (kr) eit + 2P0

n=1

inJn (kr) cos(n)eit (4)

and of the scattered wave

r a, ps(r, , t) = P0 J1(ka)H

(2)1 (ka)

H(2)0 (kr)e

it +

n=1

BnH(2)n (kr) cos(n)e

it, (5)

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where k = /c0 stand for the wave number, where Jn and H(2)n are the Bessel function of order

m and the Hankel function of the second kind and order m, and where

Bn = 2inP0Jn1(ka) nkaJn(ka)H

(2)n1(ka) nkaH

(2)n (ka)

. (6)

2. Calculate the intensity Ir =12< (pvr) of the scattered wave. Calculate the amplitude of this

intensity for the cases = 0 and pi in the limit cases ka >> 1 and ka

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the non-uniform frequency response of the loudspeaker. Store your data into the analyser and

proceed to the display of the frequency response corrected by the non-uniform response of the

loudspeaker.

3. Repeat this measurement when choosing 0 = 5pi/6, 0 = 2pi/3, 0 = pi/2, 0 = pi/3, 0 = pi/6,

and 0 = 0 (NB : if time is at your disposal, a more accurate discretization, e.g. = pi/12

instead of pi/6 would even be better . . .)

4. Compare your results with theory (use the Matlab program diffraction-cylindre-f.m).

5. From your measurements, calculate approximate directivity patterns for different frequencies,

and notably for ka > 1. Compare the measurements with theory(use the Matlab program diffraction-cylindre-theta.m), and provide justifications to explain the

difference between experiment and theory.