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General Acoustics

Catherine POTEL, Michel BRUNEAU Université du Maine - Le Mans - France

These slides may sometimes appear incoherent when they are not associated to oral comments. Slides based upon C. POTEL, M. BRUNEAU, Acoustique Générale - équations différentielles et intégrales, solutions en milieux fluide et solide, applications, Ed. Ellipse collection Technosup, 352 pages, ISBN 2-7298-2805-2, 2006 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

The world of acoustics OF THE EARTH AND OF SCIENCES THE ATMOSPHERELES GRANDS DOMAINES DE L'ACOUSTIQUEmechanical companies, buildings, solid and mechanical transportation (air, road, railway), , geology, aircraft , engineering communication, telecommunications, bathymetry, fishing,... nuclear,... chemical of the Earth engineering, and of the aero- materials and atmosphere seismic , acoustics, structures propagation vibro- imaging, in the atmosphere acoustics ultrasonic non destructive electrical evaluation and engineering, testing electronics oceanography Chapter 1 underwater sonorisation, civil engi- acoustics, fundamental electro-acoustics neering and and applied architecture acoustics, building, rooms, medicine measurement, , auditoria, comfort , signal,... cells imaging , Acoustics and its applications: , instruments phonation psycho- communi- generalities acoustics cation

noise , annoyance, biomedicine, art, comfort, architecture, culture, town planning, , society,... environment, noise pollution, telecommunications, ... LIFE AND HEALTH SCIENCES HUMAN AND SOCIAL SCIENCES Synoptic view of acoustic skills: the four fields of activity, the engineering domains, the specialised aeras (from outside to center). Adapted from R.B. Lindsay, J. Acoust. Soc. Am., 36, 1964, by Michel Bruneau, Pr., and Catherine Potel, Pr., French Acoustical Society

Music at prehistory Caves cavity+tubular hole ≡ Helmholtz's tuned to approximatively 80 Hz Among many bones and food reliefs of the men of Paleolithic, one finds some drilled phalanges of reindeer. They make whistles emitting clear and powerful . These phalanges always carry the cave 2 print of animal bites, men contenting themselves to arrange and make regular these holes. Oldest cave 1 phalanges are approximately 100.000 years old. The fundamental ≈ 160 Hz smaller the phalanges are, the higher the produced Whilstling phalanges of 1st sub- = 80 Hz sounds were. reindeer

very amplified Entrance If the speaker is close to the resonator, the voice is enhanced at the tuning http://perso.wanadoo.fr/palladia/prehistoire/mediatheque.htm of the resonator (modification of the impedance of the speaker)

Pythagorean school Acoustics of the open-air theaters

All the universe is

Pythagore de Samos (-570?/- 500?, Greek astronomer, philosophical, musicologist )

Study of acoustics ≡ Study of musical acoustics

- Relation between the length of a vibrating cord and the pitch of the emitted - “Mathematical” construction of the musical scale. Arenas of Nîmes

3 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

Vitruve One millenium and a half later (1/4)

Acoustics, which is considered under many aspects as a branch of , is henceforth detached from the musical art to become a real of the sound phenomenon.

Galileo Galilei Vitruve Marcus Pollio, (-88? to - Italian, (1565-1642) French, (1588-1648) 26?, Roman architect and military engineer) steps

Orchestrates: reflective stone slab which Sound radiation of a small clock enclosed in a vessel remains open and plays the part of where Boyle makes a partial vacuum reflectors of sounds. These reflections and those of the back wall being added to the shows the need for the presence of air for direct sound reinforce the transmitted orchestrate the production and the transmission of noise. energy and consequently improve the speech intelligibility. Experiment of Epidaure theater (Grèce) Robert Boyle (1627-1691)

One millenium and a half later (2/4) One millenium and a half later (3/4)

The mathematical theory of the sound propagation The sound results from a current of atoms emitted starts. Physics of continuous media or fields theory by the sound object; celerity and frequency of the (of which the sound field) started to reach its final sound being interpreted respectively like the speed mathematical structure. of the atoms and their number emitted per unit of times…( cf. scientific context of the time) Since then, the theories, so complex are they, are Pierre Gassendi considered (for the greatest part) as refinements of French, (1592-1655) English, (1642-1727) those which date from this period.

Global explanation of the sound and luminous phenomena; Huygens interprets both of them as being due to the propagation of longitudinal waves, associated to the of the molecules of elastic media in the case of the sound, and to the oscillating movements of ether, hypothetical substrate of the luminous phenomena, in the case of the light. Joseph Louis Lagrange Jean le Rond Deutch, (1629-1695) Swiss, (1707-1783) (Comte de-) d'Alembert Jean Bernoulli's French, (1736-1813) French, (1717-1783 ) student

One millenium and a half later (4/4) The

z Measurements made about 1860-1870 in tubes Victor Regnault The analysis of the complex sounds was lengths going up to 4900 meters (sewers of French, (1810-1878) experimentally carried out by the German Paris) physiologist and , (1821-1894), by means of which bear its z Experiments made in 1826 on the Geneva Lake by the Colladon and Sturm name : fire hose

Hermann L. von any musical sound with a given frequency, is Helmholtz associated to a timbre which results from the German, (1821-1894) superposition of a series of to the fundamental sound. 13 km The mathematical analysis of these complex sounds is based on famous work of the French mathematician Joseph Fourier (1768-1830), which by night always makes authority.

Jean-Baptiste Joseph Fourier trumpet French, (1768-1830) bronze bell, struck by an articulated hammer

4 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

Visualization of the vibrations Visualization of the vibrations of the acoustic (1/3) of the (2/3) Transmission Any sound , here transmitted by a tube to tube of the sound Gas burner the back of the membrane (on the left on the The left part of the apparatus is made up of disturbance figure) makes the membrane vibrating, which Helmholtz's resonators of increasing size (back of the makes the pressure of gas contained in the cavity in top to downwards. Each one of them is membrane) Cavity front of the membrane varying. Consequently, the connected to the back with a manometric flow of gas at the exit of the nozzle (burner) is cell which transmits the vibrations of the modulated at the frequency of the sound air in the resonators to a gas supplying the disturbance, which produces a height of flame flames which are thus modulated. Using which fluctuates at the same frequency. The the device of the turning mirror, often used variations of height of this flame allow, by an in the laboratories of physics at that time, appropriated method, in particular by the use of a the structure of the sound was visually turning mirror, a visual appreciation of the analyzed. The rotation of the mirror with 4 of the analyzed sounds. The manometric cells are, faces (on the right) will cut out the with Helmholtz's resonators, one of the essential movements of the flame and thus will slow Gas connection elements of Koenig's apparatus for the analysis of down it: the vibration of the flame Cell sensitive to the pressure variations the timbre of the musical sounds. becomes visible for the eye. (manometric cell) from Karl Rudolf Koenig (1832-1901) Manometric analyzer Specimen of the Montesquieu college, in Le Mans from Karl Rudolf Koenig (1832-1901) http://www.inrp.fr/she/instruments/instr_aco_capsule_koenig.htm http://www.inrp.fr/she/instruments/instr_aco_analyseur_koenig.

Visualization of the vibrations Lord Rayleigh of the acoustic wave (3/3)

Rudolf Koenig was born in At the dawn of XXth century, the top of Koenigsberg in Eastern Prussia, but is research in acoustics was marked by the established in Paris in 1851. Initially masterly work of the English scientist apprentice in the famous violin maker John William Strutt, Lord Rayleigh Jean Cambric Villaume (1798-1875), (1842-1919), who, in particular, he left him in 1858 to found his own synthesized the knowledge obtained company and to manufacture acoustic before in its treaty A theory of sound, apparatuses. One says of him: "The John William Strutt (Lord whose first edition appeared in 1877 name of Koenig is synonymous with Rayleigh) (T.I) and 1885 (T. II). The bases of the brilliant days of acoustics of the English, (1842-1919) acoustics then were posed. 19th century. Its instruments were among most beautiful, most effective and most precise of this time".

http://misha1.u-strasbg.fr/AMUSS/OdS311.htm

MECHANICAL WAVE (1/5) (2/5)

z A mechanical wave is an oscillatory motion which is gradually transmitted in a material medium, by vicinity,likeinformation, a change of position which one transmits to his neighbor.

http://www.kettering.edu/~drussell Animation courtesy of Dr. Dan Russell, Kettering University molecule

Schematic representation of matter made up of molecules (of The water particle at the centre moves and given masses) with elastic interactions. transmits its motion to the others

5 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

MECHANICAL WAVE: PRESSURE WAVE (3/5) MECHANICAL WAVE: SHEAR WAVE (4/5)

in a gas

http://www.kettering.edu/~drussell Animation courtesy of Dr. Dan Russell, Kettering University

→ discrete system F in a spring

→ Continuous system: propagation of a pulse along a F spring. The sections of the spring move from top to bottom as the pulse moves frome left to right.

MECHANICAL WAVE: FLEXURE WAVE (5/5) PROPAGATION CELERITY

GAS LIQUID SOLID

http://www.kettering.edu/~drussell Animation courtesy of Dr. Dan Russell, Kettering University V air = 340 m/s V water = 1500 m/s V metal ≅ 6000 m/s

flexure waves in a vibrating string Schematic look of the three fundamental states of the matter, and order of magnitude of the propagation velocity of pressure waves for each one of them

FROM DISCONTINUOUS MATTER...... TO CONTINUOUS MATTER

SHEAR WAVE polarisation

propagation particle λ Unique authorized No information is Informations are transmiited all the motion transmitted more quickly as the stiffness of the springs is great polarisation

PRESSURE WAVE propagation λ

6 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

FREQUENCY AUDIBLE AREA m/s dB V λ = 140 λ m f (Hz) ≡ 1/s 120 Audible area Wavelength 100 80

60 Speech I N F R A S O U D S The frequency of the wave is the speed with 40 U L T R A S O N D S 20 which the particle oscillates around its mean Threshold of audibility position. 0 1 Hz 20 Hz 1 kHz 2 kHz 20 kHz frequency

f = numbers of to and fro second motions of the particle

SONIC WAVE - ULTRASONIC WAVE CHARACTERISTICS OF A WAVE

⎡ ⎛ x ⎞ ⎤ ⎡ ⎛ t x ⎞ ⎤ In air: u()x;t = Acos ⎢ ω⎜t − ⎟ ⎥ = Acos⎢ 2π⎜ − ⎟ ⎥ = Acos()ωt − k x ⎣ ⎝ c ⎠ ⎦ ⎣ ⎝ T λ ⎠ ⎦ f > 20 000 Hz wave number 20 Hz ≤ f ≤ 20 000 Hz amplitude angular célérité wavelength frequency plane piston air plane

kx 0 x Audible noise Inaudible noise (for human being) k = ω c = 2π λ Sonic wave Ultrasonic wave λ = cT

A cos()ωt A cos ( ω t - k x )

ACOUSTIC TRANSMISSION PHENOMENON (1/2) ACOUSTIC TRANSMISSION PHENOMENON (2/2) Example 2: ultrasonic non destructive testing Example 1: noise path Z2 > Z1

c 2 e ∆ t = flaw V e d

c front or defect interface back Emission Propagation Reception echo echo Sources of noise Atmosphere, ground, Hearing meteorology t t phase ∆ t inversion

7 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

GENERATION OF A SONIC WAVE (1/4) GENERATION OF A SONIC WAVE (2/4) z Electrodynamic z baffle The Magnet Motor Drive System

Aim: establish a symmetrical magnetic field in which the voice coil will operate broad band loudspeakder 215 RTF bicone (Supravox) http://www.supravox.fr/haut-parleurs/intro_hp.html

http://fr.audiofanzine.com/apprendre/ glossaire/index,popup,,id_mot,49.html

closed diaphram loudspeaker with acoustic http://en.wikibooks.org/wiki/Engineering_Acoustics/Print_version horn GNU licence

GENERATION OF A SONIC WAVE (3/4) GENERATION OF A SONIC WAVE (4/4) z Electrodynamic loudspeaker z Electrodynamic loudspeaker

The Magnet Motor Drive System The Magnet Motor Drive System

The Loudspeaker Cone System The Loudspeaker Cone System

The cone serves the purpose of creating a larger radiating area The Loudspeaker Suspension allowing more air to be moved when excited by the voice coil The combination of the two flexures allows the voice coil to maintain linear travel as the voice coil is energized and provide a restoring force for the voice coil system. http://en.wikibooks.org/wiki/Engineering_Acoustics/Print_version http://en.wikibooks.org/wiki/Engineering_Acoustics/Print_version GNU licence GNU licence

AUDIBLE AREA (1/4) AUDIBLE AREA (2/4) z Orders of magnitude (1/3) z Orders of magnitude (2/3)

9IF homogeneous fluid, the characteristics of which do not depend on time 10 mm pressure (hPa) p < 2 hPa 1/10 mm ≅ 1000 1/1.000.000 mm P 0 ≅ 1000 h Pa 10 m ∆ P ≅ 160 h Pa T 0 P0 period p = P tot − P 0 < 2 h Pa atmospheric pressure Static pressure = constant time (s) Atmospheric 1 pascal acoustic pressure pressure 9IF non homogeneous fluid, the characteristics of which depend on time r P0 ⇒ PE (r;t) 2.10-5 Pa < p < 200 Pa very wide variation range Driving

8 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

AUDIBLE AREA (3/4) AUDIBLE AREA (4/4) z Orders of magnitude (3/3) z Fundamental variables p d = 200 Pa v ≈ 5⋅10 −1 m.s −1 dB Threshold of pain d 9 acoustic pressure p()r;t ξ ≈10 −4 m d 9 density ρ()r;t 140 r 9 vr ()r;t particle displacement ξ()r; t 120 9 entropy s()r;t 100 9 temperature τ()r;t 80 Variations around a reference state "E" : 60 P p p(r;t)()()= P r;t − P r;t tot 40 I N F R A S O U D tot E U L T R A S O N D r r r P0 20 ρ()r;t = ρ tot ()r;t − ρ E ()r;t Threshold of audibility ρ 0 vr ()r;t = vr ()r;t − vr ()r;t tot E r 0 r r r v 0 s()r;t = S tot ()r;t − S E ()r;t 1 Hz 20 Hz 1 kHz 2 kHz 20 kHz frequency S PE −5 r r r 0 p = 2⋅10 Pa τ()r;t = Ttot ()r;t − TE ()r;t s T v ≈ 5⋅10 −8 m.s −1 0 s time (s) −11 order of magnitude of the atomic radius of H homogeneous fluid, the characteristics of ξ s ≈10 m 2 which do not depend on time

ELEMENTARY VARIATION - INSTANTANEOUS DIFFERENCE DISSIPATIVE EFFECTS z z Elementary variation when two adjacent layers of fluid are animated with different v C speeds, the viscosity generates reaction forces between these two 1 F(t + d t) v layers which tend to oppose the displacements and are responsible 2 d F = lim []F()t + d t − F(t) for the damping of the waves. d t→0 ∆ F 9 v -v very small d F BUT 2 1 F(t) 9 very weak "viscous diffusion" velocity very weak viscosity effects t t + d t z Thermal conduction z Instantaneous difference, at any given time, from a given origin F heat transfers between two fluid regions (with slightly different E τ temperatures due to slightly different pressures) which tend to 1 τ F f disrupt their motions (acoustical origin); this disruption is 2 responsible for damping. F f ()t = ∫ d F = F ()t − FE (t) FE 9 τ2-τ1 very small BUT 9 very weak heat "propagation" quasi adiabatic transformations velocity (diffusion) Application: FE r Ptot r r p()r;t = d P = Ptot ()r;t − PE (r;t) z Molecular relaxation ∫PE t delay of return to equilibrium due to the fact that te effect of the input excitation is not instantaneous.

Some vocabulary SOUND LEVELS (1/2)

dB z Threshold of pain Transformation of acoustical energy (ordered) into heat (disordered). 9 The sensitivity of human ear depends 140 Ordered: the motion can be random (forced by a source), but gets a certain kind of order on the frequency 120 Audible area which is recognized by the brain 9 The auditive sense is proportional to 100 Disordered: entirely erractic motion, which is not a vibratory motion; macroscopic the of the acoustical 80 60 Speech temperature and pressure notions are associated to this motion. intensity I of the wave U N D S O A I N F R S

40 U N D O S R A U L T S 20 Threshold of audibility z Dissipation (dissipatio(onis) : dissolution, annihilation, destruction) −12 −2 L = 10log 10 (I I s ) with I s =10 W.m 0 Energy dissipation results from absorption. 1 Hz 20 Hz 1 kHz 2 kHz 20 kHz frequency A medium can be called a dissipative or an absorbant medium (synonymous terms). 2 p = ρ c I = 400⋅10 −12 = 2⋅10 −5 Pa I ∝ p L = 20 log 10 (p r ms p s ) s 0 0 s z Attenuation rms Decreasing of the amplitude of the considered phenomenon, due to one or several phenomena P p averages over time, on an acoustic period T : tot r r r r r ∀ r Examples: geometrical attenuation of a spherical wave (in one direction), p ()r = 0 v ()r = 0 ρ ()r = 0 attenuation due to dissipation, etc. +T 2 (diffusio(onis): act of spreading) ⌠ z Diffusion T 2 r 2 1 r 2 p ()r = []p (r;t ) with []p()r; t = lim ⎮ []p()r;t d t Phenomenon of redistribution of energy in all r ms T→∞ T ⎮ ⌡−T 2 directions (scattering) PE z Scattering (diffractum: pulled to pieces) 2 Behaviour of waves when they interact with an obstacle time (s) []p()r; t 2 ≠ []p()r; t which is not entirely transparent to them scattering by a square hole

9 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

SOUND LEVELS (2/2) INTERFERENCE OF TWO WAVES z sound level which results from two non-correlated sources

⎛ I 1 + I 2 ⎞ L 10 L 10 p p L =10 log ⎜ ⎟ =10 log 10 1 +10 2 1 2 res 10 ⎜ ⎟ 10 () p + p = 2A cos()k x cos ()ωt ⎝ I s ⎠ 1 2 3 standing (stationnary) wave Lres-L1 (dB) 2.5 A cos(-kx+ωt) A cos(kx+ωt)

2

1.5

1

0.5

L2-L1 (dB) 0 -10-9-8-7-6-5-4-3-2-10

(L 2 −L1 ) 10 L res = (L res − L 1 )+ L 1 with L res − L 1 =10 log 10 (1+10 )

INTERFERENCE OF TWO WAVES GENERATION OF AN ULTRASONIC WAVE (1/2) z Laser interferometry

p1 p2 p 1 + p 2 = 2A cos(k x)cos(ωt)

A cos(-kx+ωt) A cos(kx+ωt)

Interferometer which permits to measure the deformations of a piece between two laser exposures Laboratoire Charles Fabry de l'Institut d'Optique, Orsay http://www.iota.u-psud.fr/~roosen/capteur_interferometrique.html

GENERATION OF AN ULTRASONIC WAVE (2/2) Behavior law: example of traction test

A AA z Ultrasonic section S A 0 F L 0 σ = S connexion Test piece non loaded → A → failure -F F AA section S ≈ S0 case A L backing Test piece loaded before appearance of necking

→ A B → BB material -F F AA

L A B I II III section S ≈ S section S < S0 0 0 Test piece loaded after appearance of necking ∆ piezoelectric ε = L L 0 piece A AA section Su << S0 A L protecting piece u zone I Linear Test piece put together again after failure zone II Non zone III Plasticity Transforms an electrical signal into a mechanical vibration and conversely Hooke's law: T = E S

10 C. Potel, M. Bruneau, "Acoustique générale (...)", Ed. Ellipse, collection Technosup, ISBN 2-7298-2805-2, 2006

Relations in isotropic solids SEISMIC WAVES (1/2)

E,ν E,µ λ,µ c 11,c 12

E ν µ()E − 2µ λ λ c ()()1+ ν 1− 2ν 3µ − E 12

E c 11 − c 12 µ µ µ 2()1+ ν 2 2 µ()3λ + 2µ c 12 E E E c 11 − 2 λ + µ c 11 + c 12

E µ E 2 c 11 + 2c 12 B λ + µ 3()1− 2ν 3()3µ − E 3 3

E − 2µ λ c 12 ν ν 2µ 2()λ + µ c 11 + c 12

c 11 , c 12 : Rigidity constants (Pa) 2 E : Young modulus (Pa) B : Voluminal elasticity modulus (Pa/m ) λ , µ : Lamé coefficients (Pa) ν : Poisson ratio (no unit) http://www.ens-lyon.fr/Planet-Terre/Infosciences/Geodynamique/Structure-interne/Sismologie/pendulum.html

SEISMIC WAVES (2/2) Modal waves (1/4)

z Bulk waves z Surface waves local waves Mechanical waves modal waves

propagation sub-space

O x 1 acoustical energy: - propagates along the layers x 2 modal sub-space - bounded in x3

x 3

Š guided waves x Š surface waves 3 Š interface waves http://eost.u-strasbg.fr/pedago/fiche1/ondes_sismiques.fr.html

Modal waves (2/4) Modal waves (3/4):

guided waves surface waves interface waves vacuum Vacuum/ rigid wall / reactive impedance fluid or solid vacuum a) a) solid solid a) Vacuum/ rigid wall / reactive impedance fluid or solid

vacuum fluid or solid

solid b) b) fluid or solid http://www.kettering.edu/~drussell Animation courtesy of Dr. Dan Russell, Kettering University b) Osborne and Hart waves -Rayleighwave a) - Scholte wave, Stoneley wave, Lamb wave - anti-modal wave b) Rayleigh-Cezawa wave, etc... a) - anti-modal wave b)

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k0a = 60, kLa = 15, kTa=30, θ = θRayleigh Displacements of Lamb waves (1/2)

Vacuum

Isotropic solid

Vacuum

pressure waves (symmetric mode) flexure waves (antisymmetric mode) vectorial particle displacement field at the surface of the plate; its effect on the shape of the plate

figures from:

ρ 0 ρ 1 = 0,1 D. Royer et E. Dieulesaint, "Ondes élastiques dans les solides", tome 1 : propagation libre et guidée, Masson, (1996) J.L. Rose, "Ultrasonic waves in solid media",Cambridge Univ. Press, 1999 Programs realised by Ph. Gatignol, Pr., Université de Technologie de Compiègne

Displacements of Lamb waves (2/2) Eau / Aluminium / Eau; ka=170 ; H=5 mm; before the 1st critical angle

pressure waves (symmetric mode) flexure waves (antisymmetric mode) vectorial particle displacement field at the surface of the plate; its effect on the shape of the plate

mode S0 mode A0

animations realised by Patrick Lanceleur, Université de Technologie de Compiègne http://www.utc.fr/~lanceleu/links_CT04.html Lamb mode

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