Chapter 4

ANALYTICAL FORMULATION OF FUNDAMENTAL LINEAR PROBLEMS OF ACOUSTICS, IN HOMOGENEOUS, TIME INDEPENDENT FLUID MEDIA AT REST:

FUNDAMENTAL SOLUTIONS IN CARTESIANCOORDINATES

Solutions of the propagation equation

( ) ( )ω−=ω⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+∆ ;rF;rP

c 20

2 rr( ) ( )t;rft;rp

tc1

2

2

20

rr−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∆

Propagation equation Helmholtz equation

No general solution is known, except in the case of one-dimensional propagation

If the boundaries of the domain coincide with surfaces of separable curvilinear coordinates

solutions with separable variables

"basis" on which any solution can be expanded

complete family

Choice of the coordinate system

plane piston

( )tcosA ω

0 x

A cos ( ω t - k x )

plane of air x

yz

a

source S

x

y

z

Source

O L

a

Cartesian coordinates Cylindrical coordinates

Spherical coordinates

Legendre polynomialscircular functions

Bessel functions

L

Amplitude of waves in Cartesian, cylindrical and spherical coordinates (1/3)

Energy flux SA2

∝Φ proportional to the crossed surface and to the squared amplitude of the wave

constant=Φ 22

212

1 SASA =

Plane wave

S1 S2

1A 2A 21 SS = 21 AA =

Any wave the amplitude of which is independent of the point, in a given space, has a plane character in this space.

constant=Φ

Amplitude of waves in Cartesian, cylindrical and spherical coordinates (2/3)

Cylindrical wave

constantrA2

=

Any wave the amplitude of which decreases such as 1/√r in a given space, has a cylindrical character in this space.

r1r2

pulsating wire

SA2

∝Φ with r2S π∝

rA2

∝Φ and

r1A ∝

Application: going on holidays using a motorway

As a first approach, the noise emitted by the motorway can be modeled by a field having a cylindrical character

Amplitude of waves in Cartesian, cylindrical and spherical coordinates (3/3)

Spherical wave

constantrA 22=

Any wave the amplitude of which decreases such as 1/r in a given space, has a spherical character in this.

SA2

∝Φ with 2r4S π=

22rA∝Φ and

r1A ∝

Application: single car in open country

As a first approach, the car is a punctual source with respect to the house (field with a spherical character)

r1r2

solid angle Ω

OR conservation of the energy flux in a sector of solid angle Ωif r then S and amplitude : divergent wave if r then S and amplitude : convergent wave

punctual source

constant=Φ

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

1

One-dimensional problemsAll the variable of the problem depend on only one coordinate

( ) ( ) ( )t;xˆ,t;xv,t;xp ϕr

fields are uniform in a plane perpendicular to the coordinate

fields of plane waves

x

yc0

t fixed

A cos(-kx+ωt)

Examples

Good choice of coordinate system ( )zyx e,e,e,O rrr=R coordinate x

Good choice of coordinate system ( )zyx e,e,e,O rrr=R coordinate x

Direction of the velocity vector (1D)

( ) ( ) ( ) ( ) zzyyxx et;xvet;xvet;xvt;xv rrrr++=

0pgradtv

0

rr

=+∂∂

ρ

Euler equation

( ) ( ) 0ex

t;xpt

t;xvx0

rrr

=∂

∂+

∂∂

ρ

( ) ( ) 0x

t;xpt

t;xvx0 =

∂∂

+∂

∂ρ

( )0

tt;xvy

0 =∂

∂ρ

( ) 0t

t;xvz0 =

∂∂

ρ

( ) ( )xKt;xv yy =

( ) ( )xKt;xv zz =

( ) ( ) xx et;xvt;xv rr=

( )t;xv

null time averages

= 0

= 0denoted

Equations of acoustics for one-dimensional problems

0pgradtv

0

rr

=+∂∂

ρ

0vdivtˆ

0 =ρ+∂ρ∂ r

ρ= ˆcp 20

Fundamental laws

0ptc

12

2

20

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∆

Propagation equation

( ) ( ) 0x

t;xpt

t;xv0 =

∂∂

+∂

∂ρ

( ) ( ) 0x

t;xvt

t;xˆ0 =

∂∂

ρ+∂ρ∂

( ) ( )t;xˆct;xp 20 ρ=

( ) 0t;xptc

1x 2

2

20

2

2=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

General solution for one-dimensional problemsPropagation equation ( ) 0t;xp

tc1

x 2

2

20

2

2=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

0cxtu −=

0cxtv +=

vp

up

tv

vp

tu

up

tp

∂∂

+∂∂

=∂∂

∂∂

+∂∂

∂∂

=∂∂

vup2

vp

up

tp 2

2

2

2

2

2

2

∂∂∂

+∂∂

+∂∂

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=∂∂

∂∂

+∂∂

∂∂

=∂∂

vp

up

c1

xv

vp

xu

up

xp

0 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂

+∂∂

=∂∂

vup2

vp

up

c1

xp 2

2

2

2

2

20

2

2

0vu

p2v

pu

pc1

vup2

vp

up

c1 2

2

2

2

2

20

2

2

2

2

2

20

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

+∂∂

+∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

−∂∂

+∂∂

0vu

p2=

∂∂∂ ( ) ( )vgufp +=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

00 cxtg

cxtft;xp

( ) ( ) ( )tcxgtcxft;xp 00 ++−=

( ) ( )[ ] ( )[ ]tcxgtcxft;xp 00 +κ+−κ=

Change of variables and

Substitution into propagation equation

General solutionor

or

Particular case: progressive plane waves (1D)

progressive plane wave which propagates following increasing values of x

progressive plane wave which propagates following decreasing values of x

x

yc0

( ) ( )tcxft;xp 0−=

t fixed

→

( ) ( )tcxgt;xp 0+=

x

yc0

t fixed

Particular case (1D): standing plane waves (1/2)

x

( )t;xp at t1at t2

at t3

at t4

( ) ( ) ( )tTxXt;xp =

( )( )

( )( ) t,x,tTt"T

xXx"Xc 2

0 ∀∀=

0ck ω=

( ) ( ) 0xXkx"X 2 =+

( ) ( ) 0tTt"T 2 =ω+

Succession of nodes (here zeroes of pressure) and of antinodes (here maxima of pressure), which evolves with time following the law imposed by the function T(t).

( ) t,x,0t;xptc

1x 2

2

20

2

2∀∀=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

(real acoustic pressure)

depends only on x

depends only on t

2ω−

with Separated variable method

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

2

Particular case (1D): standing plane waves (2/2)

( ) ( ) 0xXkx"X 2 =+

( ) ( ) 0tTt"T 2 =ω+

A standing wave is necessarily sinusoidal with t.

( ) ( ) ( )xksinBxkcosAxX +=

( ) xkixki eDeCxX −+=

( ) ( ) ( )tsinFtcosEtT ω+ω=

tie ω tie ω−

or

or( ) titi eHeGtT ω−ω +=

Practically, choice of a time convention or

One or the other convention leads to the same real result.

progressive plane wave

standing plane wave

Progressive and standing plane waves

( ) ( ) ( ) ( ) ( )[ ]txkcostxkcos2AtcosxkcosAt;xp ω++ω−=ω=

( ) ( ) ( ) ( ) ( ) ( )tsinxksinAtcosxkcosAtxkcosAt;xp ω+ω=ω−=

( ) tixkitxki eee ωω+ =

x0pa

pb

standing plane wave progressive plane wave

BUT ( )[ ] ( ) ( ) ( )tixkitxki eeeetxkcosee ωω+ ≠ω+= RRR

example:

progressive plane wave

standing plane wave

Plane waves (1/4)

( ) ( ) ( )tcrngtcrnft;ru 00i +⋅+−⋅=rrrrr

MOr =r

( )[ ]rntcF 0rr⋅−κ

0OMcosOMMOnrn =θ=⋅=⋅rrr

θ n→

n→

M 0

MO r→

n→

x

y

z

O

R

M

r→

At a given time, at any given point M such as

constantrn =⋅rr

the value of the field variable (physical quantity) is the same.

These points are located in the same plane, which is termed wave plane (plane wave surface), perpendicular to the direction n :→

Plane waves (2/4)

( ) constantrntc 0 =⋅−rr

( ) 0rntcd 0 =⋅−rr

0ctdrdn =⋅r

r

When time varies, follow a given value of

speed to which a geometrical point M must move in order to follow a given value of F

i.e. i.e.

if r // n nctdrd

0r

r

=→→

n→

c0

c0

c0

The wave planes which convey a given value of the field variable F, move parallel to themselves, in the direction n which is perpendicular to them, and with the propagation velocity c0.

→

Plane waves (3/4)Particular case of a periodic plane wave: F periodic with period U

λ

λ

λ

n→

( ) Urrn 12 =−⋅κrrr

κ=λ

U

( )rntcu 0rr⋅−κ=( )[ ]rntcF 0

rr⋅−κ

u

F(u) with

At a given time t, and for two given values r1 and r2 of the position vector r such that

→ → →

F = constantthen

two wave plane which are distant from one to another with a distance equal to U/κ convey the same value of the field

particular case of monochromatic fields: ( )α+⋅−ω rnktcosCrr π= 2U

k2π

=λ

wavelength

λ

M

≈ plane wave

r

Plane waves (4/4)Validity of the plane wave hypothesis

Approximation " quasi plane wave" of a field in the neighboring of an observation point M : case of a spherical field with a far field hypothesis (r >> λ).

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

3

Monochromatic plane waves (1/4)Monochromatic wave: propagation at 1 frequency (angular frequency ω given)

frequency driven by a source emitting permanently

driven motion, steady state, linear acoustics

Any quantity can be written ( ) ( ) tiexUt;xu ω=

( ) ( ) tiexPt;xp ω=

time convention

( ) ( ) tiexVt;xv ω=rr ( ) ( ) tiexˆt;xˆ ωΦ=ϕ, , , ...

Helmholtz equation

( ) 0t;xptc

1x 2

2

20

2

2=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

−∂∂ ( ) t,x,0exP

cxti

2

02

2∀∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+

∂∂ ω

( ) x,0xPcx

2

02

2∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+

∂∂

00 ck ω=

( ) ( ) x,0xPk 20

2xx ∀=+∂

Notation:

Monochromatic plane waves (2/4)

Search for solutions ( ) x,0xPcx

2

02

2∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+

∂∂

equation in the form: (factor independent of x) (function of x) x,0 ∀=

( ) xkixki eBeAxP += −

( ) ( ) x,0xPkk 20

2 ∀=+−

20

2 kk = 0kk = 0kk −=

xkie − xkie

( ) ( ) ( )txkitxki eBeAt;xp ω+ω+− +=

Substitution into Helmholtz equation: 00 ck ω=with

dispersion equation i.e. or

redundant with andchoice: 0kk =

progressive plane wave towards x

progressive plane wave towards x

Monochromatic plane waves (3/4)

Notation

( )T00c χργ=

( ) x,0xPc

Rix

2

02

2∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+ω+

∂∂

( ) ( ) x,0xPkRik 20

2 ∀=+ω+−

Rikk 20

2 ω+=

00 c

k ω=

0kk =Dispersion equation in non dissipative fluid:

the wave number k is not always equal to k0

tpR

∂∂Example: dissipative term in the form

source

propagation medium

density compressibility

( ) t,x,0t;xptc

1t

Rx 2

2

20

2

2∀∀=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂∂

+∂∂

( ) xkixki eBeAxP += −

dispersion equation

Monochromatic plane waves (4/4)

λ

λ

λ

n→

κ=λ

U

( )α+⋅−ω rnktcosCrr

π= 2U

k2π

=λ

wavelength

( )[ ]rntcF 0rr⋅−κ in the form

periodic function with period

Wave number vector nkk rr=

or

wave vector

wave plane ≡ equiphase plane

Interaction of a monochromatic plane wave with a wall characterized by its non-zero admittance, normal incidence (1/5)

x

y

incident wave

O

A

reflected wave

B

material characterized by its impedance Z

monochromatic source at the infinity( )tcosA ω

x

y

incident wave

O

AA

reflected wave

BB

material characterized by its impedance Z

Interaction of a monochromatic plane wave with a wall characterized by its non-zero admittance, normal incidence (2/5)

( ) t,0y,0t;yptc

1y 2

2

20

2

2∀≥∀=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

( ) ( ) tieyPt;yp ω=

( ) 0y,0yPcy

2

02

2≥∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+

∂∂

( ) 00Pˆkin 0 =⎥

⎦

⎤⎢⎣

⎡β+

∂∂ Zcˆ

00ρ=β

( ) 0y,0yPˆkiy 0 ==⎥

⎦

⎤⎢⎣

⎡β+

∂∂

−

yen rr−=

Propagation equation

Incident monochromatic field

Helmholtz equation

Boundary conditions with and

Return wave B which propagates at the infinity^

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

4

Interaction of a monochromatic plane wave with a wall characterized by its non-zero admittance, normal incidence (3/5)

Solutions of the problem

( ) ( ) ( ) ( ) tiykiykiba eeBeAt;ypt;ypt;yp ω−+=+=

( )yP( ) ( )ykiyki eBeAkiyyP −−=

∂∂

0kk =

( ) ( ) 0BAˆkiBAki 0 =+β+−−

( ) ( )β+=β− ˆ1Bˆ1A

β+β−

== ˆ1

ˆ1ABˆ

pR

( ) ( ) ( ) ( )[ ]ykipp

ykip

yki eˆ1ykcosˆ2AeˆeAyP RRR −+=+= −

( ) ( ) ( )[ ] tiykipp eeˆ1ykcosˆ2At;yp ω−+= RR

( ) 0y,0yPˆkiy 0 ==⎥

⎦

⎤⎢⎣

⎡β+

∂∂

−Boundary conditions:

Dispersion equation:

Reflection coefficient

Acoustic pressure

standing part propagating part

If β=0 (perfectly rigid medium):total reflection

^

x

y

incident wave

O

AA

reflected wave

BB

material characterized by its impedance Z

Rp = 0.9: great standing part (ka = 3)

incident field alone reflected field alone

total field

Rp = 0.26: weak standing part (ka = 3)

incident field alone reflected field alone

total field

Rp = 0.99-0.02 i; oblique incidence (ka = 4)

The whole field has a stationary character along y, but has a propagating character along x

incident field alone reflected field alone

Interaction of a monochromatic plane wave with a wall characterized by its non-zero admittance, normal incidence (4/5)

Particle velocity

Euler equation

00 ckk ω==dispersion equation:

Acoustic pressure

( ) ( ) 0t;ypgradt

t;yv0

rr

=+∂

∂ρ

( ) ( ) tieyVt;yv ω=rr

( ) ( ) ( ) ti

00e

yyPi

yt;ypit;yv ω

∂∂

ωρ=

∂∂

ωρ=

( ) ( ) ( ) tiykip

yki

00

tiykip

yki

0eeˆe

cAeeˆeAkt;yv ω−ω− −

ρ−

=−ωρ

−= RR

( ) ( )t;ypgradit;yv0 ωρ

=r

( ) ( )ykip

yki eˆeAyP −+= R

x

y

incident wave

O

AA

reflected wave

BB

material characterized by its impedance Z

Interaction of a monochromatic plane wave with a wall characterized by its non-zero admittance, normal incidence (5/5)

Acoustic intensity ( )** vpvp41I rrr

+=

( )** vpvp41I +=

( )( ) ( ) ( )[ ]yki*p

ykiykip

ykiykip

ykiyki*p

yki

00

*eˆeeˆeeˆeeˆe

c4AAI RRRR −++−+

ρ−

= −−−−

⎥⎦⎤

⎢⎣⎡ −

ρ

−=

2p

00

2

ˆ1c2

AI R

1ˆp =R

here

( ) ( ) tiykip

yki eeˆeAt;yp ω−+= R

( ) ( ) tiykip

yki

00eeˆe

cAt;yv ω−−

ρ−

= Rwith

If total reflection, I = 0

x

y

incident wave

O

AA

reflected wave

BB

material characterized by its impedance Z

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

5

Acoustic field in a finite length waveguide (1D) (1/5)

L x0

pa

pb

t < 0 : acoustic sources in the waveguide

t = 0 : extinction of sources

the solutions are searched in the form of a superposition of monochromatic plane waveswhich satisfy the boundary conditions at the two extremities at x=0 and x=L

perfectly rigid wall perfectly rigid wall

Acoustic field in a finite length waveguide (1D) (2/5)

Propagation equation

Source turned off at t = 0

Boundary conditions

( ) [ ] 0t,L,0x,0t;xptc

1x 2

2

20

2

2≥∀∈∀=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂∂

−∂∂

Boundary conditions

( ) 0t;0v x = ( ) 0t;Lv x =L x

0 pa^

pb^

0t ≥∀ 0t ≥∀

LxTxnand0xTxn =∂∂+=∂∂=∂∂−=∂∂with

ati.e. ( ) 0t,0n

t;xp≥∀=

∂∂ Lxet0x ==

n→ n→

Searched solution: harmonic character

Helmholtz equation

( ) ( ) tiexPt;xp ω=

( )[ ] ( ) [ ]( )⎪⎩

⎪⎨⎧

===∂∈∀=ω+∂

,Lxet0x,0xP,L,0x,0xPc

x

20

2xx

Acoustic field in a finite length waveguide (1D) (3/5)

Solutions of the problem L x0 pa

^

pb^

( ) ( ) ( ) ( ) [ ]L,0x,eeBeAt;xpt;xpt;xp tixkixkiba ∈∀+=+= ω−

( )xP 0kk =with dispersion equation:

( ) tixkixki

00x

0x eeBeA

c1p

i1v ω− +−

ρ−

=∂ρω−

=Particle velocity

Boundary conditions

( )[ ] ( ) [ ]( )⎪⎩

⎪⎨⎧

===∂∈∀=ω+∂

,Lxet0x,0xP,L,0x,0xPc

x

20

2xx ( )

( )⎩⎨⎧

∀=+−∀=+−

ω−

ω

.t,0eeBeA,t,0eBA

tiLkiLki

ti

( )⎩⎨⎧

=+−=

− .0eeA,BA

LkiLki( ) 0Lksini2 =i.e. π= mLk , m ∈i.e.

Lmk mπ

= , m ∈

m,0A m ∀=Trivial solution EXCEPT IF

eigenvalues

eigen angular frequenciesLcm

ck 00mm

π==ω ; eigenfrequencies L2

cm2

f 0mm =

π

ω=

( )Lksini2

Acoustic field in a finite length waveguide (1D) (4/5)

Complex field associated to each mode m

pressure

velocity

( ) ( ) ( ) timm

tixkixkimm

mmmm exkcosA2eeeAt;xp ωω− =+=

( ) ( ) ( ) tim

00

mtixkixki

00

mx

mmmm

mexksin

cAi2

eeec

At;xv ωω−

ρ

−=+−

ρ

−=

Total field: superposition of eigenmodes

pressure

velocity

( ) ( ) ( )∑∑∞

=

ω∞

===

0m

timm

0mm

mexkcosA2t;xpt;xp

( ) ( ) ( )∑∑∞

=

ω∞

= ρ

−==

0m

tim

00

m

0mxx

m

mexksin

cAi2

t;xvt;xv

mimm eAA α=with

Real field associated to each mode m

Real total field: superposition of eigenmodes

Acoustic field in a finite length waveguide (1D) (5/5)

pression

vitesse

pression

vitesse

( ) ( ) ( ) ( )∑∑∞

=

∞

=α+ω==

0mmmmm

0mm tcosxkcosA2t;xpt;xp

( ) ( ) ( ) ( )∑∑∞

=

∞

=α+ω

ρ==

0mmmmm

000mm tsinxksinA

c2t;xvt;xv

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 10 L/4 L/2 3L/4 L

m=1

m=2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 10 L/4 L/2 3L/4 L

m=2

m=1

( ) ( )[ ] ( ) ( )mmmmmm tcosxkcosA2t;xpRet;xp α+ω==

( ) ( )[ ] ( ) ( )mmm00

mmm tsinxksin

c

A2t;xvRet;xv α+ω

ρ==

pressure

velocity

Solutions of 3 dimensional problems (1/5)

Propagation equation ( ) t,r,0t;rptc

12

2

20

∀∈∀=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∆ Vrr

Helmholtz equation ( ) V∈∀=ω⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+∆ r,0;rP

c

2

0

rr

No general solution is known, except in the case of one-dimensional propagation:

( ) ( ) ( )tcrngtcrnft;rp 00 +⋅+−⋅=rrrrr

Solutions with separated variables or integral representation

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

( ) ( ) ( ) ( ) ( )tTzZyYxXt;z,y,xp =

( ) ( ) ( ) ( ) ( )tTzZˆrRt;z,,rp ψΨ=ψ

( ) ( ) ( ) ( ) ( )tTˆˆrRt;,,rp ψΨθΘ=ψθ

Solution with separated variables ≡ Basis on which any solution of the problem can be projected

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

6

Wave equation in Cartesian coordinates

x

y

z

O

M

x

y

z zyx ezeyexMOr rrrr++==

( ) zzyyxx eAeAeAz,y,xA rrrr++=

zyx ezdeydexdMOdrd rrrr++==

zyx ezUe

yUe

xUUgrad

rrr

∂∂

+∂∂

+∂∂

=

zA

yA

xA

Adiv zyx∂

∂+

∂

∂+

∂

∂=

r

zxy

yzx

xyz e

yA

xA

ex

Az

Ae

zA

yA

Arotrrrr

⎥⎦

⎤⎢⎣

⎡∂

∂−

∂

∂+⎥

⎦

⎤⎢⎣

⎡∂

∂−

∂

∂+⎥

⎦

⎤⎢⎣

⎡∂

∂−

∂

∂=

( ) 2

2

2

2

2

2

zU

yU

xUUgraddivU

∂

∂+

∂

∂+

∂

∂==∆ ( ) ( )ArotrotAdivgradA

rrr−=∆

( ) 0t;z,y,xptc

12

2

20

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂−∆ ( ) 0t;z,y,xp

tc1

zyx 2

2

20

2

2

2

2

2

2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂−

∂

∂+

∂

∂+

∂

∂

Propagation equation (out sources)

Usual operators in Cartesian coordinates

and

Solutions of 3 dimensional problems (2/5)

Propagation equation

Solutions with separated variables ( ) ( ) ( ) ( ) ( )tTzZyYxXt;z,y,xp =

( ) ( ) t,z,y,x,0t;z,y,xptc

1zyx 2

2

20

2

2

2

2

2

2∀∈∀=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+∂∂

+∂∂ V

( ) t,z,y,x,tT

c1

T1

zZ

Z1

yY

Y1

xX

X1

2

2

20

2

2

2

2

2

2∀∈∀

∂∂

=∂∂

+∂∂

+∂∂ V

function of x,y,z function of t

20k−= 22

020 ck ω=let

t,0TtT 22

2∀=ω+

∂∂

( ) tieGtT ω=

tie ω

tie ω−

choice of a time convention

Solutions with separated variables - solution as a function of x

Solutions of 3 dimensional problems (3/5)

( ) V∈∀−∂∂

−∂∂

−=∂∂ z,y,x,k

zZ

Z1

yY

Y1

xX

X1 2

02

2

2

2

2

2

function of y,zfunction of x

2xk−=

x,0XkxX 2

x2

2∀=+

∂∂

( ) xkixki xx eBeAxX += −

( ) ( ) ( )xksin'Bxkcos'AxX xx +=

BA'A += ( )ABi'B −=

or

with and

Solutions of 3 dimensional problems (4/5)Solutions with separated variables - solution as a function of y

function of zfunction of y

2yk−=

or

Solutions with separated variables - solution as a function of z

( ) V∈∀+−∂∂

−=∂∂ z,y,kk

zZ

Z1

yY

Y1 2

x202

2

2

2

y,0YkyY 2

y2

2∀=+

∂∂

( ) ykiyki yy eDeCyY += −

( ) ( ) ( )yksin'Dykcos'CyY yy +=

constantfunction of z

2zk−=

or

V∈∀++−=∂∂ z,kkk

zZ

Z1 2

y2x

202

2

z,0ZkzZ 2

z2

2∀=+

∂∂

( ) zkizki zz eFeEzZ += −

( ) ( ) ( )zksin'Fzkcos'EzZ zz +=

Solutions of 3 dimensional problems (5/5)Solutions with separated variables

( ) ( )( ) ( ) tizkizkiykiykixkixki eeFeEeDeCeBeAt;z,y,xp zzyyxx ω−−− +++=

or

X(x) Y(y) Z(z) T(t)

( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( ) .tcoszksin'Fzkcos'E

yksin'Dykcos'Cxksin'Bxkcos'At;z,y,xp

zz

yyxx

ω+⋅

++=

i.e.

( ) ( )( )( ) tizki3

zkiyki2

ykixki1

xki0 eeˆeeˆeeˆeAt;z,y,xp zzyyxx ω−−− +++= RRR

( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( ) .tcoszksin'ˆzkcos

yksin'ˆykcosxksin'ˆxkcos'At;z,y,xp

z3z

y2yx1x0

ω+⋅

++=

R

RRor

only if: 2y

2x

20

2z kkkk ++−=−

20

2z

2y

2x kkkk =++

00 ck ω=with

dispersion equation

or

Waves in 3D infinite space

x

y

z

( )tzkykxki zyxe ω−++−

x

y

z

( )tzkykxki zyxe ω−+−−−( )tzkykxki zyxe ω−−+−x

y

z

( )tzkykxki zyxe ω−++−−x

y

z

( )tzkykxki zyxe ω−−−−x

y

z

( )tzkykxki zyxe ω−−−−−x

y

z

( )tzkykxki zyxe ω−+−−x

y

z

( )tzkykxki zyxe ω−−+−−x

y

z

(1) (2) (3) (4)

(5) (6) (7) (8)

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( ) ,eˆAeˆˆA

eˆˆAeˆA

eˆˆAeˆA

eˆˆˆAeAt;z,y,xp

tzkykxki20

tzkykxki310

tzkykxki210

tzkykxki30

tzkykxki320

tzkykxki10

tzkykxki3210

tzkykxki0

zyxzyx

zyxzyx

zyxzyx

zyxzyx

ω−+−−ω−−+−−

ω−+−−−ω−−+−

ω−−−−ω−++−−

ω−−−−−ω−++−

++

++

++

++=

RRR

RRR

RRR

RRR

( ) ( )tOMkiii

ieAt;z,y,xp ω−⋅−=r

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

7

Reflection and transmission at the interface between two different fluid media (1/3)

x

y

incident wave

O

A1^

reflected wave

B1^

medium 1ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

pa^

pb^

p2^

( )1a1 crntf rr⋅−source at the infinity:

na→

Reflection and transmission at the interface between two different fluid media (2/3)

Medium 1: well-posed problem

( )

t,0y,x

,0t;y,xptc

1yx 12

2

21

2

2

2

2

∀≤∀∀

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+∂∂

( ) ( ) t,0y,x,t;y,xpt;y,xp 21 ∀=∀=

( ) ( ) t,0y,x,nt;y,xvnt;y,xv 21 ∀=∀⋅=⋅rrrr

( ) ( ) t,0y,x,t;y,xy

p1t;y,xy

p1 2

2

1

1∀=∀

∂

∂

ρ=

∂

∂

ρ

( )1a1 crntf rr⋅−

Propagation equation

Boundary conditions

Driven incident field

i.e.

Medium 2: well-posed problem

( )

t,0y,x

,0t;y,xptc

1yx 22

2

22

2

2

2

2

∀≥∀∀

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+∂∂

Propagation equation

Radiation conditions at infinity

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−+⎟

⎟⎠

⎞⎜⎜⎝

⎛ ⋅−=

1

b1

1

a11 c

rntg

crn

tft;y,xprrrr

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−=

2

222 c

rntft;y,xp

rr

Medium 1: solution Medium 2: solution

x

y

incident wave

O

A1^

reflected wave

B1^medium 1

ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

pa^

pb^

p2^

x

y

incident wave

O

A1A1^

reflected wave

B1B1^medium 1

ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

papa^

pbpb^

p2p2^

Reflection and transmission at the interface between two different fluid media (3/3)

Snell-Descartes law

Boundary conditions ( ) ( ) t,0y,x,t;0,xpt;0,xp 21 ∀=∀=

factorization of a function of x and t

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−+⎟

⎟⎠

⎞⎜⎜⎝

⎛ ⋅−=

1

b1

1

a11 c

rntg

crn

tft;y,xprrrr

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−=

2

222 c

rntft;y,xp

rr

with and

equal functions:

and equal arguments: t,0y,x,c

rnt

crn

tc

rnt

2

2

1

b

1

a ∀=∀⋅

−=⋅

−=⋅

−rrrrrr

211 fgf ==

0y,x,c

xn

c

xn

c

xn

2

x

1

x

1

x 2ba =∀==

2

x

1

x

1

x

c

n

c

n

c

n2ba ==

2

2

1

1

1

1

csin

c'sin

csin θ

=θ

=θ

11 'θ=θ

x

y

incident wave

O

A1^

reflected wave

B1^medium 1

ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

pa^

pb^

p2^

x

y

incident wave

O

A1A1^

reflected wave

B1B1^medium 1

ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

papa^

pbpb^

p2p2^

θ1 θ'11c

1

fluid 1

fluid 2

θ2

2c1

x

y

Slowness surfaces

1

1

csinθ

2

2

1

1

csin

c'sin θ=

θ

case c1 < c2

Example: f(X) = cos(kX), ka = 4, θ<θc

transmitted wave

total field having a standing character following z but a propagating character following x

transmitted wave

Reflection and transmission at the interface between two different fluid media - harmonic source (1/12)

angular frequency ω

yyxxa1

a1a ekeknc

nkk1

rrrrr+=

ω== yyxxb

1b1b ekekn

cnkk

1

rrrrr−=

ω==

yyxx22

222 ekeknc

nkk2

rrrrr+=

ω==

( ) ( )tOMki1a

aeAt;y,xp ω−⋅−=r

( ) ( )tOMki1b

beBt;y,xp ω−⋅−=r

( ) ( )tOMki22

2eAt;y,xp ω−⋅−=r

yx eyexMO rr+=

x

y

incident wave

O

A1^

reflected wave

B1^

medium 1ρ1, c1

medium 2ρ2, c2

transmitted wave

θ1 θ'1

θ2

pa^

pb^

p2^

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

8

Reflection and transmission at the interface between two different fluid media - harmonic source (2/12)

( ) ( ) ( ) ( ) ( )tykxki1

tykxki1ba1

1yx1yx eBeAt;y,xpt;y,xpt;y,xpω−−−ω−+−

+=+=

( ) ( )tykxki22

2yxeAt;y,xpω−+−

=

( ) ( ) ( ) t,x,eAeBeA txki2

txki1

txki1

xxx ∀∀=+ ω−−ω−−ω−−

211 ABA =+

Equality of pressures at y = 0

( ) ( )txkiyki1

yki1y

1 x1y1y

1eeBeAkit;y,x

yp ω−−−

⎥⎦⎤

⎢⎣⎡ +−=

∂

∂

( ) ( )txkiyki2y

2 x2y

2eeAkit;y,x

yp ω−−−

−=∂

∂

1ˆˆpp =+− TR with 11p ABˆ =R 12p AAˆ =Tand

Equality of normal velocities at y = 0

( ) ( ) ( ) t,0y,x,eAki

eBAki txki

22

ytxki11

1

y x2x1 ∀=∀ρ

−=+−

ρω−−ω−−

1

yp

2

yp

1

y 121kˆkˆk

ρ=

ρ+

ρTR

Reflection and transmission at the interface between two different fluid media - harmonic source (3/12)

Reflection and transmission coefficients (pressure amplitude)

1ˆˆpp =+− TR

1

yp

2

yp

1

y 121kˆkˆk

ρ=

ρ+

ρTR

1y2y

1y2yp

12

12

kk

kkˆρ+ρ

ρ+ρ−=R

1y2y

1yp

12

1

kk

k2ˆρ+ρ

ρ=T

11y coskk1

θ= 22y coskk2

θ=111 cZ ρ= 222 cZ ρ=

1122

1122p ZcosZcos

ZcosZcosˆθ+θ

θ+θ−=R

1122

11p ZcosZcos

Zcos2ˆθ+θ

θ=T

2211

1122p cosZcosZ

cosZcosZˆθ+θ

θ−θ=R

2211

22p cosZcosZ

cosZ2ˆθ+θ

θ=T

or or

with

;; characteristic impedances

Example: ρ1=2000 kg/m3, c1=750 m/s, ρ2=2500 kg/m3, c2=1500 m/s

critical angle: 30°

reflection coefficient (pressure amplitude)

transmission coefficient (pressure amplitude)

0

0.2

0.40.6

0.81

1.2

-180

-90

0

90

180

incident angle (degree)0 10 20 30 40 50 60 70 80 90

phas

e (d

egre

e)

0

0.2

0.40.6

0.81

1.2

-180

-90

0

90

180

0 10 20 30 40 50 60 70 80 90

phas

e (d

egre

e)m

odul

us

mod

ulus

Z1 < Z2

incident angle (degree)

Example: ρ1=3000 kg/m3, c1=750 m/s, ρ2=1000 kg/m3, c2=1500 m/s

reflection coefficient (pressure amplitude)

Z1 > Z2 critical angle: 30°

Example: ρ1=3000 kg/m3, c1=750 m/s, ρ2=1000 kg/m3, c2=1500 m/s

Z1 > Z2

transmission coefficient (pressure amplitude)

critical angle: 30°

Reflection and transmission at the interface between two different fluid media - harmonic source (4/12)

Evanescent waves (1/4)

θ1 θ'1 1c1

fluid 1

fluid 2

θ2 2c1

x

y

θc θc 1c1

2c1

fluid 1

fluid 2

y

x

1c1

2c1

fluid 1

fluid 2

y

x

θ1 θ'1

θ2c1 > c2

21c ccsin =θc1 < c2

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

9

Reflection and transmission at the interface between two different fluid media - harmonic source (5/12)

m→ = kω

→slowness vector:

2x2

2

xx kk

kc1k

m >⇒ω

=>ω

=

Dispersion relation:0kkkkkk 2

x22

2y

22

2y

2x 22

<−=⇒=+

k y2 = i k"y222

2xy kk"k

2−−=

y

xk'2→

k"2

→

equiphase plane

equiamplitudeplane

k"y2< 0

k"y2> 0radiation to the infinity criterion

k x > k 2

y

x

k'2→

k"2

→

x

y

m x =k x

ω

m a→ mb

→1c

1

2c1

θ1 θ1

( ) ( )txkiy"kp12

x2y eeˆAt;y,xp ω−−= T

22yyxx2 "ki'kekekk2

rrrrr+=+=

Evanescent waves (2/4)

Example: f(X) = cos(kX), ka = 4, θ=θc

total field having a standing character following z but a propagating character following xevanescent

transmitted wave

evanescent transmitted wave

Example: f(X) = cos(kX), ka = 4, θ>θc

evanescent transmitted wave

total field having a standing character following z but a propagating character following x

evanescent transmitted wave

Reflection and transmission at the interface between two different fluid media - harmonic source (6/12)

( ) ( )[ ] ( ) ( )[ ]y2yx2x2y2yx2x2222 ek"kiekkVekkekkVnVV22

rrrrrr+=+==

( )αiexpV2

( ) ( ) ( )[ ] ( ) ( )α+ω+−+= txkiexpy"kexpek"kiekkVt;y,xv xyy2yx2x22 22

rrr

( ) ( ) ( ) ( )α+ω+−== txkcosy"kexpkkVvRev xy2x2xx 222

( ) ( ) ( ) ( )α+ω+−−== txksiny"kexpk"kVvRev xy2y2yy 2222

( ) ( ) ( ) ( ) 1y"kexpk"kV

v

y"kexpkkV

v2

y2y2

y

2

y2x2

x

22

2

2

2 =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

2y"ki

2y

2y

22

2x 22

"k"kkk >+=α+ω+−=θ txk x

θ=0 0v;0v22 yx =>

θ=π/2 0v;0v22 yx >= ( )0"k

2y <θ=0

θ=π/2

θ=π

θ=-π/2

Evanescent waves (3/4)

Reflection and transmission at the interface between two different fluid media - harmonic source (7/12)

1y2y

1y2yp

12

12

k"ki

k"kiˆρ+ρ

ρ+ρ−=R 1ˆ

p =R

1y2y

1y2yp

12

12

kk

kkˆρ+ρ

ρ+ρ−=Rreflection coefficient: with 22 yy "kik =

BUT presence of acoustic energy in the transmission medium 2.

Evanescent wave ≡ Accompanying wave in medium 2, which accompanies the total reflection phenomenon in the medium 1, and which has it own energy..

Monochromatic problem ≡ steady state problem: during the transient state, the energy incoming in the medium 2 is permanently located in this medium, during an infinite time.

Thus, in reality, the monochromatic problem is an asymptotic situation, for a very high time t, which models too strongly the physical reality, which implies the existence of an evanescent wave, which infinite extension in the direction of the interface..

total reflection in medium 1

Evanescent waves (4/4)

Reflection and transmission at the interface between two different fluid media - harmonic source (8/12)

Incident energy flux

( ) ( )txkiyki1a

x1y eeAt;y,xp ω−−−=

( ) ( )txkiykip1b

x1y eeˆAt;y,xp ω−−= R

( ) ( )txkiykip12

x2y eeˆAt;y,xp ω−−−= T

( ) ( ) ( )txkiyki1

1

ya

1y

x1y1

aeeA

k

yt;y,xpit;y,xv ω−−−

ωρ=

∂

∂

ωρ=

( ) ( ) ( )txkiykip1

1

yb

1y

x1y1

beeˆA

k

yt;y,xpit;y,xv ω−−

ωρ

−=

∂

∂

ωρ= R

( ) ( ) ( )txkiykip1

2

y2

2y

x2y2

2eeˆA

k

yt;y,xpit;y,xv ω−−−

ωρ=

∂

∂

ωρ= T

Reflected energy flux

Transmitted energy flux

( )ωρ

=+=1

y21

*yay

*ay

1

aaa

kA

21vpvp

41I

( ) 2pyy

E ˆIIab

RR =−=

0IIa2 yy

E ==T

( )ωρ

−=+=1

y2p

21

*yby

*by

1

bbb

kˆA21vpvp

41I R

( ) ( ) ykki*yy

2

2

p2

1*y2y

*2y

*2y2y

22222ekk1ˆA

41vpvp

41I

⎟⎠⎞⎜

⎝⎛ −−

+ωρ

=+= T

reflection coefficient (energy)

2

p22

11yy

E ˆcosZcosZ

IIa2

TTθ

θ==

transmission coefficient (energy)

θ < θc θ > θc

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

10

Example: ρ1=2000 kg/m3, c1=750 m/s, ρ2=2500 kg/m3, c2=1500 m/scoefficients (energy)

0

0.2

0.40.6

0.81

1.2

0

0.2

0.40.6

0.81

1.2

angle (degree)0 10 20 30 40 50 60 70 80 90

reflection

transmission

Z1 < Z2 critical angle: 30°

Example: ρ1=3000 kg/m3, c1=750 m/s, ρ2=1000 kg/m3, c2=1500 m/s

Z1 > Z2

reflection coefficient (energy)

critical angle: 30°

transmission coefficient (energy)

Reflection and transmission at the interface between two different fluid media - harmonic source (9/12)

Conservation of energy

( )a2b yyy III =+−0III

2ba yyy =+−−

x

y

O

Ia

→medium 1ρ1, c1

medium 2ρ2, c2

Ib

→

I2

→

n1→

n3→

n4→

n2→ (Σ2)σ

(Σ1)0Idiv =

r 0SdnIS

tot =⎮⌡⌠

⎮⌡⌠ ⋅=Φ

rr

0dnIdnI21

222111 =⎮⌡⌠

⎮⌡⌠ Σ⋅+⎮⌡

⌠⎮⌡⌠ Σ⋅

ΣΣ

rrrr

( ) 0dIdII2

21

ba 2y1yy =⎮⌡⌠

⎮⌡⌠ Σ+⎮⌡

⌠⎮⌡⌠ Σ+−

ΣΣ

(S) = Σ1+Σ2+σ

i.e.

i.e.

∞ 0

with

Reflection and transmission at the interface between two different fluid media - harmonic source (11/12)

Transmitted energy flux

2

22

p2

1y ZcosˆA

21I

2

θ= T

2

1

1

2

1

22

p2

p kk

coscosˆˆ1

ρ

ρ

θ

θ+= TR

2

p22

112p

ˆcosZcosZˆ1 TR

θ

θ+=

θc θc 1c1

2c1

fluid 1

fluid 2

y

x

θ1 θ'1 1c1

fluid 1

fluid 2

θ2 2c1

x

y

( ) ( ) ykki*yy

2

2

p2

1*y2y

*2y

*2y2y

22222ekk1ˆA

41vpvp

41I

⎟⎠⎞⎜

⎝⎛ −−

+ωρ

=+= T

θ < θc θ > θc

0I2y =

or

conservation of energy:

("Light") 1/2 infinite fluid

"light" fluid Fo / "heavy " fluid F1

("Heavy") 1/2 infinite fluid

Fo

F1

Gaussian emitter: ⎥⎦

⎤⎢⎣

⎡−= 2

2

oo axexpW)x(W

a

)x(Wo

x

Gaussian beam incident onto an interface

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Sub-critical incidence, high frequency (k0a = 40)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

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

11

Sub-critical incidence, low frequency (k0a = 10)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Almost critical incidence, high frequency (k0a = 40)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Almost critical incidence, low frequency (k0a = 10)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Critical incidence, high frequency (k0a = 40)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Critical incidence, low frequency (k0a = 10)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Overcritical incidence, high frequency (k0a = 40)

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

5,010 =ρρ

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

12

Overcritical incidence, low frequency (k0a = 10)

5,010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Reflection and transmission at the interface between two different fluid media - harmonic source (12/12)

Interface air - water

21 ZZ << 1ˆp ≈R

2ˆp ≈T

21 ZZ >> 1ˆp −≈R

0ˆp ≈T

1122

1122p ZcosZcos

ZcosZcosˆθ+θ

θ+θ−=R

1122

11p ZcosZcos

Zcos2ˆθ+θ

θ=T

2211

1122p cosZcosZ

cosZcosZˆθ+θ

θ−θ=R

2211

22p cosZcosZ

cosZ2ˆθ+θ

θ=T

But 0IIay2

E ≈=TInterface water - air

Sub-critical incidence, high frequency (k0a = 40), Z0<<Z1

0,0000110 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Sub-critical incidence, low frequency (k0a = 10), Z0<<Z1

0,0000110 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Sub-critical incidence, high frequency (k0a = 40), Z0>>Z1

50010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Sub-critical incidence, low frequency (k0a = 10), Z0>>Z1

50010 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

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

13

z

x

l A BMonochromatic

source S ω

zs 0

Bidimensionnel waveguide (1/8) Bidimensionnel waveguide (2/8)

Propagation equation

Given incident monochromatic field (not specified)

Helmholtz equation

Boundary conditions at and

Field propagating in the direction of increasing z (no return wave)

( ) [ ] t,0z,,0x,0t;z,xptc

1zx 2

2

20

2

2

2

2∀≥∀∈∀=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

+∂∂

l

z

x

l A B

z

x

l AA BB

( ) ( ) tiez,xPt;z,xp ω=

( ) [ ] 0z,,0x,0z,xPczx

2

02

2

2

2≥∀∈∀=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ ω+

∂∂

+∂∂

l

( ) t,0z,0n

t;z,xp∀≥∀=

∂∂ 0x = l=x

( )l==∀≥∀=

∂∂ xand0xat,t,0z,0

xz,xP

Bidimensionnel waveguide (3/8)

Form of the field

Dispersion equation

Boundary conditions

z

x

l A B

z

x

l AA BB

( )l==∀≥∀=

∂∂ xand0xat,t,0z,0

xz,xP

( ) ( ) ( )[ ] tizkxkizkxki eeBeAt;z,xp zxzx ω+−+−− +=

20

2z

2x kkk =+ 00 ck ω=with

with( ) ( ) ( )[ ] tizkxkizkxki

x eeBeAkix

t;z,xp zxzx ω+−+−− −=∂

∂

( ) t,0z,0xen,eeBAki tizkix

z ∀≥∀=− ω−

( ) t,0z,xen,eeeBeAki tizkikikix

zxx ∀≥∀=− ω−−l

ll

0BA =−

0eBeA xx kiki =− − ll

BA=

( ) 0eeA xx kiki =− − ll

( )lxksini2

, m∈π= mk x l

Bidimensionnel waveguide (4/8) Solutions of the problem

m,0A m ∀= EXCEPT IF kx takes a series of eigenvaluesl

π=

mkmx , m∈

to which a series of wave numbers kzmis associated (dispersion equation) such that

2x

20

2z mm

kkk −= i.e.22

0

2z

mc

km

⎟⎠⎞

⎜⎝⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ω=

l

depends on m does not depend on m depends on m

2 cases : 0k 2zm

> 0k 2zm

<or

Bidimensionnel waveguide (5/8) Pressure associated to each mode m

( ) tizkixkixkimm eeeeAt;z,xp mzmxmx ω−−

⎟⎠⎞⎜

⎝⎛ += ( ) ( ) tizki

xmm eexkcosA2t;z,xp mz

m

ω−=

standing wave field having a modal character following Ox, and which is moved parallel to Ox, following Oz.

( ) ( ) ( ) ti

0m

zkixm

0mm eexkcosA2t;z,xpt;z,xp mz

m

ω∞

=

−∞

=⎟⎠⎞

⎜⎝⎛== ∑∑Total field

x

y

z

0

t fixedwalls of the guide

λzm/2

λzm/2

( )t;z,xvmx

Bidimensionnel waveguide (6/8) Velocity associated to mode m

( ) ( )t;z,xpgradit;z,xv m0

m ωρ=

r( ) ( ) tizki

xm0

xx eexksinA

ki2t;z,xv mz

m

m

m

ω−

ωρ−=

( ) ( ) tizkixm

0

zz eexkcosA

k2t;z,xv mz

m

m

m

ω−

ωρ=

Total velocity field

( ) ( ) ( ) ti

0m

zkixxm

00mxx eexksinkAi2t;z,xvt;z,xv mz

mmm

ω∞

=

−∞

=⎟⎠⎞

⎜⎝⎛

ωρ−== ∑∑

( ) ( ) ( ) ti

0m

zkixzm

00mzz eexkcoskA2t;z,xvt;z,xv mz

mmm

ω∞

=

−∞

=⎟⎠⎞

⎜⎝⎛

ωρ== ∑∑

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

14

Bidimensionnel waveguide (7/8) Propagating and evanescent modes

l

π=

mkmx , m∈

22

0

2z

mc

km

⎟⎠⎞

⎜⎝⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ω=

l

0k 2zm

>

0k 2zm

<

i.e.

with

0cml

π>ωmodes m such that propagating modes m

( ) ( )[ ] ( ) ( )mzxmmm zktcosxkcosA2t;z,xpRet;z,xpmm

α+−ω==and

i.e. 0cml

π<ωmodes m such that evanescent modes m

and

mimm eAA α=

( ) ( )20

2"zz cmikik

mmω−π−== l

( ) ( )[ ] ( ) ( )mxzk

mmm tcosxkcoseA2t;z,xpRet;z,xpm

"mz α+ω==

with

0cl

π0 0c2l

π0c3

l

π ( )0c1m

l

π−0cm

l

π ( )0c1m

l

π+

ωsource

propagating modes evanescent waves

with

evanescent modes

Bidimensionnel waveguide (8/8) Plane mode

0z c

k0m

ω=

=

0cl

π0 0c2l

π0c3

l

π ( )0c1m

l

π−0cm

l

π ( )0c1m

l

π+

ωsource

propagating modes evanescent modes

( ) ( )0z00m zktcosA2t;z,xp0

α+−ω==

0k0mx =

=

Propagating plane wave in the direction of increasing z

0cl

π0 0c2l

π0c3

l

π ( )0c1m

l

π−0cm

l

π ( )0c1m

l

π+

ωsource

propagating plane mode

Phase velocity (propagating modes): 1/4

wave cross sectionl

zxy

λzm

λzm

cϕm→λzm

( ) ( )220zz mc2k2

mmlπ−ωπ=π=λ

constantezkt mz m=α+−ω

mzktdzd ω=

( ) ( )220

zmc

kc

mm

lπ−ωω=ω

=ϕ

( ) ( )[ ] ( ) ( )mzxmmm zktcosxkcosA2t;z,xpRet;z,xpmm

α+−ω==

mc ϕ

displacement velocity of a particular observer for whom the wave appears to be fixed (observer linked to a "wave cross section") :x

y

z

0

t fixedwalls of the guide

λzm/2

λzm/2

( )t;z,xvmx

x

y

z

0

t fixedwalls of the guide

λzm/2

λzm/2

( )t;z,xvmx ( )t;z,xvmx →

c ϕmx

z

c 0 n→

k zm

→k

Phase velocity (propagating modes): 2/4

x

z

c 0 n→k zm

=0

→k

→c ϕm

→∞

wave plane

( ) ( )220

zmc

kc

mm

lπ−ωω=ω

=ϕ

Phase velocity (propagating modes): 3/4( ) ( )22

0z

mck

cm

mlπ−ωω=

ω=ϕ

220 mcc

m−νν=ϕ

cωω=ν l0c cπ=ω

or else

with and

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9 10

ω/ωc

c ϕm

/c0

m=1 m=2 m=3m=4

m=5 m=6m=0

1

Phase velocity (propagating modes): 4/4

M

any observation direction

O

direction perpendicularto wave planes

wave plane

k→

kωn→

"Canonical" or intrinsic phase velocity: estimated in the direction perpendicular to the wave planes

ncnk

c 0canorrr

=ω

=ϕ

0ck

cm

=ω

≥ϕr

kcn

m

ω=⋅ ϕ

rr

guided wave:x

→c ϕm

z

c 0 n→

k zm

→k

Phase velocity estimated parallel to the guide

mm

zkc ω

=ϕ

→c ϕm

Generalization

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

15

x→c ϕm

z

c 0 n→

k zm

→k

→c gm

k

kcencc m

m

z0z0g =⋅=

rr ( ) ( )220

20

g mcc

cm

lπ−ωω

=

Group velocity (1/2)

⎟⎟⎠

⎞⎜⎜⎝

⎛

ω∂

∂=

∂ω∂

= m

mm

z

zg

k1

kcand

Group velocity (2/2)

( ) ( )220

20

g mcc

cm

lπ−ωω

= ν−ν= 220g mcc

m

cωω=νl0c cπ=ωor else

withand

0

1

2

3

0 1 2 3 4 5 6 7 8 9 10

ω/ωc

m=1

m=2m=3 m=4 m=5 m=6

m=0

0g ccm

0ccmϕ

("Heavy") fluid layer

1/2 infinite ("light") fluid

("heavy") fluide layer immersed in a "light" fluidGaussian beam incident onto a fluid layer

Fo

Fo

F1

null pressure condition

θm

z

x

l

π=

mkmx , m∈

22

1z

mc

km

⎟⎠⎞

⎜⎝⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ω=

l

mzk

22

000

zm

mck

1k

ksin m ⎟

⎠⎞

⎜⎝⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ω==θ

l

the more m increases, the more θm decreases propagating modes

Gaussian emitter:⎥⎦

⎤⎢⎣

⎡−= 2

2

oo axexpW)x(W

a

)x(Wo

x

null pressure condition

1/2 infinite ("light") fluid

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

high frequency (k0a = 40), θ = θm5

2,010 =ρρ 2,012 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

low frequency (k0a = 40), θ = θm4

2,010 =ρρ 2,012 =ρρ

Softwares realized by Ph. Gatignol, Pr., Université de Technologie de Compiègne - France

Horns (1/2)Phonographs

Edison Home Phonograph (1910)http://perso.wanadoo.fr/jlf/phonos.htm http://perso.wanadoo.fr/jlf/phonos.htm

Gramophone N° 9 (1904)

Megaphones

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

16

Horns (2/2)Acoustic cornets

http://www.inrp.fr/she/instruments/lyc_bdb/acoustique.htm

http://www.beethoven-france.org/Beethoven/Luwig-van-Beethoven.html

Experiments made in 1826 on the Geneva Lake by the physicists Colladon and Sturm

Propagation in horns, 1 parameter theory (1/6)Hypotheses

quasi-plane equiphase surfacesslow variation of the radius as a function of xparticle velocity orientated following x

r x

Series of discrete elementary cylindrical waveguide with perfectly rigid walls

d x

x

Propagation in horns, 1 parameter theory (2/6)Conservation of the velocity flow at the discontinuity

S1

S2

h

x

δ S

δ S

0Sdv =⎮⌡⌠

⎮⌡⌠ ⋅

Σ

r

SSS 21 δ++=Σ

⎮⌡⌠

⎮⌡⌠ σ⋅ρ−=⎮⌡

⌠⎮⌡⌠

⎮⌡⌠

∂ρ∂

Σdnvd

trr

VV

conservation of mass law:

0t

Sh 2 →∂ρ∂

≈

with h→0

⎮⌡⌠

⎮⌡⌠ σ⋅ρ+⎮⌡

⌠⎮⌡⌠ σ⋅ρ+⎮⌡

⌠⎮⌡⌠ σ⋅ρ=⎮⌡

⌠⎮⌡⌠ σ⋅ρ

δΣ S0

S0

S0 dnvdnvdnvdnv

21

rrrrrrr

0→

( )22110 SvSv +−ρ≈

11 Sv−≈ 22 Sv+≈

2211 SvSv = constantSv =i.e. i.e.

Propagation in horns, 1 parameter theory (3/6)Propagation equation

Conservation of the velocity flow at the discontinuity

constantSv = i.e.SSd

vvd

−= xdxdSd

Svvd −=

Use of the velocity potentialx

v∂ϕ∂

=

xdxS

xdSdx

d ⎟⎟⎠

⎞⎜⎜⎝

⎛∂ϕ∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂ϕ∂

x'1 ∂

ϕ∂=ϕ

xdSd'S = xd'

S'S'd 1 ϕ−=ϕLet and

"responsibility" of the section change for the elementary variation of the derivative dϕ'1 of the velocity potential

Between two changes of section

2

2

20

2

2

tc1

x ∂ϕ∂

=∂

ϕ∂i.e. xd

tc1'd 2

2

20

2 ∂ϕ∂

=ϕx

'2 ∂ϕ∂

=ϕwith

Propagation in horns, 1 parameter theory (4/6)Propagation equation - follow-up

Total variation dϕ' along the length dx

xdtc

1'S'S'd'd'd 2

2

20

21⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂ϕ∂

+ϕ−=ϕ+ϕ=ϕ

0tc

1'S'S" 2

2

20

=∂ϕ∂

−ϕ+ϕ

tp 0 ∂

ϕ∂ρ−= 0

xp

xSln

tp

c1

xp

2

2

20

2

2=

∂∂

∂∂

+∂∂

−∂∂

Equation known as Webster's equation (suggested first by Lagrange and Bernoulli)

0tc

1xtS

'Sxt 3

3

20

2

2=

∂

ϕ∂−

∂ϕ∂

∂∂

+∂

ϕ∂∂∂

( )01 ρ−×

t∂∂

Propagation in horns, 1 parameter theory (5/6)Solutions for infinite exponential horns ( ) x2

0 eSxS α=

e eikx i t± ω

0c

ki2k 20

22 =

ω−αm

220

2

cik α−

ω±α±=

tix

ci

x eee2

20

2

ωα−

ω−

α−tix

ci

x eee2

20

2

ωα−

ω

α−

Form of solutions:

0tc

1'S'S" 2

2

20

=∂ϕ∂

−ϕ+ϕSubstitution in the propagation equation:

Non divergent physic solutions of the problem

and

xamplitude increases

acoustic cornets

xamplitude decreases

megaphone

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

17

Propagation in horns, 1 parameter theory (6/6)Solutions for infinite exponential horns ( ) x2

0 eSxS α=

2c

0

220

2

ff

1

c

c

c

⎟⎠⎞

⎜⎝⎛−

=

α−ω

ω=ϕ

π

α=

2c

f 0c2

c0g f

f1c

kc ⎟

⎠⎞

⎜⎝⎛−=

∂∂ω

=

Phase velocity:

Group velocity:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

ρ=

2c

*2

c

00

*

ff

1ff

1c4

ppIAcoustic intensity:

cff ≤

cff >

0I =2

c

00

x2

ff

1c2

eI ⎟⎟⎠

⎞⎜⎜⎝

⎛−

ρ=

α−

200

x2

200

**0

c2e

c4ppvv

4E

ρ=

ρ+

ρ=

α−

( ) g2

c0 cf/f1cEI

=−=

If :

If :

energy density:

cut frequency:

Example of a "fold-up" horn (1/2)

Example of a "fold-up" horn (2/2)

Figure from Mario Rossi, Traité d'électricité, Volume XXI, Electroacoustique, Presses Polytechniques Romandees, 1986

Example of tuba

http://www.gleblanc.com/

Antonín DvořákSymphony of new world, beginning of the 2nd movement (Largo)

Example of Alp-horn

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

18

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

19