Solutions of the propagation...
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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