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DoctoraatDoctoraat Siegfried Siegfried VanaverbekeVanaverbeke::
Mode Theory applied to the reflection and transmission of boundeMode Theory applied to the reflection and transmission of bounded inhomogeneous d inhomogeneous ultrasonic waves on small inclusions in plates and on thin coatiultrasonic waves on small inclusions in plates and on thin coating layersng layers
•• Geboren op 21 Januari 1975 in TorhoutGeboren op 21 Januari 1975 in Torhout•• Kandidaturen Natuurkunde aan Kandidaturen Natuurkunde aan KULeuvenKULeuven Campus Campus KortrijkKortrijk(1993(1993--1995)1995)•• Licenties Natuurkunde aan KUL(1995Licenties Natuurkunde aan KUL(1995--1997)1997)•• IWTIWT--bursaalbursaal KULeuvenKULeuven Campus Campus KortrijkKortrijk(1999(1999--2002)2002)•• Verdediging op 13 december 2002Verdediging op 13 december 2002•• Promotor: Prof. Dr. Promotor: Prof. Dr. Dr.Dr. H. C. O. H. C. O. LeroyLeroy
Relevante referenties: Relevante referenties:
S. S. VanaverbekeVanaverbeke, O. , O. LeroyLeroy, G. , G. ShkerdinShkerdin, , JASAJASA, 114, 601(2003), 114, 601(2003)
S. S. VanaverbekeVanaverbeke, O. , O. LeroyLeroy, , JASAJASA, 113, 73(2003), 113, 73(2003)
SiegfriedSiegfried VanaverbekeVanaverbeke
• Normal Mode Theory
REFLECTED AND TRANSMITTED BEAM PROFILES IN A LAMB ANGLE
⎥⎦⎤
⎢⎣⎡ −= )(erfce
2h1)z(u)z(u n
2n
ir γγπ )(erfce2h)z(u n
2n
t γγπ±=
2iph
wz
0n
++−=γ 0nwh α= 0ni w)kk(p −=
RAYLEIGH ANGLE INCIDENCE: 0down =α
⎥⎦⎤
⎢⎣⎡ −= )(erfche1)z(u)z(u n
2n
ir γγπ
• Normal Mode Theory
REFLECTED AND TRANSMITTED BEAM PROFILES IN A LAMB ANGLE
⎥⎦⎤
⎢⎣⎡ −= )(erfce
2h1)z(u)z(u n
2n
ir γγπ )(erfce2h)z(u n
2n
t γγπ±=
2iph
wz
0n
++−=γ 0nwh α= 0ni w)kk(p −=
RAYLEIGH ANGLE INCIDENCE: 0down =α
⎥⎦⎤
⎢⎣⎡ −= )(erfche1)z(u)z(u n
2n
ir γγπ
• Normal Mode Theory
MODE AMPLITUDE EQUATION
updownn ααα +=
2
y,nin
2
up )2d(u
cosP4Zθ
ωα =2
y,nin
2
down )2d(u
cosP4Z
−=θ
ωα
)z(a)2d(u)
2d(u
cosP4Z
P2
)z(u)2d(uZ
)z(aikz
)z(a
n
2
y,n
2
y,nin
2
n
i*
y,n2
nnn
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+−
−=−∂
∂
θω
ω
• Normal Mode Theory
MODE AMPLITUDE EQUATION
updownn ααα +=
2
y,nin
2
up )2d(u
cosP4Zθ
ωα =2
y,nin
2
down )2d(u
cosP4Z
−=θ
ωα
)z(a)2d(u)
2d(u
cosP4Z
P2
)z(u)2d(uZ
)z(aikz
)z(a
n
2
y,n
2
y,nin
2
n
i*
y,n2
nnn
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+−
−=−∂
∂
θω
ω
INTRODUCTION: REFLECTION AND TRANSMISSION OF BOUNDED ULTRASONIC WAVES
θi=not a critical angle
transducer
θi
PLATE
LIQUID
reflected amplitude distribution
reflected phase distribution
LIQUID
transmitted sound field
x
z
y
INTRODUCTION: REFLECTION AND TRANSMISSION OF BOUNDED ULTRASONIC WAVES
θi=critical angle; Lamb angle of the plate
transducer
θi
PLATE
LIQUID
reflectedamplitude distribution: 2 lobes
transmittedsound field
reflectedphasedistributionphaseshift of 180 degrees
LIQUID
z
yx
transducer
θi=Rayleigh angle
SUBSTRATE
• Rayleigh Phase Technique
Basic principle
COATING
With coatingNo coating
Amplitude distribution
Phase distribution
RAYLEIGH
WAVE
transducer
θi=Rayleigh angle
SUBSTRATE
• Rayleigh Phase Technique
Basic principle
COATING
With coatingNo coating
Amplitude distribution
With coatingNo coating
Amplitude distribution
Phase distribution
Phase distribution
RAYLEIGH
WAVE
• Rayleigh Phase Technique
Comparison with Fourier theory
[ ]
)(erfc)exp(h1
(erfc)exp(1hp
)(erfc)exp(h1Arg
0n20n0
)0n20n0n
0
0n20n0
γγπ
γγγπΔ
γγπφ
−
−+
−=First order approximation for the
phase
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80
Pha
se s
hift(
degr
ees)
Coating thickness(µm)
w=4 mm,f=4 MHz,copper/steel, θi=30.968°
Fourier model
NMT
First order approx.
• Rayleigh Phase Technique
Comparison with Fourier theory
[ ]
)(erfc)exp(h1
(erfc)exp(1hp
)(erfc)exp(h1Arg
0n20n0
)0n20n0n
0
0n20n0
γγπ
γγγπΔ
γγπφ
−
−+
−=First order approximation for the
phase
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60 70 80
Pha
se s
hift(
degr
ees)
Coating thickness(µm)
w=4 mm,f=4 MHz,copper/steel, θi=30.968°
Fourier model
NMT
First order approx.
• Description of bounded inhomogeneous waves
Incident profile
0
0.25
0.5
0.75
1
-40 -30 -20 -10 0 10 20 30 40
z’(mm)
u i(z
’,0)
p/p)wz(zeN)0,z(u
'''
i
−=
β
Full line: gaussian beam(β=0 1/m, p=2)
Dashed line: square profile(β=0 1/m, p=8)
Dotted line: inhomogeneous wave(β=50 1/m, p=8)
β: inhomogeneity parameter
w: half beamwidth
p: behaviour at the edges of the profile
• Description of bounded inhomogeneous waves
Incident profile
0
0.25
0.5
0.75
1
-40 -30 -20 -10 0 10 20 30 40
z’(mm)
u i(z
’,0)
p/p)wz(zeN)0,z(u
'''
i
−=
β
Full line: gaussian beam(β=0 1/m, p=2)
Dashed line: square profile(β=0 1/m, p=8)
Dotted line: inhomogeneous wave(β=50 1/m, p=8)
β: inhomogeneity parameter
w: half beamwidth
p: behaviour at the edges of the profile
• Reflection at a liquid/solid interface
Reflected beam profiles: theory
z
y
LIQUID
SOLID
Fourier Model
∫+∞
∞−
= zike)k(V)k(Rdk21)0,z(u z
zzzr π
Normal Mode Theory
∫∞−
+−++−=
z
RRii0 tt)kk(itp/p)w/t(edt)z(F
αβ
)z(FzzikeN2)0,z(u)0,z(u RRRir
αα −−=
• Reflection at a liquid/solid interface
Reflected beam profiles: theory
z
y
LIQUID
SOLID
Fourier Model
∫+∞
∞−
= zike)k(V)k(Rdk21)0,z(u z
zzzr π
Normal Mode Theory
∫∞−
+−++−=
z
RRii0 tt)kk(itp/p)w/t(edt)z(F
αβ
)z(FzzikeN2)0,z(u)0,z(u RRRir
αα −−=
• Reflection at a liquid/solid interface
Reflected beam profiles: simulations
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Ampl
itude
z(mm)
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Ampl
itude
z(mm)
f=4 MHz, w=25 mm, p=8, Rayleigh angle incidence, water/steel
β=50 1/m β=-50 1/m
Incident profile
Reflected profile Incident profile
Reflected profile
• Reflection at a liquid/solid interface
Reflected beam profiles: simulations
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Ampl
itude
z(mm)
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Ampl
itude
z(mm)
f=4 MHz, w=25 mm, p=8, Rayleigh angle incidence, water/steel
β=50 1/m β=-50 1/m
Incident profile
Reflected profile Incident profile
Reflected profile
• Reflection at a liquid/solid interface
Reflected beam profiles: simulations
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Am
plitu
de
z(mm)
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Am
plitu
de
z(mm)
f=3 MHz, w=20 mm, water/aluminium, Rayleigh angle incidence(30.2°),
β=45 1/m β=-45 1/m
Incident profiles, p=5: Incident profiles, p=10:
Reflected profiles, p=5: Reflected profiles, p=10:
• Reflection at a liquid/solid interface
Reflected beam profiles: simulations
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Am
plitu
de
z(mm)
0.25
0.5
0.75
1
1.25
-40 -20 0 20 40 60 80
Am
plitu
de
z(mm)
f=3 MHz, w=20 mm, water/aluminium, Rayleigh angle incidence(30.2°),
β=45 1/m β=-45 1/m
Incident profiles, p=5: Incident profiles, p=10:
Reflected profiles, p=5: Reflected profiles, p=10:
• Reflection at a liquid/solid interface
Simulation of the reflection coefficient
0.6
0.7
0.8
0.9
1
0 15 30 45 60 75
|R|
incidence angle(degrees)
f=3 MHz, w=20 mm, p=5, water/aluminium
β=45 1/m
Infinite plane inhomogeneous wave theory:
Fourier model: �
0.6
0.8
1
1.2
1.4
1.6
0 15 30 45 60 75
|R|
incidence angle(degrees)
β=-45 1/m
• Reflection at a liquid/solid interface
Simulation of the reflection coefficient
0.6
0.7
0.8
0.9
1
0 15 30 45 60 750.6
0.7
0.8
0.9
1
0 15 30 45 60 75
|R|
incidence angle(degrees)
f=3 MHz, w=20 mm, p=5, water/aluminium
β=45 1/m
Infinite plane inhomogeneous wave theory:
Fourier model: �
0.6
0.8
1
1.2
1.4
1.6
0 15 30 45 60 75
|R|
incidence angle(degrees)
β=-45 1/m
• Interaction with a solid plate
Reflection and transmission coefficients
f=4 MHz, w=20 mm, p=5, aluminium plate, d=2 mm, β=45 1/m
Infinite inhomogeneous plane wave theory:
Fourier model: ◊
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
|R|
incidence angle(degrees)
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
|T|
incidence angle(degrees)
• Interaction with a solid plate
Reflection and transmission coefficients
f=4 MHz, w=20 mm, p=5, aluminium plate, d=2 mm, β=45 1/m
Infinite inhomogeneous plane wave theory:
Fourier model: ◊
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
|R|
incidence angle(degrees)
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40
|T|
incidence angle(degrees)
• Interaction with thin coatings
Rayleigh angle incidence
0
0.2
0.4
0.6
0.8
1
1.2
-40 -20 0 20 40 60
ampl
itude
z(mm)
-200
-150
-100
-50
0
-40 -20 0 20 40 60
phas
e(de
gree
s)
z(mm)
Amplitude Phase
water/steel, 10 µm copper coating, w=20 mm, p=5
β=50 1/m without coating:
β=50 1/m with coating:
β=-50 1/m without coating:
β=-50 1/m with coating:
• Interaction with thin coatings
Rayleigh angle incidence
0
0.2
0.4
0.6
0.8
1
1.2
-40 -20 0 20 40 60
ampl
itude
z(mm)
-200
-150
-100
-50
0
-40 -20 0 20 40 60
phas
e(de
gree
s)
z(mm)
Amplitude Phase
water/steel, 10 µm copper coating, w=20 mm, p=5
β=50 1/m without coating:
β=50 1/m with coating:
β=-50 1/m without coating:
β=-50 1/m with coating:
• Interaction with thin coatings
Rayleigh angle incidence
20
40
60
80
100
120
140
0 5 10 15 20
Rel
ativ
e am
plitu
de c
hang
e(%
)
d(µm)
Relative amplitude change in specular direction
water/steel, 10 µm copper coating, w=20 mm, p=5
β=50 1/m
β=-50 1/m
Phase shift in specular direction
25
50
75
100
125
0 4 8 12 16 20
Pha
se s
hift(
degr
ees)
d(µm)
β=50 1/m
β=-50 1/m
• Interaction with thin coatings
Rayleigh angle incidence
20
40
60
80
100
120
140
0 5 10 15 20
Rel
ativ
e am
plitu
de c
hang
e(%
)
d(µm)
Relative amplitude change in specular direction
water/steel, 10 µm copper coating, w=20 mm, p=5
β=50 1/m
β=-50 1/m
Phase shift in specular direction
25
50
75
100
125
0 4 8 12 16 20
Pha
se s
hift(
degr
ees)
d(µm)
β=50 1/m
β=-50 1/m
• BOUNDED BEAM DIFFRACTION ON A RECTANGULAR INCLUSION IN A SOLID PLATE
transd
ucer
θi
PLATE
LIQUID
transmitted sound field
LIQUID
θi=Lamb angle
inclusion
Influence of the inclusion on the reflected and transmitted beam profiles ?
• BOUNDED BEAM DIFFRACTION ON A RECTANGULAR INCLUSION IN A SOLID PLATE
transd
ucer
θi
PLATE
LIQUID
transmitted sound field
LIQUID
θi=Lamb angle
inclusion
Influence of the inclusion on the reflected and transmitted beam profiles ?
• Radiation Mode Model
The radiation modes for a simple plate
z
y
Type 1
R
T
Type 2
R
T
• Radiation Mode Model
The radiation modes for a simple plate
z
y
Type 1
R
T
Type 2
R
T
• Division in substructures
Structure 1a: ∑ ∫=
=
a,sn
k
0
i,yi,yn
1i,yn
ii dk)z,y,k(u)k(C)z,y(u
∑ ∫=
=
21 r,rm
k
0
2,y2,ym
22,ym
22 dk)z,y,k(u)k(C)z,y(uStructure 2:
Structure 1b: ∑ ∫=
=
a,sp
k
0
1,y1,yp
11,yp
11 dk)z,y,k(u)k(C)z,y(u
Boundary conditions at z=0 and z=L)k(C 2,y
m2
)k(C 1,yp
1
• Division in substructures
Structure 1a: ∑ ∫=
=
a,sn
k
0
i,yi,yn
1i,yn
ii dk)z,y,k(u)k(C)z,y(u
∑ ∫=
=
21 r,rm
k
0
2,y2,ym
22,ym
22 dk)z,y,k(u)k(C)z,y(uStructure 2:
Structure 1b: ∑ ∫=
=
a,sp
k
0
1,y1,yp
11,yp
11 dk)z,y,k(u)k(C)z,y(u
Boundary conditions at z=0 and z=L)k(C 2,y
m2
)k(C 1,yp
1
• Interaction of a Gaussian beam with an inclusion
Influence of an inclusion at a known position
f=4 MHz, w=12 mm, y0=91 mm, θi=24.96°(A1), L=15 mm, d1=25 µm,y1=0.75 mm
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Tran
smitt
ed a
mpl
itude
z ’ ( m m )
full line: z*=-10 cm; dashed line: z*=-1.5 cm; dotted line: z*=0 cm; dashdot line: z*=1.5 cm
• Interaction of a Gaussian beam with an inclusion
Influence of an inclusion at a known position
f=4 MHz, w=12 mm, y0=91 mm, θi=24.96°(A1), L=15 mm, d1=25 µm,y1=0.75 mm
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Tran
smitt
ed a
mpl
itude
z ’ ( m m )
full line: z*=-10 cm; dashed line: z*=-1.5 cm; dotted line: z*=0 cm; dashdot line: z*=1.5 cm
• Interaction of a Gaussian beam with an inclusion
Influence of an inclusion at a known position
Influence of the thickness of the inclusion(z*=0 cm,L=15 mm, y1=0.75 mm)
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed p
hase
(deg
rees
)
z ’ ( m m )m M
d1=5 µmd1=15 µm
d1=25 µm
no inclusion
d1=5 µm
d1=15 µmd1=25 µm
Thickness can be inferred from both amplitude and phase
• Interaction of a Gaussian beam with an inclusion
Influence of an inclusion at a known position
Influence of the thickness of the inclusion(z*=0 cm,L=15 mm, y1=0.75 mm)
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0
Ref
lect
ed p
hase
(deg
rees
)
z ’ ( m m )m M
d1=5 µmd1=15 µm
d1=25 µm
no inclusion
d1=5 µm
d1=15 µmd1=25 µm
Thickness can be inferred from both amplitude and phase
• Interaction of bounded inhomogeneous waves with an inclusion
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 0 2 0 4 0 6 0 8 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
d1=25 µm, y1=0.75 mm, y0=92 mm, L=15 mm, p=5, A1 Lamb angle
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-40 -20 0 20 40 60 80
Reflected amplitude
z’(mm)
β=50 1/m β=-50 1/m
Full lines: intact plate; dashed lines: z*=-7.5 mm;
dotted lines: z*=0.0 mm; dashdot lines: z*=15 mm
• Interaction of bounded inhomogeneous waves with an inclusion
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
- 2 0 0 2 0 4 0 6 0 8 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
d1=25 µm, y1=0.75 mm, y0=92 mm, L=15 mm, p=5, A1 Lamb angle
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-40 -20 0 20 40 60 80
Reflected amplitude
z’(mm)
β=50 1/m β=-50 1/m
Full lines: intact plate; dashed lines: z*=-7.5 mm;
dotted lines: z*=0.0 mm; dashdot lines: z*=15 mm
• Interaction of bounded inhomogeneous waves with an inclusion
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
Subsurface inclusion: z*=0.0 mm, d=3 mm, L=15 mm, β=50 1/m, p=5, θi=Rayleigh angle
Full line: no inclusion; dashed line: d1=5 µm; dotted line: d1=10 µm
• Interaction of bounded inhomogeneous waves with an inclusion
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
- 2 0 - 1 0 0 1 0 2 0 3 0 4 0
Ref
lect
ed a
mpl
itude
z ’ ( m m )
Subsurface inclusion: z*=0.0 mm, d=3 mm, L=15 mm, β=50 1/m, p=5, θi=Rayleigh angle
Full line: no inclusion; dashed line: d1=5 µm; dotted line: d1=10 µm
•Radiation Mode Model Origin in electromagnetism: Shevchenko, Marcuse(1970‘s)Adapted to acoustics: G. N. Shkerdin, O. Leroy, R. Briers(1993)•Radiation Mode Model Origin and historyOrigin in electromagnetism: Shevchenko, Marcuse(1970‘s)Adapted to acoustics: G. Shkerdin, O. Leroy, R. Briers(1993)Prof. G. ShkerdinRMM: Mathematical model which permits to calculate the interaction of a sound field with a structure by decomposing the sound field in the
complete and orthogonal set of radiation modes and eigenmodes of the structure
Prof. O. Leroy Dr. R. Briers