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