Part 6

ATTENUATION

Signal Loss

Loss of signal amplitude:

1 1

2 2[Neper] ln or [dB] 20logA AL L

A A= =

A1 is the amplitude without loss

A2 is the amplitude with loss

Proportional loss of signal amplitude with increasing propagation distance:

L d= α

d is the propagation distance

α is the attenuation coefficient

Major classes of attenuation:

absorption (viscosity, relaxation, heat conduction, elastic hysteresis, etc)

scattering (inhomogeneities), also causes incoherent material noise

absorption scatteringα = α + α

For example: 2water [dB/m] 0.2 [MHz]fα ≈

Plexiglas [dB/m] 100 [MHz]fα ≈

f denotes frequency

Scattering Induced Attenuation

Single-scatterer:

sP I= γ

Ps scattered power by the inhomogeneity

γ is the scattering cross section of the average scatterer

I intensity of the incident wave

Single-scattering approximation:

Let us consider a volume of given cross section A and length d. The coherent acoustic power P A I= transmitted through this region decreases by an amount of

s s1

Ni

idP P P N I n Ad I n d P

=− = = = γ = γ = γ∑

N number of scatterers in volume Ad

n the number density of the scatterers

dP P n d= − γ

i n dP P e− γ=

½i n du u e− γ=

½ nα = γ

0lim 0ω→

γ =

slim 2 Aω→∞

γ ≤

General Considerations on Scattering

Similarity:

d1

D1

λf ,1 1

d2

D2

λf ,2 2

Scaling:

2 2 2 1

1 1 1 2

d D fd D f

⎛ ⎞λξ = = = =⎜ ⎟λ ⎝ ⎠

Scattering Loss:

1 2L L=

1 1 1

2 2 2

//

L dL d

α= = ξ

α

1 1

2 21α λ

=α λ

Normalized Attenuation:

n n( , ) ( )DDα = λα λ = αλ

Power relationship:

n ( , ) i iD f D fα ∝

1( , ) i iD f a D f +α =

a is a constant determined mainly by the "degree" (relative deviation from the host medium) of

the inhomogeneity and the nature of the interaction (e. g., shear or longitudinal wave, etc.)

Polycrystalline material:

D is the grain size

a is a function of anisotropy

Low-frequency (Rayleigh) region:

3 4Rayleigh R( , )D f a D fα =

Intermediate (stochastic) region:

2stochastic s( , )D f a D fα =

High-frequency (geometrical) region:

1geometrical g( , )D f a D−α =

Surface wave attenuation on a slightly rough surface:

4 5roughness r( , )h f a h fα =

h is the rms roughness

ar is a function of the rms roughness-to-autocorrelation length ratio

Scattering Induced Attenuation in Polycrystalline Materials

Low-Frequency (Rayleigh) Region ( Dλ >> )

r

Scattered Wave

Scatterer

Incident Wave

uiu s

( )s s0( , ) ( )

i k r teu r u Fr

− ωθ = θ

s0u r/ is the amplitude of the wave at a distance r from the scatterer

( )F θ is the directivity function (θ denotes the polar angle)

Linear superposition:

s0 iu V u∝ Δ

iu is the incident wave amplitude

Δ is the relative change of the elastic properties

V is the scatterer volume

For example, for cubic crystals:

44

11 12

2 1CC C

Δ = −−

The total scattered power from a single scatterer:

2 2 2i2 2 2

s s i2S S

V uP u dS dS V I

rΔ

∝ ∝ ∝ Δ∫ ∫

2i iI u∝ denotes the intensity of the incident wave

½ nα = γ

2 2Vγ ∝ Δ is the scattering cross section

1n V −= and 3V D∝

2Rayleigh Vα ∝ Δ

3 4 2 3 4Rayleigh Ra D f D fα = ∝ Δ

Ra is a constant that is proportional to 2Δ

Intermediate (Stochastic) Region ( Dλ ≈ )

Incident Ray θdivergence

θrefraction

Refracted Ray

In the case of weak anisotropy:

refractionθ ≈ Δ

divergence Dθ ≈ λ/

Geometrical region:

divergence refractionθ ≤ θ

i. e., above a frequency where Dλ ≤ Δ/

Stochastic region (forward scattering):

weak random phase perturbation ( , )x yΦ multiplies the coherent (average) wave

2( , ) ( , ) ½ ( , )i x y i x y x ye e eΦ <Φ > − <Φ >< > ≈

Loss of the coherent wave:

2½L ≈ ϕ

2 2( , )x yϕ = < Φ >

D fϕ ∝ Δ

2 2 2L D f∝ Δ

2 2 2stochastic sa D f D fα = ∝ Δ

sa is a constant that is proportional to 2Δ

High-Frequency (Geometrical) Region ( Dλ << )

d

Incident Plane Wave

Transmitted Plane Wave

1geometrical D−α ∝

Summary:

Regime Functional Dependence

Rayleigh ( D << λ ) 2 3 4D fα ∝ Δ

stochastic ( D DΔ≤ λ ≤ ) 2 2D fα ∝ Δ

geometrical ( Dλ ≤ Δ ) 1D−α ∝

Grain Scattering Induced Attenuation In Polycrystalline Iron

(100 μm grain diameter)

log{Frequency [MHz]}

log{

Atte

nuat

ion

Coe

ffic

ient

[dB

/cm

]}

-5

-4

-3

-2

-1

0

1

2

3

-1 0 1 2 3

Rayleigh region stochastic region geometrical region

shear

longitudinal

Experimental Grain Scattering Induced Attenuation

longitudinal wave in SAE 1020 steel

Frequency [MHz]

Atte

nuat

ion

Coe

ffic

ient

[dB

/cm

]

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

57 µm48 µm38 µm31 µm18 µm10 µm

Complicating effects:

preferred orientation between neighboring grains (e. g., prior austenite grain structure, columnar grain structure, severe plastic flow, etc.)

shape of the grains can also be very different from the ideal uniaxial shape (e. g., needle-like

alpha (hexagonal) grains in titanium)

Ultrasonic Grain Size Assessment Typical welded zone:

Over-heated welded zone:

transmission C-scan of an app. 4"-wide electric resistance welded butt joint between two 0.25"-thick steel plates at 15 MHz

electrode

grain coarsening

plastic flow

high-pressure side

low-pressure side

interface

Experimental Aspects of Ultrasonic Attenuation Measurements

time

“main bang” A0 A1 A2 • • •

time

“main bang” A0 A1 A2 • • •

0imp diff surf

120log 2AL d L L L

A= = α + + +

Impedance Mismatch:

R0R1 R2

R3

T1

T2T3

d

I

Medium 1

Medium 1 (3)

(sample)Medium 2

Reflection coefficient:

2 112

2 1

Z ZRZ Z

−=

+

12 21R R R= = −

2 212 21 1T T T R= = −

Z1 and Z2 are the acoustic impedances of the first and second media, respectively

Front surface reflection:

0R R=

Multiple-reflection:

21R R T= , 3 22R R T= , ... 2 1 2nnR R T−=

Transmission:

21T T= , 2 22T R T= , ... 2( 1) 2nnT R T−=

where n = 1, 2, ...

0 2imp

120log 40log 20log(1 )RL T R

R= = − = − −

Diffraction Correction

the acoustic field of a circular piston radiator at a /λ = 10

z = a

near-field

far-field

z = a = N10

z = a20

Near-field / far-field transition: 2aN =λ

a denotes the radius of the transducer λ is the acoustic wavelength in the medium

Simplified Sound Field of a Circular Piston Radiator

-2

-1

0

1

0 1 2 3 4

2-10dB contour"searchlight"

model

simplified model

θ-10 dB aξ

zN

Two identical transducers in a pitch-catch mode:

rL

r

( )( )( 0)p zD z

p z=

=

z is the distance between the transducers (in a pulse-echo operation with a normally aligned mirror, the distance between the transducer and the mirror is only z /2)

Lommel diffraction correction:

2L 0 1( ) 1 [ (2 ) (2 )]i sD s e J s i J s− π= − π + π/ / /

s = z /N In the far-field:

Llim ( )s

D s i s→∞

= π/

2Llim ( )

z

aD zz→∞

π=

λ

Lommel Diffraction Correction for a Circular Piston Radiator

z / N

Diff

ract

ion

Cor

rect

ion

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

far-field asymptote

circular piston radiator of 0.5"-diameter and z = 20 cm separation in water

Frequency [MHz]

Diff

ract

ion

Cor

rect

ion

[dB

]

-10

-8

-6

-4

-2

0

0 10 20 30 40 50

Refraction Correction Snell’s Law:

2 2

1 1

sinsin

cc

θ=

θ

fluid solid

For slightly divergent beams:

sin tanθ ≈ θ ≈ θ

21 2

1

cz z zc

= +

Pulse-echo configuration:

1L

1diff

1 2 1L

1

2

20log2 2 /

zDN

Lz d c cD

N

⎛ ⎞⎜ ⎟⎝ ⎠≈

⎛ ⎞+⎜ ⎟⎝ ⎠

2 2

11 1

a a fNc

= =λ

Surface Roughness

Rough Surface

Transducer

Flaw

Liquid

Solid

Incident Wave Coherent Reflection

z

x

θI θR

Incoherent Reflection

Incoherent Transmission

Coherent ShearTransmission

CoherentLongitudinalTransmission

θT

θL

Phase-Screen Approximation

( , )s x y is the surface height distribution

h is the rms height

Λ is the correlation length

2 2( , )h s x y= < >

2( , ) ( , ) ( , ) ( , )C s x y s x y h cξ η = < − ξ − η > = ξ η

transverse isotropy:

2 2 2ρ = ξ + η

Gaussian distribution:

2 2/( )c e−ρ Λρ =

Logarithmic distribution:

/( )c e−ρ Λρ =

small curvature: h << Λ

Phase perturbation (without the common e-iωt term)

( , , )0( , , , 0 ) ( , , , ) i x yx y z x y z e− Φ ωω = = ωu u

u denotes the displacement field just inside the rough solid

0u denotes the displacement field just inside the smooth solid

L,T w I L,T L,T( , ) [ cos( ) cos( )]s x y k kΦ = θ − θ

R w I2 ( , ) cos( )s x y kΦ = θ

kL, kT and kw are wavenumbers

“Coherent” Transmission Coefficients

2½L ≈ < Φ > Reflected compressional wave:

0R ( , ) 20log RL

Rω θ =

Longitudinal transmitted wave:

L0L

L( , ) 20log

TL

Tω θ =

Shear transmitted wave:

T0T

T( , ) 20log

TL

Tω θ =

Phase-screen approximation:

2 2R,L,T R,L,T = 8.686 [dB] L h Cω

2R I w= 2 [cos( ) ]C cθ /

1 2-L L L I w2 = [cos( ) cos( ) ]C c cθ θ/ /

1 2-T T T I w2 = [cos( ) cos( ) ]C c cθ θ/ /

cw, cL and cT are sound velocities

Surface Roughness Induced Attenuation of the Reflected Ultrasonic Wave at Normal Incidence

Frequency [MHz]

Atte

nuat

ion

[dB

]

0

5

10

15

20

25

30

35

40

0 5 10 15 20

45.6 µm25.6 µm15.2 µm12.8 µm11.4 µm9.9 µm

8.7 µm5.6 µm

(solid lines are best fitting f 2 curves)

Surface Roughness Induced Attenuation of the Double-Transmitted Longitudinal and Shear Waves

Frequency [MHz]

Atte

nuat

ion

[dB

]

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20

0º reflection 12º long. tr.10º long. tr.

0º long. tr.26º shear tr.24º shear tr.22º shear tr.

(solid lines are best fitting f 2 curves).