Wave Propagation Fundamentals -3
Ultrasonic Attenuation
Learning Objectives
Attenuation definitions
Simple method for measuring average attenuation
Model-based approach for determining attenuation
as a function of frequency
Ultrasonic Attenuation
A1
plane wave amplitudes:
A2
attenuating
medium
d
A2
= exp ⎡⎣ −α ( f ) d ⎤⎦
A1
attenuation (Np/unit length)
Example: water
α w ( f ) = 25.3 ×10−15 f 2
Np / m
Attenuation is affected by many variables so that
generally it is not possible to give generic values
that are useful for quantitative analyses.
Thus, attenuation generally must be obtained
experimentally for the material in question
When attenuation values are quoted they are often
given as average values in decibels/unit length
α dB / l = 8.686α Np / l
plane wave
pulse traveling
in an attenuative
medium:
A2
A1
d
average attenuation (Np/l)
A2
= exp ⎡⎣ −α Np / l d ⎤⎦
A1
average attenuation (dB/l)
α dB / l
⎛A ⎞
20log10 ⎜ 2 ⎟
A1 ⎠
⎝
=−
d
= ( 20log10 e )α Np / l
= 8.686 α Np / l
Simple method to determine average attenuation
To model the effects of attenuation on the characteristics of a signal, we must be
able to find the attenuation of a material as a function of frequency. However, in
some applications we may be content to obtain an average attenuation value
which describes the overall behavior of the amplitude of the signal. We can
obtain such the average attenuation of a solid sample, for example, by using the
pulse-echo setup shown.
D
On the oscilloscope screen we will see a series of evenly spaced pulses
that are decreasing in amplitude. These are the waves that have been
reflected from the back surface of the block one or more times.
A1
A2
A3
A4
Let the measured amplitudes of these signals be A1, A2, … etc. as shown
If the back surface of the block is in the far-field of the transducer, the incident
and reflected waves will behave like attenuated spherical waves so we could
express the voltage of these signals, v(t), in the form:
v (t ) =
+
g (t − 2D / c )
2D
g (t − 6D / c )
6D
exp ( −2α Np / l D ) +
g (t − 4D / c )
4D
exp ( −4α Np / l D )
exp ( −6α Np / l D ) + ...
where the waveform shape is given by the g(t) function and α Np / l
average attenuation of the block (in Np/l)
is the
If we let gmax be the maximum amplitude of the g function then the amplitudes shown
are given by
g max
A1 =
exp ( −2α Np / l D )
2D
g max
A2 =
exp ( −4α Np / l D )
4D
g max
A3 =
exp ( −6α Np / l D )
6D
g
A4 = max exp ( −8α Np / l D )
8D
etc.
If we take the ratio of the first two reflections we have
A1
= 2 exp ( 2α Np / l D )
A2
Now
⎛A ⎞
20 log10 ⎜ 1 ⎟ = 20 log10 ( 2 ) + 20 log10 ⎡⎣ exp ( 2α Np / l D ) ⎤⎦
⎝ A2 ⎠
= 6 dB + ( 2 D ) 20 log10 ( e ) α Np / l
= 6 dB + ( 2 D ) α dB / l
which gives
α dB / l
⎛ A1 ⎞
20log10 ⎜ ⎟ − 6 dB
A2 ⎠
⎝
=
2D
Measurement of attenuation as a function
of frequency
ρ1 , c p1
ρ 2 , c p 2 , cs 2
v0
a
voltages received
from front, back
surface reflections
material
to be measured
D2
D1
V f s (ω ) = s (ω ) t Afs (ω )
Vb s (ω ) = s (ω ) t Abs (ω )
system function
acoustic/elastic transfer functions that
describe the waves received from the
front and back surface reflections
front surface transfer function
t Afs (ω ) = t%Afs (ω ) exp ⎡⎣ −2α w (ω ) D1 ⎤⎦
ideal transfer
function model
t%Afs (ω ) = D% p ( k p1a 2 / 2 D1 ) R12 exp ( 2ik p1 D1 )
plane wave reflection coefficient
back surface transfer function
t Abs (ω ) = t%Abs (ω ) exp ⎡⎣ −2α w (ω ) D1 − 2α p 2 (ω ) D2 ⎤⎦
ideal transfer
function model
t%Abs (ω ) = D% p ( k p1a 2 / 2 D ) T12 R21T21 exp ( 2ik p1 D1 + 2ik p 2 D2 )
plane wave reflection, transmission coefficients
(normal incidence)
D% p ( u ) = 2 ⎡⎣1 − exp ( iu ) { J 0 ( u ) − i J1 ( u )}⎤⎦
diffraction coefficient
In the back surface signal, the distance D appearing
in the diffraction correction is:
D = D1 +
cp2
c p1
D2
Vb s (ω )
t%Abs (ω )
exp ⎡⎣ 2α p 2 (ω ) D2 ⎤⎦
= fs
V f s (ω ) t%A (ω )
measure
model
usually fit to a simple polynomial
function of frequency over the
bandwidth of the system
Vbs (ω )
exp ⎡⎣ 2α s (ω ) D2 ⎤⎦ =
V fs (ω )
t%Afs (ω )
t%Abs (ω )
Remarks:
To measure attenuation, it is not necessary to measure
the system function (it cancels out).
Setup shown is for measurement of P-wave attenuation.
S-wave attenuation measurements must be done in
a different setup.
This approach is ad-hoc. Actual mechanisms of
attenuation are rather complex.
For high attenuation, wave speed as well as amplitude
is affected (material dispersion).