Acoustic/Elastic Transfer Function
Learning Objectives
Modeling the acoustic/elastic wave processes as
a transfer function
Blocked force concept
Special cases where the transfer function can be obtained
analytically
Effects of material attenuation
Acoustic/Elastic Transfer Function
FB
Ft
All the complex wave propagation and scattering processes occurring
in an ultrasonic measurement system can be characterized in terms of
an acoustic/elastic transfer function
transmitted Ft
force
t A (ω )
FB received
"blocked" force
The Blocked Force on Reception
On reception the receiving transducer face moves and
scattered waves are also generated
pscatt
pinc
The Blocked Force on Reception
If the incident waves are treated locally like plane waves
and diffraction effects are ignored, we can consider the
interactions of the incident waves with the transducer face
as those of plane wave with an infinite planar interface.
Now, consider the case where the interface is held rigidly
fixed:
y
reflected
wave
incident
wave
θ
θ
x
rigid
boundary
The Blocked Force on Reception
y
reflected
wave
θ
θ
x
rigid
boundary
incident
wave
In this case it can be shown that at the interface the
blocked force is just double the force due to the incident
wave only as if the transducer were absent, i.e
FB (ω ) = 2 Finc (ω )
pinc
where
Finc = ∫∫ pinc dS
S
S
The Blocked Force on Reception
proof for normally incident waves
vx = 0
x
pinc = Pi exp ( ikx − iω t ) p = P exp ( −ikx − iω t )
reflt
r
Equation of motion:
∂p
−
= −iωρ vx
∂x
p = pinc + preflt
from vx = 0 at x = 0 ikPi − ikPr = 0
Pr = Pi (reflected pressure = inc. pressure)
or
Thus, also at x = 0
F = Finc + Freflt = Pi S + Pr S = 2 Finc
Acoustic/elastic processes model
For a number of calibration setups the acoustic/elastic
transfer function can be obtained explicitly:
front surface reflection
reflection from flat-bottom hole
back surface reflection
reflection from flat cylinder
Acoustic Transfer Function
Example: the acoustic/elastic transfer function of two aligned
circular piston transducers of different radii can be found.
b
a
D
t A (ω ) =
π /2
⋅∫
0
2
2 2
Θ
ik
D
−
a
b
exp
16
(
)
{
1
2
πa
sin 2 u cos 2 u
⎡ ik D 2 + ( a − b )2 + 4ab cos 2 u
exp
1
2
( a − b ) + 4ab cos 2 u ⎢⎣
⎧⎪πb 2
Θ= ⎨ 2
⎪⎩πa
⎤ du ⎪⎫
⎥⎦ ⎬
⎪⎭
a≥b
b≥a
Acoustic Transfer Function
Special Cases
4
⎧
t A (ω ) = 2 ⎨exp ( ik1 D ) −
π
⎩
a=b:
π /2
⋅∫
0
⎫
2
2
2 ⎤
⎡
sin u exp ik1 D + 4a cos u du ⎬
⎣
⎦ ⎭
2
At the high frequencies found for most NDE transducers
the above integral can be evaluated approximately to yield:
t A (ω ) = 2 exp ( ikD ) ⎡⎣1 − exp ( ika 2 / D )
{
}
⋅ J 0 ( ka 2 / D ) − iJ1 ( ka 2 / D ) ⎤
⎦
Acoustic Transfer Function
Other special cases:
{
}
a >> b :
b2
t A (ω ) = 2 2 exp [ikD ] − exp ⎡ik D 2 + a 2 ⎤
⎣
⎦
a
a << b :
t A (ω ) = 2 exp [ikD ] − exp ⎡ik D 2 + a 2 ⎤
⎣
⎦
D >> a, b :
{
}
exp ( ik1 D )
t A (ω ) = −ika
D
2
Acoustic Transfer Function
Attenuation of ultrasound in the fluid can be added
to this model:
t A (ω ) → t A (ω ) exp ⎡⎣ −α (ω ) D ⎤⎦
α(ω) … attenuation coefficient
For water
α = 25.3 ×10−6 f 2
f… frequency in MHz
D … distance traveled
1/mm
Acoustic Transfer Function
Example acoustic transfer function (including fluid
attenuation effects)
0.4
-40
0.35
-45
-50
2a
2a
0.25
-60
0.2
-65
0.15
-70
-75
phase
0.1
D = 444 mm
a = 3.175 mm
-55
amplitude
-80
0.05
0
0
-85
5
10
15
Frequency (MHz)
-90
20
Phase (Deg)
D
Amplitude
0.3
Acoustic Transfer Function
Goldstein, A., Gandhi, D.R., and W.D. O’Brien,” Diffraction
effects in hydrophone measurements,” IEEE Trans. Ultrasonics,
Ferroelectrics, and Freq. Control, 45, 972-979, 1998.