SIMILARITY OF GAUSSIAN AND
PISTON TRANSDUCER VOLTAGES
. D. E. Chimenti1 and O. I. Lobkis2
1
2
Center for NDE and Aerospace Eng Dept, Iowa State Univ, Ames IA 50011
Theoretical & Applied Mechanics Dept, Univ of Illinois, Urbana, IL 61801
ABSTRACT. A plane-wave decomposition of collimated beams and electromechanical reciprocity relations are used to demonstrate fundamental differences and unusual similarities about
transducer fields and transducer voltages under various conditions. It is shown that the voltage
induced by a transmitting acoustic piston transducer (constant particle velocity over the transducer surface) radiating into an ideal fluid medium on a second identical piston transducer,
operating as a receiver, is nearly identical to the voltage observed when the two transducers
have instead a Gaussian radial surface velocity distribution. The strong similiarity in induced
voltage begins when the two devices are separated by only several acoustic wavelengths, still
well within the nearfield of both transducers, and the similarity increases with separation.
INTRODUCTION
We calculate here the voltage induced on a second transducer resulting from the
radiation of a transmitting transducer and demonstrate several novel features of transducer voltage at the receiver as a function of the relative transducer positions, their
size, and the distribution of particle velocities on their surfaces. Surprisingly, the voltage induced by a transmitting piston transducer on a second identical piston receiving
transducer is nearly identical to the voltage observed when the particle velocities of two
transducers have a radial Gaussian dependence. The strong similiarity in the induced
voltage is already apparent when the two transducers are separated by only a dozen
or so acoustic wavelengths from each other, that is, while they are still well within
each other's nearfield. As the separation increases, the two lateral voltage dependences
become more and more similar, until the combined directivity functions of the piston
fields can be justifiably replaced by Gaussian directivity functions. The origin of this
phenomenon is in the averaging effect of the second piston radiator, when it acts as a
coherent detector of the acoustic field.
Many authors have undertaken analyses of bounded acoustic beams using a spectrum of plane waves, e.g., Brekhovskikh [1,2]. Numerous applications involving collimated acoustic beams have relied on similar methods. The initial rigorous treatment
of interacting acoustic beams on fluid-solid halfspaces was performed by Bertoni and
Tamir [3]. In Chimenti, et al. [4], the method was extended to curved layered plates,
and Lobkis, et al. [5] exploited reciprocity to develop an accurate model of the voltage
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
836
FIGURE 1: Geometry of transducer placement and relative motion of the two transducers.
detected on a finite piston receiving transducer.
THEORETICAL DEVELOPMENT
Let two planar transducers (transmitter and receiver) be located facing each
other in an acoustic coupling fluid, such as water. The range distance between them
is 2, and their acoustical axes are displaced by a distance r. A closely related and
essentially equivalent geometry is that of two transducers deployed on one side (or
opposite sides in transmission) of a uniform plate. The transducers have radii of a and
6, respectively, as illustrated in Fig. 1.
Incident Field ^
According to the Rayleigh integral formula for the field of a planar transducer,
the particle velocity potential of the incident field can be expressed as
M)^^,
(i)
where Vi(pi) is the axisymmetric particle velocity distribution on the transmitter surface Si. Here, p\ is the radial polar coordinate measured from the transmitter center, K
is the wavenumber in the fluid, and R is the distance between an arbitrary observation
point (p, z) in cylindrical coordinates and some point on the transmitter surface Si .
The amplitude of Vi(pi) can depend on the local radial coordinate pi in one of several
different ways. The subscript 1 refers always to the transmitter; later, subscript 2 will
refer to the receiver. The geometry of the calculation is the one normally used. The
distances R and x are given by R — \/£2 + z*2 and x 2 = p2 -f p\ — 2ppi cos fa .
The spherical wave exp(iKR)/R can be expanded in terms of cylindrical Bessel
functions in the wavenumber integration variable r. The 2D integral expression in
Eq (1) can be simpflified for an axially symmetric transducer velocity distribution by
introducing the Bessel function relation for the spherical wave ex.p(iK,R)/R. Then, we
invert the order of integration and integrate over the transmitter surface. The integral
over the azimuthal transmitter angle fa is given by 27rJ0(pr) JQ(PIT). In the resulting
expression for the field the integral over the local radial variable pi may be recognized
as the Hankel transform of the particle velocity profile function Vi(pi). This realization
permits us to write
J0(pr)Tdr ,
837
(2)
where HT denotes the Hankel transform in the wavenumber variable r. If we now
particularize the velocity distribution to one that is rectangular (i. e., Vi = constant if
pi < a, and Vi(pi) = 0 for pi > a), the Hankel transform becomes ViaJi(ar)/r, and
Eq (2) reduces to
, z ) = V,a
X
T
I J!(ar) Jo(pr) dr ,
(3)
which is exactly King's solution of the Rayleigh integral for planar transducers. In the
following section we use Eq (2) to calculate properties of field and voltage for arbitrary
velocity distributions, and these results are applied to several special cases.
Receiver Voltage and Reciprocity
According to the Auld-Kino electromechanical reciprocity relation, we can write
the voltage V(z,r) induced on a receiving transducer (device 2) that samples the
acoustic field produced by the transmitter (device 1) in the following manner,
(4)
where V^p^) is the surface velocity distribution on the receiving transducer (when it
operates as a transmitter), ^ is the incident field produced by device 1, and dS^ is
the differential surface area on the receiver, given by dS^ — Pz dp^ dfa. The geometry
for the receiver is entirely equivalent to that for the transmitter. The radial distance
from the transmitter axis to a point on the receiver is given analogously by p2 =
r 2 + Pz — 2rp2 cos 02. Now, replace the incident field \^ in Eq (4) by its representation
from Eq (2) to find
_
-
^ [VM]
JO(T,T)T dr
V2(p2}f>2 df)2
jo(T/9(02)) d(/)2
(5)
Following the procedure outlined earlier for integration over the transmitter, the integral
over the local receiver coordinate 02 is 27rJo(rr)^o(p2/7"), and the subsequent integral
over the remaining local receiver coordinate p2 is the Hankel transform of the receiver
velocity distribution V^p^). With these simplifications we finally arrive at the result
where (z, r) are the cylindrical coordinates of the receiver center with respect to the
transmitter center. Similar expressions have been derived by other workers as well.
Of course, the receiver voltage is the result of a coherent sum over its face of the
incident field ^; and is therefore expected to be quite different from the acoustic field
detected at the center of the receiving transducer, as can be verified by a comparison
of Eqs (2) and (6). These two quantities will approach each other only as the radius
of the receiver shrinks to zero, i.e., as the receiver approximates a point detector. In
838
that case, the velocity distribution approaches a delta function V^p^) -> 5(p 2 ), and
HT[V2(p2)] is constant.
The "Equivalent Transducer"
We have seen that the field and voltage of finite radiators are, in general, quite
different. There does exist, however, a subtle relationship between the voltage on the
receiver and the field of the transmitter. To examine this behavior in more detail, we
introduce the concept of an "equivalent transducer" (ET). What is meant here is the
equivalence between the actual situation — finite transmitter and finite receiver of some
dimensions and velocity distribution — and a conceptually equivalent transmitter, of
some radius and having some velocity distribution, detected by a point receiver.
We can find the particle velocity distribution on the surface of an equivalent
transmitter by differentiating the voltage with respect to the coordinate z and evaluating the resulting expression on the surface of the transmitter, where the integral can be
recognized as the inverse Hankel transform of the product of the Hankel transforms of
the actual transmitter and receiver velocity distributions Vi(pi) and V^pz). The fact
that the kernels of the Hankel transform and its inverse are both Jo(rr) permits us to
rewrite the integral in formal notation as
VET(r) = 2^H-l{HT[Vl(p1}}HT(V,(p,)}}(r).
(7)
For each pair of transducers with specific velocity distributions V\ and V% , we can find
an equivalent distribution VET such that the field of the ET is exactly the same as the
voltage of the actual transducer pair. For example, the Rayleigh formula can be used
to calculate the voltage directly using the ET formalism.
A careful examination of the above relations reveals that this phenomenon of
the ET is actually only a direct consequence of reciprocity. The roles of the two
transducers are completely equivalent and interchangeable. When one of the pair of
virtual transducers is a point receiver, the other device will be a single, finite "equivalent
transducer" whose velocity distribution, calculated in Eq (7), yields a voltage at the
receiver that is identical to the voltage of the physical pair. In principle, there is any
number of transducer pairs whose velocity profiles could be considered equivalent to
the ones actually used in a given measurement.
APPLICATION TO TRANSDUCER VELOCITY PROFILES
In this section we apply the results of the preceding theoretical development
to several special cases of transducer velocity profiles (all having axial symmetry) for
which analytical solutions are possible.
Multiple Point Receivers
Before we compare calculated fields and voltages from expressions derived above,
let us examine the averaging effect at the receiver of several discrete point receivers
deployed along a line coincident with the diameter of a piston receiver. Three point
receivers are arrayed from the center of the local receiver coordinate system to the edge
of where an identical piston receiver would be located. The geometry is illustrated
schematically in Fig. 2a), where a virtual piston receiver is shown in dashed lines for
comparison. Figure 2b) shows the voltage profile when a piston radiator at a range of
839
Transmitter
/cz=143
- - -- ———
———
Left PR
Right PR
Center PR
Sum
FIGURE 2: a) Geometry of three point receivers to illustrate averaging effect of finite
receiver. Virtual piston receiver is shown in dashed lines for comparison, b) Fields of three
point receivers measuring piston radiator (dashed and dotted) and their coherent sum (solid) .
one Rayleigh distance (kz = ka2/X) is detected by three coherent point receivers. The
field of the central point receiver is plotted as a dotted curve, and the signals of the
left and right point receivers are plotted as dashed curves. The coherent sum is the
solid curve and is clearly smoother than the field functions. Extending this summation
analytically to the entire face of a finite receiver yields the results demonstrated in the
following section.
Gaussian Beams
This case is perhaps the most important of all from the standpoint of its impact on theoretical modeling because of the widespread use of Gaussian beams in the
modeling of finite-aperture acoustic phenomena.
We represent the velocity distribution of a Gaussian-beam transducer using the
conventional formulation for the transmitter and receiver, respectively. To be a good
approximation to a Gaussian beam, the physical transducer diameter must be large
enough that the velocity amplitude at the edge is small compared with its value at
the center, or the transducer must be otherwise apodized. Then, the integration limit
representing the Hankel transform can be extended to infinity, yielding
1
f°
exp[-rV/4] .
JQ
(8)
Inserting this intermediate result into Eq (7), we obtain
VET
= 7ra2
2
/>o°
/
JQ
exp[-(a 2
62)r2/4] JQ(rr)r dr
-VlV2exp[-r2/(a2 + b2)}.
840
(9)
a)
LLl
-
\
-4
r/a
r/a
FIGURE 3: a) Velocity profiles for two Gaussian-beam transducers and their ET. b) Effective velocity profiles rectangular-beam pair: equal radii a — b (solid curve); a > b (dashed
curve) profile is rectangular, then gradually decreasing. Insets show geometric overlap functions for the cases when a = b and when b < a.
We have demonstrated here that the acoustic system consisting of a Gaussian-beam
transmitter of beam waist a and a receiver of beam waist b behaves the same (has
the same lateral and range dependence) as the combination of an equivalent Gaussianbeam transducer of waist ^/a2 -f 62 and a point receiver, as illustrated in Fig. 3a).
Here, the profile of each Gaussian-beam transducer is shown along with the profile of
the equivalent transducer, also a Gaussian in this case. Alternatively, one may say that
the field of a single Gaussian-beam transducer of waist \/o? -f b2 is equivalent to the
voltage of a pair of Gaussian-beam transducers of beam waist a and 6, respectively. So,
in this particular case, both the original two radiators and the ET have Gaussian-beam
profiles, as we would expect.
Rectangular Velocity Distributions
From a practical viewpoint the rectangular velocity distribution is by far the one
most often met with in experiments. Its behavior, therefore, is particularly valuable
to examine in detail. Such a comparison between the field values of a piston radiator
(rectangular velocity profile) and the voltage, as measured by a second identical probe,
is shown in Fig. 4 for ka = 30. In frame a) is shown the behavior quite close to the
transmitter, at a range of kz = 50, or alternatively when z/7Z = 0.35, where 71 is the
Rayleigh distance a 2 /A. Here, the field is characterized by rapid oscillations owing to
successive interference of edge and face waves, whereas the voltage function at the same
range already resembles a Gaussian function. In frame b), at one Rayleigh distance
(kz = 140 or z/7£ = 1.03), the field and voltage are still quite different.
Examining this case analytically, the velocity profile is given by a step function,
whose Hankel transforms are straightforward because the profile is a constant from
r — 0 to radius a or 6,
a
Hr[Vi(pi)]
= Vi j JQ(rr)rdr
841
= VioJi(oT)/T,
(10)
(b)
•So.6-
*p
FIGURE 4: Comparison of field (dashed) and voltage (solid) functions for piston radiator
transducers with ka — 30 at various ranges, a) z/7t = 0.35; b) z/7i — 1.03.
and a similar expression for the transducer of radius b.
Then, for the equivalent velocity profile of the two rectangular transducers we
have
oo
VET = 27ra6ViV2 / Ji(ar)J^br)J 0 (rr) — .
J
0
T
(11)
This expression can be evaluated analytically (apart from constant multipliers) with
the following result,
(
7T&2, r < \a - b\
a2 arccosp + b2 arccos # — 0,
0, r>a + b
a — b\ < r < a + b ,
(12)
where p = (r 2 + a2 - 6 2 )/(2ar), q = (r2 - a2 -f 6 2 )/(2frr), and the term Q is given
by 2172 = (r2 — (a — b)2)((a + b)2 — r 2 ). Although the expression in Eq (12) appears
complicated, its plotted value is simple and easily interpreted. The result is shown in
Fig. 3b) for the two possible cases. The solid curve shows the velocity profile for an
ET when both piston radiators have the same radius b = a, and the dashed curve is
for b < a. The dependence changes only in that the velocity is constant to a radius
equal to the absolute difference between the piston radii a — b\. Physically, this profile
corresponds to the simple overlap function of two disks, illustrated schematically in the
insets for a — b and a^b.
The radially decreasing velocity profile has the shape of a nearly linear function.
The unusual and unexpected result here is that the equivalent source we calculate to
yield the voltage described above should be a planar transducer of size a -f b with
a spatial distribution of its particle velocity according to Eq (12) and as shown in
Fig. 3b). The mathematical form of this radial velocity function is nearly triangular,
and the value drops from a maximum at the transducer center to zero at the rim
(r = a + 6). This is the so-called "Chinese hat" function, a term coined by Bracewell
[6] in conjunction with a related problem in optics.
842
b)
/cz=100
Chinese hat
Gaussian fit
|
0
1
r/a
2
o
1
r/a
2
3
FIGURE 5: Voltage comparison for equivalent "Chinese hat" function and Gaussian voltage
function versus axial transducer position for Ka = K,b = 50 and ranges of a) KZ — 50; and b)
KZ = 100.
An example of this substitution and the near equivalence of the two voltage
profile functions is shown in Fig. 5. The dashed curve is the voltage of two Gaussian
transducers, and the solid curve is the voltage profile of two pistons, both pairs at
various ranges. In Fig. 5a) we have kz — 50 (or about 1/3 Rayleigh distance), and
the two curves are close in functional form. Frame b) at kz = 100 shows the near
equivalence of these two voltage functions, although the range is still only 2/3 of the
Rayleigh distance. Already, the similarity is striking, demonstrating why substitution
of a Gaussian voltage for a rectangular one in analytical models is well justified. In
both cases of Fig. 5 the Gaussian transducer fields have been chosen with beam waist
values that minimize the deviation between the two functions for z — 0.
REFERENCES
1.
2.
3.
4.
L. M. Brekovskikh, Waves in Layered Media, (Academic Press, New York, 1975).
L. M. Brekovskikh, Usp. Fiz. Nauk 50, 539-79 (1953).
H. L. Bertoni and T. Tamir, Appl. Phys. 2, 157-72 (1973).
D. E. Chimenti, J. Zhang, S. Zeroug and L. B. Felsen, J. Acoust. Soc. Am. 95,
45-59 (1994).
5. O. I. Lobkis, A. Safaeinili, and D. E. Chimenti, J. Acoust. Soc. Am. 99, 2727-36
(1996).
6. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed., (McGrawHill, Boston, 2000).
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