On the transient response of a circular transducer

WAVE MOTION 8 (1986) 235-241

235

NORTH-HOLLAND

ON THE TRANSIENT

RESPONSE

OF A CIRCULAR TRANSDUCER

M. El B A N N A and M. E! N O K A L I Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. Received 17 May 1985, Revised 23 September 1985

The transient acoustic pressure resulting from the motion of a circular transducer mounted in an infinite bame is considered. A fast Fourier transform algorithm is used to obtain the exact pressure for cases involvingspatially uniform velocitypulses. Attention is focused on the phenomenon of "replica', whereby the pressure is a scaled version of the input velocity, in order to deduce a general criterion for this phenomenon to occur. The frequency dependence of the acoustic pressure at a given field point is also computed and shown to be a slowly varying function.

I. I n t r o d u c t i o n

The transient response of a circular transducer has undergone considerable investigations [1-4]. Although the emission from ultrasonic transducers is usually a steady-state sinusoid, considerable number of applications require pulsed ultrasound sources. Among these applications are N D E [5] and medical ultrasonic imaging [6] where it is necessary to understand the resulting image. The transient field of a circular transducer due to a spatially uniform velocity distribution is derived in [1] and [2], while the case of a spatially nonuniform transducer velocity is treated in [3, 7, 8]. The methods of solution commonly used to obtain the radiating field of a circular transducer are the Rayleigh integral, the King integral and the Schoch solution [2]. The basic idea of the Rayleigh integral is borrowed from the well-known Huyghen's principle in optics which divides the radiating surface into point sources each emitting a wavelet with the resultant radiating field being the superposition of these wavelets. The King integral results from the solution of Helmholtz equation in cylindrical coordinates, while the essence of the Schoch solution lies in the selection

of a suitable origin for the coordinate system to carry out the Rayleigh integral. Although the previous methods of solution were used for the case of sinusoidally excited radiators, they can be modified to deal with the case of pulsed radiators. Hence, it can easily be shown that, starting from the Rayleigh integral, the radiating field for a circular transducer is obtained by convolving the impulse response with any arbitrarily specified velocity function [2]. In this paper the impulse response of a circular transducer is first reviewed and the relations required to find the transient pressure are presented. The convolution integral resulting from the problem formulation is carried out using an FFT subroutine. The transient pressure due to a monosinusoidal velocity is computed for different ratios of r / a , r being the distance normal to the piston's axis, and a the piston's radius. The frequency dependence of the maximum sound pressure as well as its dependence on the radial distance are deduced for this monosinusoidal velocity input. The phenomenon of replica is investigated by selecting the input velocity to be a pair of sinusoidal pulses separated by a time r and a general criterion for its occurrence is then deduced.

0165-2125/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

M. El Banna, M. El Nokali / Transient response of transducer

236

2. The impulse response

The impulse response of a circular transducer at point M is given by

L(R) h(r,t)2art sin O(R)

(1)

where L(R) is the arc length of the intersection of the piston with a sphere of radius R centered at M, as shown in Fig. 1, t is the instantaneous time ~z -

-

r

M

The power intensity, I, is obtained by multiplying the sound pressure by the complex conjugate of the particle velocity, i.e.

l(r, t) = P(r, t) . U*(r, t),

(5)

where U* is the complex conjugate of the particle velocity U. It is worth noticing that the area in front of the circular transducer can be divided into three regions depending on the ratio r/a. Table 1 summarizes the expression for the impulse response of the transducer in each of these regions as well as its time duration. Table 1 Expressions and time durations for the impulse response of the transducer

RI

Region

l 1I

J

L(R

III

r~ a

h ( r, t)

Duration

0.0 <1.0

c c (c/'rr).ot(t) (c/~r) el(t)

z/c<~t<~Rt/c, R t = R 2 z/c<~t R t Rffc<~ t<~ R2/c

>1.0

•

× Fig. 1. Geometrical parameters used to calculate the impulse response of a circular transducer.

a ( t ) = COS-t((c2t 2 - z2+ r 2 - a2)/(2r(c2t 2 - z2)t/2)).

and O(R) is the angle between R and the normal to the piston surface from M. The impulse response is a time-limited function of duration (R2-R~)/c, where c is the speed of sound, R2 and Rt are the largest and smallest distances from the point M respectively, to the rim of the transducer. Once calculated, the impulse response is used to compute the velocity potential function 4) from the relation

3. Transient pressure

qS(r, t)= v(t) * h(r, t)

(2)

where v(t) is the velocity function fed to the transducer and * denotes the convolution process. The sound pressure P and the particle velocity U are expressed in terms of the velocity potential function ~b as a P(r, t ) = p - ~ tb(r, t), (3)

At this point it is instructive to solve for the transient sound pressure P due to an impulsive velocity of strength V0. Using eqs. (2) and (3) we get

qb(r, t)= Voh(r, t),

(6)

Oh(r, t) Ps(r, t) = pVo - (7) at Substituting for h(r, t) in (7) from Table 1, the sound pressure P~ in region III can be solved for and written as Pz(r,

t) =

_

(pVoc/rr) pc 2

a,~(t)

•- at

Vo

IT(1

(4)

X ( X 2 _ Z ~ + y 2 _ 1 ) - 2 X ( X 2 - Z 2) (X:_Z:)[4y2(x2_z 2) _ ( X 2 _ Z 2 + y 2 _ 1)]1/2

where p is the density of the irradiated medium.

(8)

U(r, t ) = - V ~ b ( r , t),

M. El Banna, M. El Nokali / Transient response of transducer

where X is the normalized time ct/a, Y is the ratio r~ a, and Z is the normalized distance z~ a. Figure 2 shows the normalized pressure ~raPs(r, t)/pc 2 Vo, ,

Q_

!

8

Y:4.0 Z = I0.0

4

0 :l.0Cm

237

infinite values in Ps, the aliasing problem cannot be avoided entirely. However, a reasonable number of samples, N = 21°, is chosen in this paper. The velocity function v(t) is then sampled with the same rate as the sound pressure and is written as M

vs(nT)= ~ v ( t ) 8 ( t - n T ) ,

(11)

n=l

where M is the number of samples of v(t). The FFT computes the circular convolution while eq. (9) represents a linear convolution. However, the FFT can easily be modified to compute the linear convolution of (9) by augmenting both Pss(r, nT) and vs(nT) by zeros such that the total number of samples in each function becomes L = M + N - 1. The FFT subroutine can then be used to calculate the discrete Fourier transform (DFT) Ps(k) and v(k) where

0

-4 -8

0

I

I

I

I

2

4

6

8

0

t (F-s) Fig. 2. Normalized transient pressure P = Ora/pc 2Vo)P~versus

time t.

L-1

Ps(r, k) = E Psi(r, nT) e -i¢2~/L)nk,

(12)

n=0

as given by (8), for a point in region Iii. Notice that the pressure approaches asymptotically +oo at Rl/c and -co at R2/c, Rl/C is taken as the origin. The time integration of P8 is equal to zero which implies that the area under the positive pressure is equal to the area under the negative pressure. From (2), (3) and (7) it is straightforward to show that the sound pressure P(r, t) due to a velocity v(t) is obtained by convolving it with the pressure resulting from the impulsive velocity Ps, i.e.

P(r, t) = v(t) * Pt(r, t).

(9)

The convolution integral of eq. (9) will be carried out by an FFT subroutine. First, the function Ps(r, t) is properly sampled to avoid the problem of aliasing. The sampled pressure can be expressed

L-I

v(k) = ~, Vs(nT) e -i(2"/L)"k.

The amount of computation needed to evaluate the DFT is approximately proportional to L 2, while it is only proportional to L Iog2 L when the FFT is used. Hence, the computation time is reduced by a factor L/log2 L, i.e., by a factor of 2'1/11 in our case ( L = 2 ' 1 ) . By multiplying point by point Ps(r, k) and v(k) as expressed in (12) and (13) one obtains the DFT of the sound pressure P(r, k) as

P(r, k) = P~(r, k) . v(k).

N

Pss(r, nT)= ~ P s ( r , t ) 8 ( t - n T )

(10)

n=l

where T is the sampling period. Because of the

(14)

Finally, the time samples of the sound pressure is obtained by taking the inverse FFT of P(r, k), i.e.,

P(r, nT)= T~-'[P(r, k)].

as

(13)

n=0

(15)

In the next section, the technique outlined above will be used to calculate the transient sound pressure for the case of a monopulse and a pair of bipolar pulse velocity.

M. El Banna, M. El Nokali / Transient response of transducer

238

4. Results and discussion

4.1. Monopulse In this section the transient sound pressure due to a monosinusoidal pulse is presented. The curves of Figs 3(a,b) show the transient pressure with water being the irradiated medium at a frequency of 2 MHz for two different values of Y = r/a, all being greater than one. From these curves it is

O. I I

i

I

Y:4.0

z ,oo

0.09

0.2

Y : 2.0 Z = I0.0 o = I.OCm

I

0.1

tl)

clear that as the field point gets farther from the transducer, the magnitude of the sound pressure decreases and the negative portion of the pressure becomes distorted. The m a x i m u m sound pressure at a given distance Y = 4.0, is plotted versus the frequency in Fig. 4(a),

'O

0 x

0.07

a_E

X

n

0.05

(a) -0.1

0

0.03

i

I

1

I

I

I

2

3

4

5

0

[

I

I

I

I

I

2

3

4

5

i

I

f (MHz)

t (/~s) i

I

I

I

I

I

0.24

i

I

I

Y=4.0 Z = I0.0

0.04

=

.

0 = I.Ocm

'_o

0.16

X

'0

K

x

o

0

a. E

0-

0.08

-0.04

-

(b)

(b) 0

0

I

I

I

I

I

2

4

6

8

I0

t

(/~s)

Fig. 3. Normalized pressure P resulting from a monosinusoidal velocity input; (a) Y = 2.0 and (b) Y = 4.0.

2

1

I

I

I

I

4

6

8

I0

12

R (cm) Fig. 4. M a x i m u m n o r m a l i z e d pressure resulting from a s i n u s o i d a l v e l o c i t y i n p u t (a) versus f r e q u e n c y f, (b) versus r a d i a l d i s t a n c e R at an a n g l e 30 ° to the vertical.

239

M. El Banna, M. El Nokali / Transient response of transducer

showing that the frequency dependence of the pressure is a slowly varying function. It can also be noticed from this curve that the bandwidth of the maximum sound pressure is less than typical transducer bandwidths of few MHz. The maximum sound pressure as a function of the radial distance R at 30 ° from the vertical axis is shown in Fig. 4(b). The curve shows the field is inversely proportional to R (far field) for distances greater than 10 cm (0.75 a2/A) which is in accordance with the well-known theories in optics.

(2) The ratio between the maximum positive and the maximum negative in each version is almost unity; (3) The two versions are separated by an interval of nearly constant pressure, the duration of this portion depends on the choice of r. Applying these three conditions to Fig. 6(a), we find that when r equals zero, the sound pressure 0.08

4.2. A pair of bipolar pulses

.

.

.

.

0.04

A pair of bipolar sinusoidal velocity pulses separated by an interval r as shown in Fig. 5 is selected

I

o X

0

n

r

-3

0.8

Y=4.0 Z = I0.0

0.4

O= I.OCm

-0.04

(a)

T=O.O

0 -0.4 -0.8

0

2

l 4

0

I

I

I

4

6

8

I0

t (Fs) f

I

i

0.04 6

8

\

I0

t (M-s) Fig. 5. Bipolar sinusoidal input velocity v.

to study the phenomenon of replica which is of considerable interest in studying the near and far fields of the transducer [4]. For the output to be a replica (scaled version of the input velocity) the following three conditions have to be met: (1) The output sound pressure consists of two scaled versions of the input velocity. The first version is in phase with the input velocity because of the positive infinite value of Ps, while the second one is out of phase with the input velocity due to the negative infinite value of Ps;

,

0

O O-

Y=4.0

-0.04

Z = I0.0 O=l.OCm 2" = 0 . 2 0 / C

(b) -0.08 0

t

t

i

t

2

4

6

8

0

t (l~s) Fig. 6. Normalized pressure resulting from a bipolar sinusoidal input velocity for different values of ~'; (a) ~'=0.0, (b) ~ = 0.2 a/c, (c) r=O.4 a / c and (d) r=O.6 a / c

240

M. El Banna, M. El Nokali / Transient response of transducer

Fig. 7(b) the second pulse comes before the convolution of the first pulse with both the positive and the negative sound pressure ends, hence, it is not a replica. This holds true for higher values of ~-.

i

0.04

o\

Itl I

om

i

0.04

X

(3_

-O.04

-

Z = 10.0 0 = 1.0 c m

(c)

I

T = 0 . 4 O/C X

-0.08

i 2

0

I

I

I

4

6

8

n

Y=4.0

I0

Z = I0.0

-0.04

t (/zs)

0 = 1.0cm

T = 0 . T O/C

(a) 0.04

-0.08

0

i

I

I

I

4

8

12

16

20

t (/~s) I

\

O

0 X

J

n

-O.O4

0.04 Y=4.0 Z = I0.0 O= I.OCm

(d) -0.08 0

I

"l" = 0 . 6 O/C

J

I

t

I

4

8

12

16

0

o X

20

t (/zs)

(3-

Y=4.0 Z = I0.0

-0.04

0 = I.Ocm

Fig. 6 (continued)

does not meet the above second condition, therefore it is not a replica of the input. While Figs. 6(b) and (c) meet all the above three conditions and hence each of them is a replica of the input velocity. Figure 6(d) is computed for r=O.6a/c and critically meets the above conditions. Figure 7(a) is not a replica because the two negative sound pressure pulses have merged into one pulse. In

"Lr = 0 . 8 0 / C

(b) -0.08 0

I

I

a

i

4

8

12

16

20

t (/zs) Fig. 7. Normalized sound pressure where it is no longer a replica of the input; (a) interference occurs, (h) second pulse comes before the convolution of the first pulse with both positive and negative pressure ends.

M. El Banna, M. El Nokali / Transient response o f transducer

5. Summary and conclusions

References

The F F T is used to c o m p u t e the transient s o u n d pressure of a circular t r a n s d u c e r resulting from the m o n o s i n u s o i d a l velocity pulse a n d a pair o f b i p o l a r s i n u s o i d a l pulse. U s i n g the m o n o s i n u s o i d a l velocity pulse the f r e q u e n c y d e p e n d e n c e of the m a x i m u m s o u n d pressure is c o m p u t e d a n d s h o w n to be a slowly v a r y i n g f u n c t i o n . The m a x i m u m s o u n d pressure as a f u n c t i o n of the radial distance is p r o d u c e d a n d used to distinguish the n e a r from the far field. The p h e n o m e n o n of replica is studied a n d three deduced.

conditions

for

241

its

occurrence

are

[1] P.R. Stepanishen, "Transient radiation from pistons in an infinite planar baffle", J. Acoust. Soc. Amer. 49 (5), 16291638 (1970). [2] G.R. Harris, "Review of transient field theory for a baffled planar piston", J. Acoust. Soc. Amer. 70 (1), 10-20 (1981). [3] G.R. Harris, "Transient field of a baffled planar piston having an arbitrary amplitude distribution", J. Acoust. Soc. Amer. 70 (1), 186-204 (1981). [4] D.E. Robinson et al., "Near field transient radiation patterns for circular pistons", IEEE Trans. Acoust., Speech and Signal Proc. 22 (6), 394-403 (1974). [5] G.V. Blessing et al., "The effect of surface roughness on ultrasonic echo amplitude in steel", Ultra. Syrup. 923-927 (1983). [6] J.F. Havlice et al., "Medical ultrasonic imaging: An overview of principles and instrumentation", Proa IEEE 62 (4), 620-639 (1979). [7] E.G. Williams and J.D. Maynard, "'Numerical evaluation of the Rayleigh integral for planar radiators using the FFF', J. Acoust. Soc. Amer. 72 (6), 2020-2030 (1982). [8] P.R. Stepanishen, "Forward and backward projection of acoustic fields using FFT methods", J. Acoust. Soc. Amer. 71 (4), 803-812 (1982).