Application of the angular spectrum approach to compute the far-field pressure

Ultrasonics

36

( 1998) 47-5 I

Application of the angular spectrum approach to compute the far-field pressure

Abstract The angular spectrum decomposition of an acoustic source can be used to compute the radiation pressure in three dimensional radiation problems. Then. the aim of this paper is to demonstrate that the discrete far-field pressure can be computed from the two-dimensional (2D) discrete Fourier transform (DFT) of the normal velocity. In fact, using the asymptotic expression of the Rayleigh integral. it will be shown that the analytical far-field pressure is given by the 2D Fourier transform of the normal velocity. Using the discrete form of this expression, guidelines for selection of the sampling interval and the size of baffle in which the source is mounted will be given. The case of a rectangular source mounted in a rigid baffle and radiating into water will be considered. 0 1998 Elsevier Science B.V. Kc~~,nvrds: Angular

spectrum

approach;

Far-field

pressures

1. Introduction The angular spectrum approach (ASA) has been widely used to calculate acoustical fields. The numerical implementation of this method using the DFT results in some errors (spatial aliasing, frequency aliasing, phase shift error) [ 1,2]. Several authors [l-5] have used the ASA to compute the field pressures radiated by rectangular and circular sources. In their work they used the same expression in the far-field and in the near-field zone. The aim of this paper is to show that, in the farfield zone. a simpler expression, which allows fast computation, can be used. Moreover, guidelines for the choice of the sampling interval and the size of the baffle in which the rectangular source is mounted will be given. In the first section the far-field is computed according to the 2D-DFT of the normal velocity. The second section will be devoted to the study of the piston mode. the errors due to the use of the 2D-DFT and to discretization will be discussed. Guidelines will be proposed to choice the necessary parameters for the numerical implementation of the ASA. Finally, the influence of the guard band (D-2A region, see Fig. 1) is given.

Fig. I. Geometry acoustic field.

and

notation

I I 89:

004l-624X.!98,!$19.00 K+ 1998 Elsevier Science B.V. All rights reserved. PI1 s()041-674x(97)00133-9

for

the

calculation

of

the

2. Far-field pressure It can be shown that the far-field pressure radiated by a rectangular source mounted in a rigid baffle (see Fig. 1) with time harmonic excitation is given by: +m

a V(.Y’.J’)

p”(s. j.. 1) =po * Corresponding author. Fax: (33) 3 27 14 e-mail: [email protected]

used

s -aj

j -W

x exp[ -,j2n(Yf,

+ .I*:/;.) d-l-‘dj,’

(la)

with po

=

-jkpc

“”

(Fig. 1). The dimension of the source is equal to (2,4,2B). In this paper B is considered equal to A. Eq. ( la) shows that the far-field pressure is proportional to the two-dimensional (2D) Fourier transform ( FT) of the normal velocity, i.e.

(lb)

27tR

where the upper index a refers to the analytical expression, v(x, :) is the normal velocity on the source at L = 0, k is the wave number and k,.?..: = 2qf;,+,__ with .f;.,.= n,.+/h (n,, n, are the direction cosines, n, = cos (I,, n =cos (I)).). R = OM where M is the field point and 0 is the centre of the rectangular source

ny

0

0

P”(& .J: :)=I)0

U.r;>.r;.).

With the help of the two-dimensional fast Fourier transform (2D FFT), the numerical implementation of Eq. ( la) can be obtained. Nevertheless, some errors

ny

nx

0 0

nx

OK

I

I\

nY

(2)

d

1

nx nx

Fig. 2. Far-field pressures for 11,~= 1 for a rectangular source of size (2A.2A ) = (8;.,8i.) mounted in a rigid wall of width D = I281 and N = 256. The normal velocity is supposed constant. (a) Analytical pressure. (b) Discrete pressure. (c) Difference between (a) and (b). (d) Solid and dash-dotted lines present the analytical and discrete Cartfield pressure for q=O.

a

nY

0

b

0

nx

0

d

c -30

%

Z-

s al-20 m I '5.30

;620 D 2 ‘E 10 0 2-0

1

-10

1

1

-40

Fig. 3. Same as Fig. 2 except that I?,~= I.!2 and .Y= 128

9

0

Q

J

"X

I

Y

1

0.5 nx Fig. 4. Same as Fig. 2 except

that

2A is taken

equal

b

a

ny

to 3..

0 --0

ny

nx

0

0

nx

C

Fig. 5. Same as Fig. 4 except that nuP=]

appear, due to discretization and the use of the 2D FFT. These errors are discussed in the second section.

,4. ,%‘=h/l

pa(/?, ?I,, n, )=pO sin c(kA cos (I,) sin &A

cos Ov)

(3)

where ~~=4A’p,. If the classical sampling and the 2D discrete Fourier transform (DFT) of V(S, ~3): 3. Piston mode When the normal velocity is constant (i.e. V(S. J’) = v(,), the far-field pressure as given by Eqs. ( 1a) and ( lb) can be easily computed and given by:

R. Talbi et al. J Ultrasonics 36 ( 1998) 47-51

50

are used, computed

the discrete far-field and given by [6]:

pressure

can be easily

For nuP= 1, Fig. 2(a) and (b) show the analytical and discrete far-field pressure (Eq. (3) and Eq. (5)). Fig. 2(c) displays the difference between the analytical and discrete far-field pressures. Fig. 2(d) shows the analytical (solid line) and the discrete (dash-dotted line) far-field pressures for ny = 0. For nuP = l/2, Fig. 3 shows the same result as Fig. 2. We note a good agreement between the theoretical and DFT result for n,* B 1 where L, is defined by 2A = L,i, (see below). A difference between these pressures is observed when n,,/2
sin(nLm,/N) p,,(R, m,AJ mYAf) =p,, L sin(rcm,/N) -Jh

x exp

(

(m,Qx+ my&)

7

>

(5)

Where Af= l/(NAd), Ad is the sampling interval (Ad= l/f,, Ji being the spatial sampling frequency) used to sample the rectangular source (i.e. 2A = LAd), m,, my = -N/2 + l...N/2. Finally the size of the guardband D-2,4 is also sampled (D = NAd, see Fig. 1). The lower index D in the above equation refers to the discrete expression. Finally, 4, and #Y represent the phase shift error with 4,=#,=0.5 if L is even and ~$,=#,.=0 if L is odd. It is interesting to note that the magnitude of the discrete pressure does not depend upon this phase shift. In the following, for comparison purposes, the analytical pressure solution will be always computed , =2n,,m:% y/N with rn:, y = - N/ ;;n”,;” .$y(“n”u,“;. “nup is the maximal direction cosine in order to maximise the value (i.e. nuP =2/(2Ad)), angular range i.e. ]nx] I 1 and ]nyj I 1. In addition, the angular resolution at which the analytical pressure farfield is calculated is also equal to An,= An,=2n,,/N. In other words, the analytical solution will be undersampled when nuP is less than unity. In order to study the influence of nUP, D and L, several cases are considered below.

3.2. Influence of the size

In order to study the influence of the size of the source (2A,2A) on the choice of nUP, 2A is taken equal to 21 (i.e. 2A=2i.=LP;1), D=l28i,. As in Fig. 2, Figs. 4 and 5 show the pressures versus nuP (nuP= 1 for Fig. 4 and nuP= l/4 for Fig. 5). Good results are obtained for n,, y 1. When this condition fails, as in Fig. 5, the DFT values become erroneous [6]. 3.3. Influence of the guard band (i.e. D-2A) In order to show the influence of the guard band D-2A (D = NAd), Fig. 3 is considered. Fig. 6 displays the analytical (Eq. (3)) and the discrete (Eq. (5)) farfield pressures for nuP = 1, 2A = 81, and N= 128. For Figs. 3 and 6, the angular resolution An, = An,=2n,,/N at which the pressures are computed, are equal to l/128 and l/64, respectively. It is clear that

3.1. Influence of nuP with D and L$xed For different values of nuP and by keeping D = 1282 and 2A = 8/2, Figs. 2 and 3 show the far-field pressures.

s D

$ -20 2 .z z-40 I

1

1 “Y

0

-0

qf the source

b

0

$ -20 3 5 g-40 z 1

1

0.5 .7y!e9?.5 0 0 ny

“X

nx

0 c zi 6 D

-10 ZT E. a,-20 -0 a g-30

2 4 2 '$2 30

r"

1

1

ny

OO

IlX

-40 -50

0

0.5

nx Fig. 6. Same as Fig. 3 except that nup=

I and D=6i, (N= 128).

1

when D decreases, keeping N constant, a maximum but the computed range (i.e. nuP= 1) can be obtained discrete pressure becomes undersampled.

4. Conclusion Application of the angular spectrum approach (ASA) to compute the far-field pressure allows fast computation. Errors due to discretization and to the 2D FFT have been analysed. Using the above examples we note a good agreement between the analytical solution and the DFT result over the interval (-n,J2, n,,!2). Guidelines for the choice of the sampling interval and the size of the guard band are given. In fact. knowing 2A in terms of i., Ad can be found by the choice of the upper angular range (i.e. nuP =i,/(2Ad)) and D can be obtained (i.e. the size of the guard band) from the desired angular resolution An, = An, = 2+,/N.

References [I] P. Wu. R. Kazys, T. Stepinski. Analysis of the numerically implemented angular spectrum approach based on the evaluation oftwodimensional acoustic fields. Part 1. Errors due to the discrete Fourier transform and discretization. J. .Acoust. Sot. Am. 99 (1996) 13.19. [2] P. Wu, R. Kazys, T. Stepinski, Analysis of the numerically implemented angular spectrum approach based on the evaluation of twodimensional acoustic fields. Part II. Characteristics as a function of angular range, J. Acoust. Sot. Am. 99 ( 1996) 1349. [3] D.P. Orofino. P.C. Pedersen. Eficient angular spectrum decompoPart I: Theory. IF,EE Trans. Ultrason. sition of acoustic sources Ferroelec. Freq. Contr. 40 (3) ( 1993) 38. [4] D.P. Orofino. P.C. Pedersen. Eficient angular spectrum dccomposition of acoustic sources Part II: Results. IEEE Trans. Illtrason. Ferroelec. Freq. Contr. 40 ( 3) ( 1993) 250. [5] P. Wu. R. Kazys. T. Stepinski. Optimal selection of parameters for the angular spectrum approach to numerically cvnluate acoustic fields. J. Acoust. Sot. Am. IO1 (1997) 125. [6] J. Assad. J.-M. Rouvaen. Numerical aoluation of the fartield directivity pattern using the FFT. J. .4couqt. Sot. Am. (1997) submitted for publication.