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The radiation impedance of a rectangular piston
Journal of Sound and Vibration (1977) 55(2), 275-288
THE RADIATION
IMPEDANCE
A RECTANGULAR
OF
PISTON
P. R. STEPANISHEN Department of Ocean Engineering, University of Rhode Island, Kingston, Rhode Island 02881, U.S.A.
(Received 29 July 1975, and in revisedform 11 July 1977)
The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindemann [l]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.
1. INTRODUCTION The evaluation of the radiation impedance for a circular piston in a rigid infinite baffle is a classical problem in linear acoustics. As a result of the interest in the problem and the relative simplicity of its solution, the classical expression for the radiation impedance of a circular piston has been derived by a wide variety of approaches, which include Green function [2], integral transform [3], and, more recently, time domain techniques [4]. Although the radiation impedance of a circular piston in a rigid infinite baffle has been the subject of several analyses, the radiation impedance of a rectangular piston has received considerably less attention as a result of the more complicated nature of the problem. The Green function or multiple surface integral representation for the impedance has been used by Swenson and Johnson [SJ, and by Arase [6] to obtain integral and series representations for the radiation resistance and reactance, whereas Wyrzykowska [7] used the method to obtain a series representation for the radiation resistance. Stenzel [8] had previously obtained a series expression for the radiation resistance by relating the radiation resistance to the far field power while, most recently, Burnett and Soroka [9] numerically evaluated integral expressions for the radiation resistance and reactance which were obtained from the work of Chetaev [lo]. In the present paper, an alternative method of evaluating the radiation impedance of a rectangular piston is presented. This method [ 1 l] is based on evaluating the Fourier transform of the impulse response which is equivalent to the inverse Fourier transform of the impedance. Since the impulse response for the rectangular piston is available from the recent work of 275
276
P. R. STEPANISHEN
Lindemann [l], the present paper is concerned with the task of evaluating the transform and presenting new expressions for the radiation resistance and reactance of a rectangular piston. Relatively simple expressions are developed for both the low and high frequency behaviour of the radiation resistance and reactance. Numerical results are then presented to illustrate the behaviour of the resistance and reactance as a function of aspect ratio and normalized frequency. Finally, numerical results are presented to illustrate the accuracy of evaluating the resistance and reactance via the use of simplified low and high frequency expressions or via the use of FFT algorithms. 2. THEORY Consider a rectangular radiator which is mounted in a rigid infinite planar baffle as indicated in Figure 1. The radiator is in contact with a fluid in the upper half space. Furthermore, the radiator is specified to be vibrating with a sinusuidal uniform velocity which precludes any bending motion of the radiator. As a result of the motion of the piston, the acoustic medium exerts a reaction force on the piston which is linearly related to the velocity of the piston via the radiation impedance. The problem of interest here is to evaluate the radiation impedance of the rectangular piston as a function of the excitation frequency of the velocity and the aspect ratio of the piston. Although the radiation impedance for a rectangular piston in an infinite rigid planar baffle can be expressed as a multiple surface integral, it follows from recent results [l 11that the radiation impedance can also be expressed as a one-dimensional Fourier transform of an impulse response which relates the force on the piston to the velocity of the piston. The radiation impedance and the impulse response of a rectangular piston with an aspect ratio n = a/b are thus a Fourier transform pair which can be represented as follows : co z,,(kb) = r z,(T) eeJkbrdr,
- 03
z,(r) = ;
Z,,(kb) eJkbrd&b),
s --m
(la, b)
where Z,(kb) is the radiation impedance, and Z,(T) is the impulse response. The quantity kb is a normalized wave number, and z = et/b is a normalized time variable where c is the speed of wave propagation in the acoustic medium. Since the normalized radiation resistance and reactance coefficients of the rectangular piston, R,(kb) and X,(kb), are related to the radiation impedance by the expression Z,(kb) = pcA,[R,(kb)
+ jX,(kb)l,
Figure 1. A rectangular piston in an infinite planar rigid baffle.
(2)
211
RADIATION IMPEDANCE OF A PISTON
where A, is the area of the piston and pcA, is the plane wave resistance, it then follows that the normalized radiation resistance and reactance coefficients of a rectangular piston can also be evaluated directly from equation (la). In order to evaluate the radiation impedance of a rectangular piston by using the impulse response method, the impulse response of the rectangular piston must first be obtained. The impulse response for this case can be obtained from the results of Lindemann [ 11, via the use of an elementary scaling transformation, as z,(r) = /‘CA, [h(T) - k%(T)1~ where A, = b2n,
6(T)
is the Dirac delta function
(3)
and g,(z) is defined as
0,
1
T
&r(T) =
I (2/?27’C){ 1+
O
n - T},
(2/n) { 1 - (1/T)
(T' -
l
1)"2},
. (44
i
(2/M){T - (n/T)[(T' - 1)"' +
[(T'/n)' -
I)]"']},
io,
Note that g,,(r) can also be simply expressed as a sum of functions g,(r) =fi(r) where they,(r)
+fi(r)
+_&(r),
T),
o
(n’ +
n
6
(d
1)"2 < 5
+
1)1'2
: i.e.,
(4b)
are defined as +n-
l
n < f2(T)
1)1’2,
-(2/rCT)(T2 -
=
1<
-(2/7'CT)[(T/n)2 - 1]1'2,
f3(T)=
The normalized impedance can now be obtained obtain the following expression : Z,,(kb)
-1 PC&
5 <
n <
(n’ + 1)“2
T <
(n2 + 1)“2,
T <
(n’ + 1)‘j2.
by substituting
equation
(3) into (1 a) to
, _(nZ+‘)1’2 g,(T)
.i
e-jbbr
dr,
(5)
0
where the sifting property of the Dirac delta function has been utilized to evaluate its integral. After substituting the expression for g,,(r) in equation (4b) into equation (5) and then integrating the first term, the normalized impedance can be expressed as follows : z,,(kb)/pcA,
= 1 - [2/m(jkb)] x [l -ee-
jkb
[(n + 1) - (n2 + 1)1/2e-jkb(n’+1)“z] + [2/lcn(jkb)2] _
e-jnkb
+
e-jkb(n2+1)1’2
I+ G%)Fl(n,kb) + (2/n)FAn,kb), (64
where cn2+ 1)’ 12
(7’ - 1)“2 _jkbrdT
F,(n, kb) =
e
s
1
T
(n*+l)'i* [(T/n)' -
F,(n, kb) = sn
1]"2 e-jkbrdr.
(6b)
5
Equation (6) can be transformed to the expressions which were numerically integrated by Burnett and Soroka [9] to obtain their extensive tables for the normalized radiation resistance and reactance coefficients of rectangular pistons.
278
P.
R. STEPANISHEN
The integral expressions for F,(n, kb) and F&z,kb) can both be expressed in terms of a new integral via elementary transformations: i.e., F,(n, kb) = I(cosh-‘n,
kb),
F&z, kb) = [(cash-’ ([n’ + 1]“2/n), nkb),
U’a, b)
where
s Y
sin2 /3
I(& Y) = Since 1(x, y) can also be expressed
jxcosh8dfi.
as Y
1(x, y) =
&
tanh B
-JX s the following
expression
is then obtained
A(e-jxcashb) dp,
by parts :
after integrating
Y
I(x,y)
(8)
dP
0
= - 1 [tanhpe-1xcosh4]~+ Jx
e-j~
cash 5
-& ~ d/3. cash' /?I Jx s0
(9)
A new expression for the normalized impedance can now be obtained from the substitution of equations (7) and (9) into equation (6a). After performing the algebra, the following expression is obtained :
ZW) -z
1_
PC1+ 4lnnl + (Jkb)2 (y4
[1
_
e-ikb _ e-jnkb + e-Jkb(n*+l)l'*]
+
jkb
PC&
+ (f/N (2/n) C(kb, cash-’ (n2 T 1)“2) + 7Jnkb C(nkb, cash-’ [(n’ + l)““/nl), Jkb
(10)
where
s Y
C&Y) = Cl(w) +jC2(x,y) =
0
e-jxcoshLi ------d/3. cosh2 /I
It then follows from equations (2), (10) and (11) that the normalized reactance coefficients can be expressed as
(11) radiation
resistance
and
R,(kb) = I - [(2/nn)l(kb)2] [ 1 - cos (kb) - cos (nkb) + cos ((n’ + 1)“2 kb)] + [(2/lr)/kb] C,(kb, cash-’ (nZ + 1)“2) + [(2/m)/kb]
C2(nkb, cash-‘Kn2
+ l)“‘/n]),
(12a)
X,(kb) = [(2/n7r)/kbl(1 + n) - [(2/nn)/(kb)21 [sin (kb) + sin (nkb) - sin ((n” + 1)‘12 kb)] - [(2/r()/kb] Cl(kb, cash-‘(n2
+ 1)‘12)
- [(2/nn)/kb] C,(nkb, cash-’ [(n’ + I)“‘/n]),
(12b)
where
C1k.v) =
s
’cos(xcoshfi)
0
cash’ /I
’
dP,
Cz(x,
Y>
=
s 0
sin (x cash fi) cosh2 /I
dP.
(13a, b)
In order to evaluate R,(kb) and X,,(kb) by using equations (10) and (12) the integral C(x, v) must first be evaluated. Since the low and high frequency cases, i.e., nkb 4 1 and kb >>1,
RADIATION
IMPEDANCE
279
OF A PISTON
are readily evaluated from the integral expressions in equations (13) and (1 l), these cases are initially discussed. Consider first the evaluation of R,(M) and X,(H) for the low frequency case. It follows from the integral expressions in equation (13) that the trigonometric functions can be approximated for the case where x cash y < 1 by the initial terms in their power series. The quantities C,(x, y) and CJx, y) can then be approximated by the expressions
JJ dp
C,(x, Y) =
Y
x2 ’ dp, 2 s 0
--s cosh2p 0
which can be readily evaluated
’ dB C,(x, y) = -x s coshp
to obtain
the following
expressions
C,(x, y) = -x sin-’ (tanhy)
-SY,
It then follows from the substitution of equations X,(M), for nkb 4 1, can be expressed as follows :
for x cash y < 1: + G sinhy.
(15) into equations
(lsa,b)
(12) that R,(kb) and
R,(kb) NN(kb)2 (n/2x), X,,(kb) “Nkb{(1/3m)
(14a,b)
0
3
X2
C,(X,Y) = tanhy
+ f coshfldj3, 3! s
0
(lea)
[1 + rz3 - (n’ + 1)3’2]
+ (l/rr)cosh-‘([n2
+ 11”‘) + (n/lr)cosh-‘([n2
+ 1]1’2/n)j.
(16b)
The expressions in equation (16) can also be obtained directly from equation (5) by again utilizing the small argument approximations for the exponential functions in equation (5) : i.e., e-Jkbr N 1 - +(kb)2 - jkbT. In order to evaluate R,(kb) and X,,(kb) for the high frequency case, where nkb 9 I, the asymptotic evaluation of the integral C(x, y) is first required. By an elementary application of stationary phase techniques [12], it is easily shown that, for x $ 1, C(x, y) N (~1/2x)r’~ em”“+ n’4) + O( 1/kb)3/2,
(17)
and therefore C,(x, y) N (rc/2x)“‘c0s
(x + X/4),
It then readily follows from equations be expressed as follows: R,(kb) N I - [(2/m)/(kb)2]
C2k
Y) _ -( 7r/2x)“2 sin (x + n/4).
(12) and (18) that R,(kb) and X,(kb),
(I 8a, b)
for kb $ 1, can
[1 - cos (kb) - cos (nkb) + cos ([n’ + 1]“2 kb)]
+ [2/7c(kb)3]1’2cos (kb + 377/4) + [2/rr(nkb)3]1’2 cos (nkb + 3n/4), X,(kb) N [{2( 1 + n)/nn}/kb]
(19a)
- [(2/nn)/(kb)2] [sin (kb) + sin (nkb)
- sin(kb[n’ + I]‘“)] - [2/x(kb)“]‘l’sin - [2/rc(nkb)3]1’2 sin (nkb + 37c/4).
(kb + 3n/4) (19h)
Although the normalized radiation resistance and reactance coefficients thus have been readily evaluated in terms of elementary functions for both the low and high frequency range, the solutions in equations (16) and (19) are only approximations to the exact solutions. In order to obtain the exact solution for the normalized radiation resistance and reactance coefficients in other than the integral form in equation (12), the integrals in equations (13) must be first evaluated.
280
P. R. STEPANISHEN
In order to evaluate are first noted:
Cl(x, y) and C2(x, y) as given in equations
cos (x cash 8) = .zO ~,,,(-l)~ J,,(x)
(13), the following
identities
cash (2mB),
sin (x cash j?) = 2 _$ (-1)” J,,,,, (x) cash ([2m + 1] B), m-0
(20a,
b)
where E, is the Neumann symbol and J,( ) is the Bessel function of the first kind of order v. The integrals CI(x, y) and C2(x, v) can then be expressed as series expansions: i.e., CL&Y) =
J,, (4 ~,CY),
,$ e,(-Om m=O
C,(x,y)
= 2 z m-0
where the coefficients
(-1)” J ~m+~(~)4m+l(~),
Wa,
b)
d,,,(y) are defined by the integral
4n(y) =
(22) 0
Although the coefficients d,,,(y) can be readily expressed in terms of either hypergeometric functions or incomplete beta functions, the integral expressions for d,,,(y)can also be integrated via the use of standard integral tables. In particular, it readily follows from the use of tables [13] that the initial coefficients can be expressed as
do(y)
=
s’
dP
___
cosh2 /?
= tanhy,
s’
dP
d,(y) =
0
0
= sin-’ (tanhy), cash /I
(23a b)
Y
cash 28 W)
-
= s 0
cosh2 fi
d/3 = 2y - tanh y,
d,(y) =
‘cash 3p ___ dfi = 4 sinh y - 3 sin-’ (tanh y). s cosh2 fi
0
G%
4
The coefficients d,(y) for m > 2 can also be expressed via the use of the tables [ 131 as a finite series of hyperbolic sine functions. The normalized radiation resistance and reactance coefficients can now be obtained by substituting the results of equations (21) into equations (12). It then follows that the radiation coefficients can be expressed as R,(kb)
= 1 - [(2/7~)/(kb)~] 4/x +kb
[l - cos( kb) - cos (nkb) + cos ([n” + l]“‘kb)]
= c
(-l)“J2,+1
(kb)d,,+,(cosh-‘[n’
4/7-w m +-
kb
c
2/n
2/W kb
dz,,,+l cosh-1[T]1’2),
- [(2/nn)/(kb)2]
[sin(kb)
Pa)
+ sin(nkb)
- sin( [nz + l]1’2kb)]
@
G kb lfl=O --
(
(-1)” J ,m+lWb)
Ill=0
X,(kb) =[(2 (1 + Q)/nnf/kb] --
+ 1]“2)
III=0
m c
III=0
E,(-1)“’ Jz,,, (kb)d,,
(cash-‘[n”
+ 11”‘)
&,,A- 1)” Jzn(nkb)d2, ( cosh-‘r+r2),
Wb)
281
RADIATION IMPEDANCE OF A PISTON
For the low frequency case where nkb 6 1, it readily follows, via the use of equations (23) and the small argument approximations for the Bessel functions, that equations (24a) and (24b) reduce to the low frequency expressions in equations (16a) and (16b), respectively.
3. NUMERICAL
RESULTS
In the preceding section an expression for the impulse response of a rectangular piston as a function of its aspect ratio n = a/b was presented in equations (3) and (4). The normalized impulse response for a rectangular piston of any aspect ratio n can thus be evaluated from the expression and the results for II = 1, 2,4 and 8 are shown in Figure 2 where each response is normalized by the plane wave impedance, pcA,, of the piston. Since the normalized impulse responses are plotted on the same time scale, the results correspond to the case of a rectangular piston with a fixed width b, whose length varies in accordance with the aspect ratio. The results in Figure 2 clearly indicate several points of interest concerning the variation of the normalized impulse response with the aspect ratio. Each impulse response has a Dirac delta function with a unit strength at z = 0, which is then followed by a negative pulse whose characteristics are dependent on the aspect ratio. Since the integral of the impulse response over all time r must be zero [I, 1 l] the integral of the negative pulse must be unity to offset the contribution of the delta function. As indicated by the expressions and the results in Figure 2, the time duration of the impulse response increases as the aspect ratio increases. Furthermore, as the aspect ratio increases, the maximum negative value of the impulse response which occurs at 7 = 0 monotonically decreases and approaches the limiting value of 2/7r. Although the behavior of the impulse response for an aspect ratio n is dependent on the aspect ratio for 0 < 7 > 1 and n < 7 < m, the behavior of the impulse response over 1 < 7 < n is independent of the aspect ratio (see equation (4a)). It thus follows that the impulse response for an aspect ratio m > n exhibits the same temporal behavior over 1 < 7 < n as the impulse response for the aspect ratio n where n > 1. As an example, it is noted from the results in Figure 2 that the impulse response for an aspect ratio of 4 is equivalent over 1 < 7 < 2 to the impulse response for an aspect ratio of 2. Several points of interest concerning the behavior of the normalized mutual radiation resistance and reactance coefficients as functions of kb and the aspect ratio can be obtained by inspection of the results in Figure 2. Since the coefficients can be expressed as the Fourier transforms of the respective impulse response functions, the characteristics of the coefficients can thus be obtained from elementary properties of Fourier transforms. In particular, note
Figure 2. The normalized impulse response for different aspect ratios. -, n=2;____,n=l.
n = 8; -
-, n = 4; - - -,
282
P. R. STEPANISHEN
kb (rodlord
Figure 3. The normalized radiation resistance coefficient for different aspect ratios.
that the resistance and reactance coefficients approach unity and zero, respectively, for kb $ 1, for all values of the aspect ratio, as a result of the delta function at r = 0. Furthermore, since the asymptotic behavior in the time domain governs the small argument behavior in the frequency domain, it follows that the coefficients will be dependent on the aspect ratio for small kb. Finally, since the time structure of the impulse response is only weakly dependent on the aspect ratio n when n is greater than approximately 4, the resistance and reactance coefficients can be expected to exhibit the same behavior as functions of kb, except for small arguments as noted above. Although the normalized impulse response of a rectangular piston is of interest for pulsed excitations, the normalized mutual radiation resistance and reactance coefficients are of more interest for harmonic excitations. For this reason, the resistance and reactance coefficients as functions of kb are presented in Figures 3 and 4 for the same aspect ratios which were previously considered. These curves have been obtained from the highly accurate tables of the coefficients which have been previously tabulated as a function of kb& by Burnett and Soroka [9]. Although the selection of kbfi as the independent variable results in an aspect-independent square law relation for the resistance coefficient at small kb (see equation (16a)), the selection of kb as the independent parameter leads to results which are more readily interpreted in the light of the present analysis. Furthermore, the similarities between the resistance and reactance coefficients as functions of aspect ratio are more readily noted. The normalized radiation resistance and reactance coefficients which are shown in Figures 3 and 4 exhibit several interesting characteristics. At low frequencies, where kb < 1, the
1.00
2.00
3.00
4.00
5.03
7.00
8.00
kb (rodlam)
Figure 4. The normalized radiation reactance coefficient for different aspect ratios.
RADIATION
IMPEDANCE
OF A
PISTON
283
resistance and reactance coefficients are aspect-dependent and are proportional to (kb)2 and kb, respectively (see equation (16)). The coefficients for aspect ratios greater than approximately 4 are noted to exhibit very similar behavior for kb > 1. Hence, for n > 4 and kb > 1, on the aspect ratio. the resistance and reactance coefficients are only weakly dependent Finally, the asymptotic or high frequency results are also somewhat evident from the results in which the resistance coefficient is seen to rapidly approach its limiting value of unity for kb greater than an approximate value of 4. In order to provide an indication of the accuracy of the approximate expressions in equations (16) and (19) for the resistance and reactance coefficients, Tables 1 and 2 present data on the accuracy of the approximate expressions for the resistance and reactance coefficients as functions of kb for n = 1 and 4. The numerical results obtained from the approximate expressions are compared to the corresponding numerical results which were published by Burnett and Soroka and are shown in Figures 3 and 4. Since equation (16) is only valid for t&b < 1, its use has been restricted to 0 < kb < 1. Furthermore, since equation (19) is only valid for kb + I, its use has been restricted to kb 2 2. The results in Table 1 illustrate the expected behavior of the approximate expressions for the resistance coefficient. Although equation (16a) yields reasonably accurate results over 0 < kb < 1 for n = 1, the percent errors are much larger for n = 4. This is, of course, to be expected since nkb = 4 at kb = 1. In contrast to the use of equation (16a) which results in monotonically increasing errors as kb is increased, the use of equation (19a) results in a decaying oscillatory error as kb is increased. Furthermore, the errors associated with the use of equations (19a) decrease as n is increased. Note that equation (16a) yields a 27% error for kb = 2 and n = 1; hence, if equation (16a) is used for 0 6 kb < 2 and equation (19a) is used for kb > 2, the resulting expressions yield results for n = 1 which are in error by less than 30 % for all kb. The results in Table 2 illustrate that the approximate expressions for the reactance coefficients in equations (16b) and (19b) exhibit the same behavior as evidenced by the resistance coefficients in Table 1. More specifically, the percent error resulting from the use of equation (16b) increases monotonically as kb is increased, whereas the percent error resulting from the use of equation (19b) exhibits a decaying oscillatory behavior as kb is increased. Once again, it is noted that the accuracy of equation (16b) deteriorates as n is increased for a fixed kb whereas the inverse holds true for equation (19b). As the final item of interest, consider the evaluation of the radiation resistance and reactance coefficients by using an FFT approach. Since the radiation coefficients are simply related to the Fourier transform of the appropriate impulse response (see equations (2) and (5)), it then follows that the coefficients can be readily evaluated via the use of standard FFT algorithms and methods. The utilization of the FFT approach thus offers an alternative method to evaluating the resistance and reactance coefficients via the use of the series in equation (24). In order to provide an indication of the accuracy in evaluating the radiation coefficients via the FFT approach the results in Table 3 show the normalized percent error between the resistance and reactance coefficients obtained from the FFT approach relative to the tabulated values of Burnett and Soroka [9]. More specifically, the percent errors for both the resistance and reactance coefficients are shown as functions of kb for aspect ratios of n = 1 and 4. Since the percent errors are dependent on the sampling time interval which was used for each case of interest, the sampling interval and the number of non-zero samples of the impulse response are indicated for each case. The results in Table 3 clearly show several trends of interest. In particular, large percent errors in the resistance coefficient are noted for small values of kb; however, the error decreases as either kb or the sampling frequency of the impulse response increases. Note that as
040000 om157 OX)0635 0.01425 0.02524 0.03724 0.05616 0.07589 0.09830 0.12325 0.15056
oaoooo
n=l Burnett and Soroka [9] 0.51012 0.70431 O-87306 0.99807 1.07100 1.09380 1.07700 I.03630 0.98870 0.94855 0.92521 0.92195 0.93638
0.70282 0.87515 0.96831 1 .01547 1.03469 I .03605 1.02540 1mO97 0.98501 0.96428 0.94957 0.9445 1 0.95077
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
n=l
Equation (19a)
O-001.59 0.00637 0.01432 0.02546 0.03979 0.05730 0.07799 0.10186 0.12892 0.15925
n=l Burnett and Soroka [9]
n=l Equation (16a)
38 24 11 2.7 3.4 -5.3 -4.8 -2.8 -0.37 1.7 2.6 2.4 1.5
Percent error
n=l
0.31 0.49 0.87 1.40 2.00 2.80 3.60 4.60 5.70
0.00
0.00
?Z=l Percent error
7.5 6.8 1.5 0.37 -0.59 -2.1 -1.4 0.73 -0.71 0.28 0.90 0.58 0.75 0.78323 0.92828 1.01732 1.05530 1.07555 1.06673 1.03473 1.00894 0.98563 0.96282 0.95796 0.96461 0.97171 0.85780 0.99170 1.03260 1.05914 1.06419 I.04473 1.01981 1m157 0.97861 0.96555 0.96659 0.97022 0.97899
-
n=4 Percent error
090 0.47 1.90 4.30 7.70 12.20 17-90 24.80 33+IO 42.00 540l n=4 Burnett and Soroka [9]
Of)0634 0.02499 0.05493 0.09454 0.14179 o-19437 0.24995 0.30635 0.36367 0.41446
090000
n=4 Percent error
of kb and n
n=4 Equation (19a)
OWOOO 0.00637 O-02546 0.05730 0.10186 0.15915 0.22918 0.31194 040744 0.51566 0.63662
as a function
n=4 Burnett and Soroka [9]
coeficient
n=4 Equation (16a)
(16a) and (19a) to evaluate the resistance
kb
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
kb
The percent error in using equations
TABLE 1
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
kh
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
0.1
0.0
kh
(I 6b)
(16b)
0.72499 0.7478 1 0.58700 0.45066 0.34066 0.2563 1 0.19613 0.15846 0.14119 0.14102 0.15298 0.17055 0.18662
n=l Equation
0~00000 0.04732 0.09464 0.14196 0.18923 0.23660 0.28392 0.33124 0.37856 0.42588 0.47320
n=l Equation
Burnett
Burnett
0.66133 0.67109 0.61995 0.52615 0.41245 0.30126 0.2 1042 0.15035 0.12323 0.12397 0.14267 0.16764 0.18842
and Soroka
n=l
oGOOOo 0.04728 0.0743 1 0.14084 0.18664 0.23 146 0.27508 0.31727 0.35783 0.37656 0.43327
and Soroka
??=I
[9]
[9]
error
40.00 11.40 -5.30 -14.30 -17.40 -14.90 -6.80 5.4 14.6 13.8 7.2 1.7 -0.96
Percent
n=l
0.00 0.08 0.30 0.80 1.40 2.20 3.20 4.40 5.80 7.40 9.20
?Z=l Percent error
-.._
(16b) 0.65179 0.50228 0.37287 0.29115 0.20094 0.13234 0.10617 0.08867 0.08220 0.09706 0.11061 0.11723 0.12554
n=4 Equation
0~00000 0.08468 0.16935 0.25403 0.33870 0.42338 0.50806 0.59273 0.67741 0.76209 0.84676
Burnett
0.53646 0.48077 0.38468 0.30007 0.22547 0.15396 0.11182 0.09408 0.08329 0.08909 0.10540 0.11371 0.11976
n=4 and Soroka
omOoo 0.08422 0.16577 024219 0.31139 0.37185 0.42265 0.46356 0.49493 0.51763 0.53295
[9]
[9]
21.50 4.50 3.07 -2:84 -10.90 -14+)0 -5.05 -5.75 -1.31 8.95 4.94 3.10 4.83
n=4 Percent error
0.00 0.55 2.20 4.90 8.80 13.90 20.20 27.90 37.00 47.00 58.00
n=4 Percent error
of kh and n
n=4 and Soroka
a.s a jimction
Burnett
cot$cient
12= 4 Equation (16b)
The percent error in using equations (16b) and ( 19b) t o evaluate the reactance
TABLE 2
286
P.R.STEPANISHEN
kb increases the percent error reaches a minimum and then increases, though not monotonically, as kb increases. In contrast to the percent errors for the resistance coefficients, the results in Table 3 show the percent errors for the reactance coefficients to be considerably less than 0.1% for the cases of n = 1 and 4 over the kb range of 0.0 to 8.0. Although the resistance and reactance coefficients as functions of frequency and aspect ratio are available for selected frequencies and aspect ratios from the tables of Burnett and Soroka, the coefficients at other frequencies and aspect ratios can be obtained either by interpolation methods or by the methods of the present paper. Several observations concerning the evaluation of the radiation resistance and reactance coefficients as functions of frequency and aspect ratio are thus noted from the preceding numerical results and analysis. For nkb < 1 the coefficients can be accurately evaluated via the use ofequations (16), whereas, for kb g 1 the coefficients can be accurately evaluated via the use of equations (19). Although
TABLE 3 The normalized percent error in the resistance and reactance coefJicients obtained via the FFT approach
Aspect ratio = 1 Sampling time interval = 0.0153398 Number of points = 93 A \ r Percent error Percent error reactance kb resistance 0.20 040 060 0.80
1.00 290 3.00 490 5.00 690 7.00 8.00
76.7431 19.3158 8.6824 4.9619 3.2405 0.9584 0.5609 0.4571 0.4534 0.4928 0.5268 0.5219
0.0037 0.0036 0.0036 0.0036 0.0036 0.0035 0.0035 0.0041 0.0090 0.0267 0.0369 0.0346
Aspect ratio = 4 Sampling time interval = 0.0153398 Number of points = 269 z% / \ Percent error Percent error resistance reactance kb 0.20 0.40 0.60 0.80
1.oo 2.00 3.00 4.00 5.00 6.00 7.00 8.00
12.2308 3.2306 1.5695 0.9947 0.7345 0.3896 3.3007 0.2847 0.2949 0.3088 0.3180 0.3140
-00013 -0.0013 -0~0010 -0.0005 0~0001 0.0034 0.0042 0.0046 0.0105 0.0220 0.0314 0.0341
Aspect ratio = 4 Aspect ratio = 1 Sampling time interval = 0.0076699 Sampling time interval = 0.0076699 Number of points = 538 Number of points = 185 h h r , , Percent error Percent error Percent error Percent error resistance reactance kb resistance reactance kb 0.20 0.40 0.60 0.80 1.oo 2.00 3.00 4.00 5.00 6.00 7.00 8.00
38.0692 9.5867 4.3121 2.4668 1.6128 0.4805 0.2826 0.2298 0.2263 0.2442 0.2611 0.2608
OW55 oGO53 0.0053 oGO52 oGO5 1 oGO40 0.0016 -0003 1 -0GO95 -0.0016 0.0200 0.0237
0.20 0.40 0.60 0.80 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
6.1322 1.6116 0.7788 0.4922 0.3639 0.1947 0.1520 0.1425 0.1477 0.1537 0.1570 0.1576
-0w53 -0.0043 -0.0025 -0.0004 0.0015 0.0016 0.0038 -0GO12 -0.0165 0.0062 0.0129 0.0169
RADIATIONIMPEDANCEOF A PISTON
287
TABLE 3-continued Aspect ratio = 4 Aspect ratio = 1 Sampling time interval = 00319174 Sampling time interval = 0.0019174 Number of points = 2151 Number of points = 738 A /. ‘, I I Percent error Percent errop Percent error Percent error reactance kb resistance reactance kb resistance _...__. 0.20 19.6373 -0~0059 0.20 3.3991 -0.0270 0.40 4.9348 -0.0059 040 0.8626 -0.0238 0.60 2.2126 -0.0057 0.60 0.3987 -00188 0.80 1.2603 -0.0056 0.80 0.2426 -0.0129 I .oo 0.8194 -00053 1.00 0.1759 -0.0072 2.00 0.2366 -0~0030 2.00 0.0937 -0.0029 3.00 0.1373 o+IOo7 3.00 0.0757 0.0082 4.00 0.1135 0.0049 4.00 0.0703 0.0077 5.00 0.1147 0.0060 5.00 0.0756 -0.0170 6.00 0.1248 -0.0051 6.00 0.0785 0.0086 7.00 0.1315 -0.0085 7.00 0.0770 -0.0077 8.00 0.1287 0.0005 8.00 0.0791 -O+K)58
the reactance coefficients can be accurately evaluated by using the FFT approach for low, mid-range and high frequencies, the accuracy of the computations for the resistance coefficients decreases with frequency at low frequencies. Since the series expansions for the resistance and reactance coefficients in equation (24) reduce to the low frequency expressions in equation (16) and are ideally suited for low frequency rather than high frequency computations, the series expansions for the resistance coefficient can be used in the low frequency range. The FFT approach when combined with the series expansions can then be used to accurately evaluate the resistance and reactance coefficients in the low- to mid-frequency range whereas the FFT approach or asymptotic expansion in equations (19) can then be used in the high frequency range.
ACKNOWLEDGMENTS A portion of this work was supported by a grant from the United States National Institute of Health. In addition, the author wishes to acknowledge the constructive comments of the reviewers.
REFERENCES 1. 0. LINDEMANN1974 Journal of the Acoustical
2. 3. 4. 5.
Society of America 55, 708-717. Transient fluid reaction on a baffled plane piston of arbitrary shape. L. E. KINSLERand A. R. FREY 1964 Fundamentals of Acoustics. New York: John Wiley and Sons, Inc. M. C. JUNGERand D. FEIT 1972 Sound Structures and their Interaction. Cambridge: The MIT Press. P. R. STEPANISHEN1971 JournaI of the AcousticaI Society of America 49, 841-849. The time dependent force and radiation impedance on a piston in a rigid infinite planar baffle. G. W. SWENSON,JR. and W. E. JOHNSON1952 Journal of the Acoustical Society of America 24, 84. Radiation impedance of a rigid square piston in an infinite baffle.
288
P. R. STEPANISHEN
6. E. M. ARASE 1964 Journal ofthe Acoustical Society of America 36, 1511-1525. Mutual radiation impedance of square and rectangular pistons in a rigid infinite baffle. 7. B. WYRZ~KOWSKA 1956 Proceedings of the 2nd Conference on Ultrasonics, Warsaw 99-106. Acoustic impedance of plane generators; rectangle, ring, ellipse. 8. H. STENZEL 1939 NRL Translation No. 130 (translated by A. R. Stickley). Handbook for the calculation of sound propagation phenomena. 9. D. S. BURNETT and W. W. SOROKA 1969 Journal of the Acoustical Society of America 51, 16181623. Tables of rectangular piston radiation impedance functions, with applications to sound transmission through deep apertures. 10. D. N. CHETAEV 1951 Prikladnaya Matematika i Mekhanka 15,439444. The impedance of a rectangular piston vibrating in an opening in a flat baffle. Il. P. R. STEPANISHEN 1973 Journal of Sound and Vibration 26, 287-298. The impulse response and time dependent force on a baffled circular piston and a sphere. 12. E. T. COPSON 1965 Asymptotic Expansions. Cambridge, England: Cambridge University Press. 13. I. S. GRAUSHTEYN and I. M. RYZHIK 1965 Table of Integrals, Series and Products. New York: Academic Press.
THE RADIATION
IMPEDANCE
A RECTANGULAR
OF
PISTON
P. R. STEPANISHEN Department of Ocean Engineering, University of Rhode Island, Kingston, Rhode Island 02881, U.S.A.
(Received 29 July 1975, and in revisedform 11 July 1977)
The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindemann [l]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.
1. INTRODUCTION The evaluation of the radiation impedance for a circular piston in a rigid infinite baffle is a classical problem in linear acoustics. As a result of the interest in the problem and the relative simplicity of its solution, the classical expression for the radiation impedance of a circular piston has been derived by a wide variety of approaches, which include Green function [2], integral transform [3], and, more recently, time domain techniques [4]. Although the radiation impedance of a circular piston in a rigid infinite baffle has been the subject of several analyses, the radiation impedance of a rectangular piston has received considerably less attention as a result of the more complicated nature of the problem. The Green function or multiple surface integral representation for the impedance has been used by Swenson and Johnson [SJ, and by Arase [6] to obtain integral and series representations for the radiation resistance and reactance, whereas Wyrzykowska [7] used the method to obtain a series representation for the radiation resistance. Stenzel [8] had previously obtained a series expression for the radiation resistance by relating the radiation resistance to the far field power while, most recently, Burnett and Soroka [9] numerically evaluated integral expressions for the radiation resistance and reactance which were obtained from the work of Chetaev [lo]. In the present paper, an alternative method of evaluating the radiation impedance of a rectangular piston is presented. This method [ 1 l] is based on evaluating the Fourier transform of the impulse response which is equivalent to the inverse Fourier transform of the impedance. Since the impulse response for the rectangular piston is available from the recent work of 275
276
P. R. STEPANISHEN
Lindemann [l], the present paper is concerned with the task of evaluating the transform and presenting new expressions for the radiation resistance and reactance of a rectangular piston. Relatively simple expressions are developed for both the low and high frequency behaviour of the radiation resistance and reactance. Numerical results are then presented to illustrate the behaviour of the resistance and reactance as a function of aspect ratio and normalized frequency. Finally, numerical results are presented to illustrate the accuracy of evaluating the resistance and reactance via the use of simplified low and high frequency expressions or via the use of FFT algorithms. 2. THEORY Consider a rectangular radiator which is mounted in a rigid infinite planar baffle as indicated in Figure 1. The radiator is in contact with a fluid in the upper half space. Furthermore, the radiator is specified to be vibrating with a sinusuidal uniform velocity which precludes any bending motion of the radiator. As a result of the motion of the piston, the acoustic medium exerts a reaction force on the piston which is linearly related to the velocity of the piston via the radiation impedance. The problem of interest here is to evaluate the radiation impedance of the rectangular piston as a function of the excitation frequency of the velocity and the aspect ratio of the piston. Although the radiation impedance for a rectangular piston in an infinite rigid planar baffle can be expressed as a multiple surface integral, it follows from recent results [l 11that the radiation impedance can also be expressed as a one-dimensional Fourier transform of an impulse response which relates the force on the piston to the velocity of the piston. The radiation impedance and the impulse response of a rectangular piston with an aspect ratio n = a/b are thus a Fourier transform pair which can be represented as follows : co z,,(kb) = r z,(T) eeJkbrdr,
- 03
z,(r) = ;
Z,,(kb) eJkbrd&b),
s --m
(la, b)
where Z,(kb) is the radiation impedance, and Z,(T) is the impulse response. The quantity kb is a normalized wave number, and z = et/b is a normalized time variable where c is the speed of wave propagation in the acoustic medium. Since the normalized radiation resistance and reactance coefficients of the rectangular piston, R,(kb) and X,(kb), are related to the radiation impedance by the expression Z,(kb) = pcA,[R,(kb)
+ jX,(kb)l,
Figure 1. A rectangular piston in an infinite planar rigid baffle.
(2)
211
RADIATION IMPEDANCE OF A PISTON
where A, is the area of the piston and pcA, is the plane wave resistance, it then follows that the normalized radiation resistance and reactance coefficients of a rectangular piston can also be evaluated directly from equation (la). In order to evaluate the radiation impedance of a rectangular piston by using the impulse response method, the impulse response of the rectangular piston must first be obtained. The impulse response for this case can be obtained from the results of Lindemann [ 11, via the use of an elementary scaling transformation, as z,(r) = /‘CA, [h(T) - k%(T)1~ where A, = b2n,
6(T)
is the Dirac delta function
(3)
and g,(z) is defined as
0,
1
T
&r(T) =
I (2/?27’C){ 1+
O
n - T},
(2/n) { 1 - (1/T)
(T' -
l
1)"2},
. (44
i
(2/M){T - (n/T)[(T' - 1)"' +
[(T'/n)' -
I)]"']},
io,
Note that g,,(r) can also be simply expressed as a sum of functions g,(r) =fi(r) where they,(r)
+fi(r)
+_&(r),
T),
o
(n’ +
n
6
(d
1)"2 < 5
+
1)1'2
: i.e.,
(4b)
are defined as +n-
l
n < f2(T)
1)1’2,
-(2/rCT)(T2 -
=
1<
-(2/7'CT)[(T/n)2 - 1]1'2,
f3(T)=
The normalized impedance can now be obtained obtain the following expression : Z,,(kb)
-1 PC&
5 <
n <
(n’ + 1)“2
T <
(n2 + 1)“2,
T <
(n’ + 1)‘j2.
by substituting
equation
(3) into (1 a) to
, _(nZ+‘)1’2 g,(T)
.i
e-jbbr
dr,
(5)
0
where the sifting property of the Dirac delta function has been utilized to evaluate its integral. After substituting the expression for g,,(r) in equation (4b) into equation (5) and then integrating the first term, the normalized impedance can be expressed as follows : z,,(kb)/pcA,
= 1 - [2/m(jkb)] x [l -ee-
jkb
[(n + 1) - (n2 + 1)1/2e-jkb(n’+1)“z] + [2/lcn(jkb)2] _
e-jnkb
+
e-jkb(n2+1)1’2
I+ G%)Fl(n,kb) + (2/n)FAn,kb), (64
where cn2+ 1)’ 12
(7’ - 1)“2 _jkbrdT
F,(n, kb) =
e
s
1
T
(n*+l)'i* [(T/n)' -
F,(n, kb) = sn
1]"2 e-jkbrdr.
(6b)
5
Equation (6) can be transformed to the expressions which were numerically integrated by Burnett and Soroka [9] to obtain their extensive tables for the normalized radiation resistance and reactance coefficients of rectangular pistons.
278
P.
R. STEPANISHEN
The integral expressions for F,(n, kb) and F&z,kb) can both be expressed in terms of a new integral via elementary transformations: i.e., F,(n, kb) = I(cosh-‘n,
kb),
F&z, kb) = [(cash-’ ([n’ + 1]“2/n), nkb),
U’a, b)
where
s Y
sin2 /3
I(& Y) = Since 1(x, y) can also be expressed
jxcosh8dfi.
as Y
1(x, y) =
&
tanh B
-JX s the following
expression
is then obtained
A(e-jxcashb) dp,
by parts :
after integrating
Y
I(x,y)
(8)
dP
0
= - 1 [tanhpe-1xcosh4]~+ Jx
e-j~
cash 5
-& ~ d/3. cash' /?I Jx s0
(9)
A new expression for the normalized impedance can now be obtained from the substitution of equations (7) and (9) into equation (6a). After performing the algebra, the following expression is obtained :
ZW) -z
1_
PC1+ 4lnnl + (Jkb)2 (y4
[1
_
e-ikb _ e-jnkb + e-Jkb(n*+l)l'*]
+
jkb
PC&
+ (f/N (2/n) C(kb, cash-’ (n2 T 1)“2) + 7Jnkb C(nkb, cash-’ [(n’ + l)““/nl), Jkb
(10)
where
s Y
C&Y) = Cl(w) +jC2(x,y) =
0
e-jxcoshLi ------d/3. cosh2 /I
It then follows from equations (2), (10) and (11) that the normalized reactance coefficients can be expressed as
(11) radiation
resistance
and
R,(kb) = I - [(2/nn)l(kb)2] [ 1 - cos (kb) - cos (nkb) + cos ((n’ + 1)“2 kb)] + [(2/lr)/kb] C,(kb, cash-’ (nZ + 1)“2) + [(2/m)/kb]
C2(nkb, cash-‘Kn2
+ l)“‘/n]),
(12a)
X,(kb) = [(2/n7r)/kbl(1 + n) - [(2/nn)/(kb)21 [sin (kb) + sin (nkb) - sin ((n” + 1)‘12 kb)] - [(2/r()/kb] Cl(kb, cash-‘(n2
+ 1)‘12)
- [(2/nn)/kb] C,(nkb, cash-’ [(n’ + I)“‘/n]),
(12b)
where
C1k.v) =
s
’cos(xcoshfi)
0
cash’ /I
’
dP,
Cz(x,
Y>
=
s 0
sin (x cash fi) cosh2 /I
dP.
(13a, b)
In order to evaluate R,(kb) and X,,(kb) by using equations (10) and (12) the integral C(x, v) must first be evaluated. Since the low and high frequency cases, i.e., nkb 4 1 and kb >>1,
RADIATION
IMPEDANCE
279
OF A PISTON
are readily evaluated from the integral expressions in equations (13) and (1 l), these cases are initially discussed. Consider first the evaluation of R,(M) and X,(H) for the low frequency case. It follows from the integral expressions in equation (13) that the trigonometric functions can be approximated for the case where x cash y < 1 by the initial terms in their power series. The quantities C,(x, y) and CJx, y) can then be approximated by the expressions
JJ dp
C,(x, Y) =
Y
x2 ’ dp, 2 s 0
--s cosh2p 0
which can be readily evaluated
’ dB C,(x, y) = -x s coshp
to obtain
the following
expressions
C,(x, y) = -x sin-’ (tanhy)
-SY,
It then follows from the substitution of equations X,(M), for nkb 4 1, can be expressed as follows :
for x cash y < 1: + G sinhy.
(15) into equations
(lsa,b)
(12) that R,(kb) and
R,(kb) NN(kb)2 (n/2x), X,,(kb) “Nkb{(1/3m)
(14a,b)
0
3
X2
C,(X,Y) = tanhy
+ f coshfldj3, 3! s
0
(lea)
[1 + rz3 - (n’ + 1)3’2]
+ (l/rr)cosh-‘([n2
+ 11”‘) + (n/lr)cosh-‘([n2
+ 1]1’2/n)j.
(16b)
The expressions in equation (16) can also be obtained directly from equation (5) by again utilizing the small argument approximations for the exponential functions in equation (5) : i.e., e-Jkbr N 1 - +(kb)2 - jkbT. In order to evaluate R,(kb) and X,,(kb) for the high frequency case, where nkb 9 I, the asymptotic evaluation of the integral C(x, y) is first required. By an elementary application of stationary phase techniques [12], it is easily shown that, for x $ 1, C(x, y) N (~1/2x)r’~ em”“+ n’4) + O( 1/kb)3/2,
(17)
and therefore C,(x, y) N (rc/2x)“‘c0s
(x + X/4),
It then readily follows from equations be expressed as follows: R,(kb) N I - [(2/m)/(kb)2]
C2k
Y) _ -( 7r/2x)“2 sin (x + n/4).
(12) and (18) that R,(kb) and X,(kb),
(I 8a, b)
for kb $ 1, can
[1 - cos (kb) - cos (nkb) + cos ([n’ + 1]“2 kb)]
+ [2/7c(kb)3]1’2cos (kb + 377/4) + [2/rr(nkb)3]1’2 cos (nkb + 3n/4), X,(kb) N [{2( 1 + n)/nn}/kb]
(19a)
- [(2/nn)/(kb)2] [sin (kb) + sin (nkb)
- sin(kb[n’ + I]‘“)] - [2/x(kb)“]‘l’sin - [2/rc(nkb)3]1’2 sin (nkb + 37c/4).
(kb + 3n/4) (19h)
Although the normalized radiation resistance and reactance coefficients thus have been readily evaluated in terms of elementary functions for both the low and high frequency range, the solutions in equations (16) and (19) are only approximations to the exact solutions. In order to obtain the exact solution for the normalized radiation resistance and reactance coefficients in other than the integral form in equation (12), the integrals in equations (13) must be first evaluated.
280
P. R. STEPANISHEN
In order to evaluate are first noted:
Cl(x, y) and C2(x, y) as given in equations
cos (x cash 8) = .zO ~,,,(-l)~ J,,(x)
(13), the following
identities
cash (2mB),
sin (x cash j?) = 2 _$ (-1)” J,,,,, (x) cash ([2m + 1] B), m-0
(20a,
b)
where E, is the Neumann symbol and J,( ) is the Bessel function of the first kind of order v. The integrals CI(x, y) and C2(x, v) can then be expressed as series expansions: i.e., CL&Y) =
J,, (4 ~,CY),
,$ e,(-Om m=O
C,(x,y)
= 2 z m-0
where the coefficients
(-1)” J ~m+~(~)4m+l(~),
Wa,
b)
d,,,(y) are defined by the integral
4n(y) =
(22) 0
Although the coefficients d,,,(y) can be readily expressed in terms of either hypergeometric functions or incomplete beta functions, the integral expressions for d,,,(y)can also be integrated via the use of standard integral tables. In particular, it readily follows from the use of tables [13] that the initial coefficients can be expressed as
do(y)
=
s’
dP
___
cosh2 /?
= tanhy,
s’
dP
d,(y) =
0
0
= sin-’ (tanhy), cash /I
(23a b)
Y
cash 28 W)
-
= s 0
cosh2 fi
d/3 = 2y - tanh y,
d,(y) =
‘cash 3p ___ dfi = 4 sinh y - 3 sin-’ (tanh y). s cosh2 fi
0
G%
4
The coefficients d,(y) for m > 2 can also be expressed via the use of the tables [ 131 as a finite series of hyperbolic sine functions. The normalized radiation resistance and reactance coefficients can now be obtained by substituting the results of equations (21) into equations (12). It then follows that the radiation coefficients can be expressed as R,(kb)
= 1 - [(2/7~)/(kb)~] 4/x +kb
[l - cos( kb) - cos (nkb) + cos ([n” + l]“‘kb)]
= c
(-l)“J2,+1
(kb)d,,+,(cosh-‘[n’
4/7-w m +-
kb
c
2/n
2/W kb
dz,,,+l cosh-1[T]1’2),
- [(2/nn)/(kb)2]
[sin(kb)
Pa)
+ sin(nkb)
- sin( [nz + l]1’2kb)]
@
G kb lfl=O --
(
(-1)” J ,m+lWb)
Ill=0
X,(kb) =[(2 (1 + Q)/nnf/kb] --
+ 1]“2)
III=0
m c
III=0
E,(-1)“’ Jz,,, (kb)d,,
(cash-‘[n”
+ 11”‘)
&,,A- 1)” Jzn(nkb)d2, ( cosh-‘r+r2),
Wb)
281
RADIATION IMPEDANCE OF A PISTON
For the low frequency case where nkb 6 1, it readily follows, via the use of equations (23) and the small argument approximations for the Bessel functions, that equations (24a) and (24b) reduce to the low frequency expressions in equations (16a) and (16b), respectively.
3. NUMERICAL
RESULTS
In the preceding section an expression for the impulse response of a rectangular piston as a function of its aspect ratio n = a/b was presented in equations (3) and (4). The normalized impulse response for a rectangular piston of any aspect ratio n can thus be evaluated from the expression and the results for II = 1, 2,4 and 8 are shown in Figure 2 where each response is normalized by the plane wave impedance, pcA,, of the piston. Since the normalized impulse responses are plotted on the same time scale, the results correspond to the case of a rectangular piston with a fixed width b, whose length varies in accordance with the aspect ratio. The results in Figure 2 clearly indicate several points of interest concerning the variation of the normalized impulse response with the aspect ratio. Each impulse response has a Dirac delta function with a unit strength at z = 0, which is then followed by a negative pulse whose characteristics are dependent on the aspect ratio. Since the integral of the impulse response over all time r must be zero [I, 1 l] the integral of the negative pulse must be unity to offset the contribution of the delta function. As indicated by the expressions and the results in Figure 2, the time duration of the impulse response increases as the aspect ratio increases. Furthermore, as the aspect ratio increases, the maximum negative value of the impulse response which occurs at 7 = 0 monotonically decreases and approaches the limiting value of 2/7r. Although the behavior of the impulse response for an aspect ratio n is dependent on the aspect ratio for 0 < 7 > 1 and n < 7 < m, the behavior of the impulse response over 1 < 7 < n is independent of the aspect ratio (see equation (4a)). It thus follows that the impulse response for an aspect ratio m > n exhibits the same temporal behavior over 1 < 7 < n as the impulse response for the aspect ratio n where n > 1. As an example, it is noted from the results in Figure 2 that the impulse response for an aspect ratio of 4 is equivalent over 1 < 7 < 2 to the impulse response for an aspect ratio of 2. Several points of interest concerning the behavior of the normalized mutual radiation resistance and reactance coefficients as functions of kb and the aspect ratio can be obtained by inspection of the results in Figure 2. Since the coefficients can be expressed as the Fourier transforms of the respective impulse response functions, the characteristics of the coefficients can thus be obtained from elementary properties of Fourier transforms. In particular, note
Figure 2. The normalized impulse response for different aspect ratios. -, n=2;____,n=l.
n = 8; -
-, n = 4; - - -,
282
P. R. STEPANISHEN
kb (rodlord
Figure 3. The normalized radiation resistance coefficient for different aspect ratios.
that the resistance and reactance coefficients approach unity and zero, respectively, for kb $ 1, for all values of the aspect ratio, as a result of the delta function at r = 0. Furthermore, since the asymptotic behavior in the time domain governs the small argument behavior in the frequency domain, it follows that the coefficients will be dependent on the aspect ratio for small kb. Finally, since the time structure of the impulse response is only weakly dependent on the aspect ratio n when n is greater than approximately 4, the resistance and reactance coefficients can be expected to exhibit the same behavior as functions of kb, except for small arguments as noted above. Although the normalized impulse response of a rectangular piston is of interest for pulsed excitations, the normalized mutual radiation resistance and reactance coefficients are of more interest for harmonic excitations. For this reason, the resistance and reactance coefficients as functions of kb are presented in Figures 3 and 4 for the same aspect ratios which were previously considered. These curves have been obtained from the highly accurate tables of the coefficients which have been previously tabulated as a function of kb& by Burnett and Soroka [9]. Although the selection of kbfi as the independent variable results in an aspect-independent square law relation for the resistance coefficient at small kb (see equation (16a)), the selection of kb as the independent parameter leads to results which are more readily interpreted in the light of the present analysis. Furthermore, the similarities between the resistance and reactance coefficients as functions of aspect ratio are more readily noted. The normalized radiation resistance and reactance coefficients which are shown in Figures 3 and 4 exhibit several interesting characteristics. At low frequencies, where kb < 1, the
1.00
2.00
3.00
4.00
5.03
7.00
8.00
kb (rodlam)
Figure 4. The normalized radiation reactance coefficient for different aspect ratios.
RADIATION
IMPEDANCE
OF A
PISTON
283
resistance and reactance coefficients are aspect-dependent and are proportional to (kb)2 and kb, respectively (see equation (16)). The coefficients for aspect ratios greater than approximately 4 are noted to exhibit very similar behavior for kb > 1. Hence, for n > 4 and kb > 1, on the aspect ratio. the resistance and reactance coefficients are only weakly dependent Finally, the asymptotic or high frequency results are also somewhat evident from the results in which the resistance coefficient is seen to rapidly approach its limiting value of unity for kb greater than an approximate value of 4. In order to provide an indication of the accuracy of the approximate expressions in equations (16) and (19) for the resistance and reactance coefficients, Tables 1 and 2 present data on the accuracy of the approximate expressions for the resistance and reactance coefficients as functions of kb for n = 1 and 4. The numerical results obtained from the approximate expressions are compared to the corresponding numerical results which were published by Burnett and Soroka and are shown in Figures 3 and 4. Since equation (16) is only valid for t&b < 1, its use has been restricted to 0 < kb < 1. Furthermore, since equation (19) is only valid for kb + I, its use has been restricted to kb 2 2. The results in Table 1 illustrate the expected behavior of the approximate expressions for the resistance coefficient. Although equation (16a) yields reasonably accurate results over 0 < kb < 1 for n = 1, the percent errors are much larger for n = 4. This is, of course, to be expected since nkb = 4 at kb = 1. In contrast to the use of equation (16a) which results in monotonically increasing errors as kb is increased, the use of equation (19a) results in a decaying oscillatory error as kb is increased. Furthermore, the errors associated with the use of equations (19a) decrease as n is increased. Note that equation (16a) yields a 27% error for kb = 2 and n = 1; hence, if equation (16a) is used for 0 6 kb < 2 and equation (19a) is used for kb > 2, the resulting expressions yield results for n = 1 which are in error by less than 30 % for all kb. The results in Table 2 illustrate that the approximate expressions for the reactance coefficients in equations (16b) and (19b) exhibit the same behavior as evidenced by the resistance coefficients in Table 1. More specifically, the percent error resulting from the use of equation (16b) increases monotonically as kb is increased, whereas the percent error resulting from the use of equation (19b) exhibits a decaying oscillatory behavior as kb is increased. Once again, it is noted that the accuracy of equation (16b) deteriorates as n is increased for a fixed kb whereas the inverse holds true for equation (19b). As the final item of interest, consider the evaluation of the radiation resistance and reactance coefficients by using an FFT approach. Since the radiation coefficients are simply related to the Fourier transform of the appropriate impulse response (see equations (2) and (5)), it then follows that the coefficients can be readily evaluated via the use of standard FFT algorithms and methods. The utilization of the FFT approach thus offers an alternative method to evaluating the resistance and reactance coefficients via the use of the series in equation (24). In order to provide an indication of the accuracy in evaluating the radiation coefficients via the FFT approach the results in Table 3 show the normalized percent error between the resistance and reactance coefficients obtained from the FFT approach relative to the tabulated values of Burnett and Soroka [9]. More specifically, the percent errors for both the resistance and reactance coefficients are shown as functions of kb for aspect ratios of n = 1 and 4. Since the percent errors are dependent on the sampling time interval which was used for each case of interest, the sampling interval and the number of non-zero samples of the impulse response are indicated for each case. The results in Table 3 clearly show several trends of interest. In particular, large percent errors in the resistance coefficient are noted for small values of kb; however, the error decreases as either kb or the sampling frequency of the impulse response increases. Note that as
040000 om157 OX)0635 0.01425 0.02524 0.03724 0.05616 0.07589 0.09830 0.12325 0.15056
oaoooo
n=l Burnett and Soroka [9] 0.51012 0.70431 O-87306 0.99807 1.07100 1.09380 1.07700 I.03630 0.98870 0.94855 0.92521 0.92195 0.93638
0.70282 0.87515 0.96831 1 .01547 1.03469 I .03605 1.02540 1mO97 0.98501 0.96428 0.94957 0.9445 1 0.95077
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
n=l
Equation (19a)
O-001.59 0.00637 0.01432 0.02546 0.03979 0.05730 0.07799 0.10186 0.12892 0.15925
n=l Burnett and Soroka [9]
n=l Equation (16a)
38 24 11 2.7 3.4 -5.3 -4.8 -2.8 -0.37 1.7 2.6 2.4 1.5
Percent error
n=l
0.31 0.49 0.87 1.40 2.00 2.80 3.60 4.60 5.70
0.00
0.00
?Z=l Percent error
7.5 6.8 1.5 0.37 -0.59 -2.1 -1.4 0.73 -0.71 0.28 0.90 0.58 0.75 0.78323 0.92828 1.01732 1.05530 1.07555 1.06673 1.03473 1.00894 0.98563 0.96282 0.95796 0.96461 0.97171 0.85780 0.99170 1.03260 1.05914 1.06419 I.04473 1.01981 1m157 0.97861 0.96555 0.96659 0.97022 0.97899
-
n=4 Percent error
090 0.47 1.90 4.30 7.70 12.20 17-90 24.80 33+IO 42.00 540l n=4 Burnett and Soroka [9]
Of)0634 0.02499 0.05493 0.09454 0.14179 o-19437 0.24995 0.30635 0.36367 0.41446
090000
n=4 Percent error
of kb and n
n=4 Equation (19a)
OWOOO 0.00637 O-02546 0.05730 0.10186 0.15915 0.22918 0.31194 040744 0.51566 0.63662
as a function
n=4 Burnett and Soroka [9]
coeficient
n=4 Equation (16a)
(16a) and (19a) to evaluate the resistance
kb
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
kb
The percent error in using equations
TABLE 1
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
kh
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
0.1
0.0
kh
(I 6b)
(16b)
0.72499 0.7478 1 0.58700 0.45066 0.34066 0.2563 1 0.19613 0.15846 0.14119 0.14102 0.15298 0.17055 0.18662
n=l Equation
0~00000 0.04732 0.09464 0.14196 0.18923 0.23660 0.28392 0.33124 0.37856 0.42588 0.47320
n=l Equation
Burnett
Burnett
0.66133 0.67109 0.61995 0.52615 0.41245 0.30126 0.2 1042 0.15035 0.12323 0.12397 0.14267 0.16764 0.18842
and Soroka
n=l
oGOOOo 0.04728 0.0743 1 0.14084 0.18664 0.23 146 0.27508 0.31727 0.35783 0.37656 0.43327
and Soroka
??=I
[9]
[9]
error
40.00 11.40 -5.30 -14.30 -17.40 -14.90 -6.80 5.4 14.6 13.8 7.2 1.7 -0.96
Percent
n=l
0.00 0.08 0.30 0.80 1.40 2.20 3.20 4.40 5.80 7.40 9.20
?Z=l Percent error
-.._
(16b) 0.65179 0.50228 0.37287 0.29115 0.20094 0.13234 0.10617 0.08867 0.08220 0.09706 0.11061 0.11723 0.12554
n=4 Equation
0~00000 0.08468 0.16935 0.25403 0.33870 0.42338 0.50806 0.59273 0.67741 0.76209 0.84676
Burnett
0.53646 0.48077 0.38468 0.30007 0.22547 0.15396 0.11182 0.09408 0.08329 0.08909 0.10540 0.11371 0.11976
n=4 and Soroka
omOoo 0.08422 0.16577 024219 0.31139 0.37185 0.42265 0.46356 0.49493 0.51763 0.53295
[9]
[9]
21.50 4.50 3.07 -2:84 -10.90 -14+)0 -5.05 -5.75 -1.31 8.95 4.94 3.10 4.83
n=4 Percent error
0.00 0.55 2.20 4.90 8.80 13.90 20.20 27.90 37.00 47.00 58.00
n=4 Percent error
of kh and n
n=4 and Soroka
a.s a jimction
Burnett
cot$cient
12= 4 Equation (16b)
The percent error in using equations (16b) and ( 19b) t o evaluate the reactance
TABLE 2
286
P.R.STEPANISHEN
kb increases the percent error reaches a minimum and then increases, though not monotonically, as kb increases. In contrast to the percent errors for the resistance coefficients, the results in Table 3 show the percent errors for the reactance coefficients to be considerably less than 0.1% for the cases of n = 1 and 4 over the kb range of 0.0 to 8.0. Although the resistance and reactance coefficients as functions of frequency and aspect ratio are available for selected frequencies and aspect ratios from the tables of Burnett and Soroka, the coefficients at other frequencies and aspect ratios can be obtained either by interpolation methods or by the methods of the present paper. Several observations concerning the evaluation of the radiation resistance and reactance coefficients as functions of frequency and aspect ratio are thus noted from the preceding numerical results and analysis. For nkb < 1 the coefficients can be accurately evaluated via the use ofequations (16), whereas, for kb g 1 the coefficients can be accurately evaluated via the use of equations (19). Although
TABLE 3 The normalized percent error in the resistance and reactance coefJicients obtained via the FFT approach
Aspect ratio = 1 Sampling time interval = 0.0153398 Number of points = 93 A \ r Percent error Percent error reactance kb resistance 0.20 040 060 0.80
1.00 290 3.00 490 5.00 690 7.00 8.00
76.7431 19.3158 8.6824 4.9619 3.2405 0.9584 0.5609 0.4571 0.4534 0.4928 0.5268 0.5219
0.0037 0.0036 0.0036 0.0036 0.0036 0.0035 0.0035 0.0041 0.0090 0.0267 0.0369 0.0346
Aspect ratio = 4 Sampling time interval = 0.0153398 Number of points = 269 z% / \ Percent error Percent error resistance reactance kb 0.20 0.40 0.60 0.80
1.oo 2.00 3.00 4.00 5.00 6.00 7.00 8.00
12.2308 3.2306 1.5695 0.9947 0.7345 0.3896 3.3007 0.2847 0.2949 0.3088 0.3180 0.3140
-00013 -0.0013 -0~0010 -0.0005 0~0001 0.0034 0.0042 0.0046 0.0105 0.0220 0.0314 0.0341
Aspect ratio = 4 Aspect ratio = 1 Sampling time interval = 0.0076699 Sampling time interval = 0.0076699 Number of points = 538 Number of points = 185 h h r , , Percent error Percent error Percent error Percent error resistance reactance kb resistance reactance kb 0.20 0.40 0.60 0.80 1.oo 2.00 3.00 4.00 5.00 6.00 7.00 8.00
38.0692 9.5867 4.3121 2.4668 1.6128 0.4805 0.2826 0.2298 0.2263 0.2442 0.2611 0.2608
OW55 oGO53 0.0053 oGO52 oGO5 1 oGO40 0.0016 -0003 1 -0GO95 -0.0016 0.0200 0.0237
0.20 0.40 0.60 0.80 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00
6.1322 1.6116 0.7788 0.4922 0.3639 0.1947 0.1520 0.1425 0.1477 0.1537 0.1570 0.1576
-0w53 -0.0043 -0.0025 -0.0004 0.0015 0.0016 0.0038 -0GO12 -0.0165 0.0062 0.0129 0.0169
RADIATIONIMPEDANCEOF A PISTON
287
TABLE 3-continued Aspect ratio = 4 Aspect ratio = 1 Sampling time interval = 00319174 Sampling time interval = 0.0019174 Number of points = 2151 Number of points = 738 A /. ‘, I I Percent error Percent errop Percent error Percent error reactance kb resistance reactance kb resistance _...__. 0.20 19.6373 -0~0059 0.20 3.3991 -0.0270 0.40 4.9348 -0.0059 040 0.8626 -0.0238 0.60 2.2126 -0.0057 0.60 0.3987 -00188 0.80 1.2603 -0.0056 0.80 0.2426 -0.0129 I .oo 0.8194 -00053 1.00 0.1759 -0.0072 2.00 0.2366 -0~0030 2.00 0.0937 -0.0029 3.00 0.1373 o+IOo7 3.00 0.0757 0.0082 4.00 0.1135 0.0049 4.00 0.0703 0.0077 5.00 0.1147 0.0060 5.00 0.0756 -0.0170 6.00 0.1248 -0.0051 6.00 0.0785 0.0086 7.00 0.1315 -0.0085 7.00 0.0770 -0.0077 8.00 0.1287 0.0005 8.00 0.0791 -O+K)58
the reactance coefficients can be accurately evaluated by using the FFT approach for low, mid-range and high frequencies, the accuracy of the computations for the resistance coefficients decreases with frequency at low frequencies. Since the series expansions for the resistance and reactance coefficients in equation (24) reduce to the low frequency expressions in equation (16) and are ideally suited for low frequency rather than high frequency computations, the series expansions for the resistance coefficient can be used in the low frequency range. The FFT approach when combined with the series expansions can then be used to accurately evaluate the resistance and reactance coefficients in the low- to mid-frequency range whereas the FFT approach or asymptotic expansion in equations (19) can then be used in the high frequency range.
ACKNOWLEDGMENTS A portion of this work was supported by a grant from the United States National Institute of Health. In addition, the author wishes to acknowledge the constructive comments of the reviewers.
REFERENCES 1. 0. LINDEMANN1974 Journal of the Acoustical
2. 3. 4. 5.
Society of America 55, 708-717. Transient fluid reaction on a baffled plane piston of arbitrary shape. L. E. KINSLERand A. R. FREY 1964 Fundamentals of Acoustics. New York: John Wiley and Sons, Inc. M. C. JUNGERand D. FEIT 1972 Sound Structures and their Interaction. Cambridge: The MIT Press. P. R. STEPANISHEN1971 JournaI of the AcousticaI Society of America 49, 841-849. The time dependent force and radiation impedance on a piston in a rigid infinite planar baffle. G. W. SWENSON,JR. and W. E. JOHNSON1952 Journal of the Acoustical Society of America 24, 84. Radiation impedance of a rigid square piston in an infinite baffle.
288
P. R. STEPANISHEN
6. E. M. ARASE 1964 Journal ofthe Acoustical Society of America 36, 1511-1525. Mutual radiation impedance of square and rectangular pistons in a rigid infinite baffle. 7. B. WYRZ~KOWSKA 1956 Proceedings of the 2nd Conference on Ultrasonics, Warsaw 99-106. Acoustic impedance of plane generators; rectangle, ring, ellipse. 8. H. STENZEL 1939 NRL Translation No. 130 (translated by A. R. Stickley). Handbook for the calculation of sound propagation phenomena. 9. D. S. BURNETT and W. W. SOROKA 1969 Journal of the Acoustical Society of America 51, 16181623. Tables of rectangular piston radiation impedance functions, with applications to sound transmission through deep apertures. 10. D. N. CHETAEV 1951 Prikladnaya Matematika i Mekhanka 15,439444. The impedance of a rectangular piston vibrating in an opening in a flat baffle. Il. P. R. STEPANISHEN 1973 Journal of Sound and Vibration 26, 287-298. The impulse response and time dependent force on a baffled circular piston and a sphere. 12. E. T. COPSON 1965 Asymptotic Expansions. Cambridge, England: Cambridge University Press. 13. I. S. GRAUSHTEYN and I. M. RYZHIK 1965 Table of Integrals, Series and Products. New York: Academic Press.