On sound absorption of finite-size absorbers in relation to their radiation impedance

Journal of Sound and Vibration (1989) 135(2), 225-262

ON SOUND ABSORPTION OF FINITE-SIZE ABSORBERS IN RELATION TO THEIR RADIATION IMPEDANCE F. P. MECHEL Fraunhofer-Znstitut fiir Bauphysik, Stuttgart 80, Germany (Received 6 September 1988, and in revised form

16 March 1989)

The variational solution of the Helmholtz integral equation for the sound field in front of sound absorbers of finite size leads to a simple analogue circuit for the sound absorption: a pressure source with the radiation impedance of the absorber area as the source impedance acts on the wall impedance of the absorber as the load. This simple relation for the sound absorption up to now was derived from individual solutions for finite-size absorbers, only. A generalized derivation is presented which not only determines necessary and sufficient conditions for the application of this approximate description, but also gives estimates for its errors. Next, algorithms and analytical expressions are described for the computation of the radiation impedances of the absorber strip and of the rectangle. Further, numerical examples of the sound absorption coefficients for directed and for diffuse sound incidence are presented. Finally, a quantity which characterizes the absorber edge effect numerically is plotted over the impedance plane of the absorber for both types of sound incidence. The results are in a good agreement with exact solutions for the absorber strip.

1. INTRODUCTION

The lateral extent of absorbers is implicitly assumed to be infinite in the evaluation of standard measurements of the sound absorption in reverberant rooms. It also is in most textbook

descriptions

of sound absorption,

which can be summarized

by the relation

(1) between

the absorption

and the normalized

coefficient

a (Si)

wall impedance

for a plane wave incident

2 = Z’+ jZ”

or the normalized

at a polar angle

6,

wall admittance

= F’+ jF” of a locally reacting absorber (normalized with the free field impedance of a plane wave Z, = poco, p. being the density and co the sound speed in air) which F = l/Z

would produce

a reflection

factor r( Si) =

It is known from experimental strong influence on the absorption.

Z-~/COS 6, Z+

l-

F/COS 6,

(2)

~/COS 6, = 1+ F/COS 6i’

experience that the finite extent of absorbers has a The experimental findings are summarized under the

term “edge effect” (see reference [l]). They are rather complex, because the edge effect depends not only on the length of the edges of the absorber, as was assumed originally, but also on its shape, on its impedance Exact analytical

solutions

only for simple geometrical

and on the angle of sound incidence.

for sound absorption

by absorbers

of finite extent are known

bodies, such as cylinders and spheres, [2], and for the absorber

strip as the only representative

of flat absorbers

[3,4];

results are given in reference

[5].

225 0022-460X/89/230225 + 38 %03.00/O

0 1989 Academic

Press Limited

226

F. P. MECHEL

These solutions, which are based on a boundary value problem in separable co-ordinates, are numerically rather complicated, however, at least for the flat absorber. That is why approximate solutions were sought for the absorber strip. These approximations start from the Helmholtz integral equation for the sound field in front of the absorber. The solution of this integral equation usually follows one of two methods of approximation. One of them is solution by iterations, which was applied in reference [6]. The number of iterations mostly is restricted to one (Born’s approximation) because the amount of numerical work would increase drastically with the number of iterations. The second method consists in the application of a variational principle on the integral equation. It was applied first on absorber strips by Levitas and Lax [7], and was described by Morse and Ingard in reference [8], to which the reader is referred for details of the method. Here only a short outline is given in the next section, Northwood, Grisaru and Medcof [9] have extended the results to diffuse sound incidence, with some corrections made by Northwood [lo]. A key role is played in this approximate variational solution by a variable C (see below), which fohows from the extremal solution of the variational variable, by means of which the sound absorption coefficient can be written as

cu(O, @)=(l/cos

0)4F’sin

@/ICF+sin

@I*.

(3)

Here we have used the angles 0, @ of sound incidence from references [7,9, lo] in order to facilitate the connections with these references. Their relation to the usual angles 6i, rpi is indicated by Figure 1 and is given by COS 4i=

sin @ COS

0,

cos Q, = sin rpisin ai,

Figure

1. Absorber

COs0 = cos Si/ Jl -sin2 cpisin’ 6,. (4)

strip and co-ordinates.

It is to the merit of Thomasson [ll, 121, to have shown that this variable C is closely related to the (normalized) radiation impedance Z, of a strip, if its surface velocity distribution corresponds to that of the trace of a plane wave incident at the angles 0, @ (or @ii, Pi): C = Z, sin Cp.

(5)

Then one can write: (Y(@,0) =

1 4F’ sin Q, cos 0 IZ,F+l[”

1 a(9i9

9,)

=-

cos 6i

4F’ IZSF+ 1)’’

(6)

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227

For very wide strips, Z, + l/cos 61, and then expression (6) becomes identical to expression (1). If one applies the normalized absorption cross-section QJA of the absorber, with the absorber surface area A, which is the ratio of the absorbed power U, to the power fli which the incident sound wave would send through the area A under normal incidence, and which is related to the absorption coefficient by Q,/A=Rb/ni=(Y(~i,

Cpi)COs6i,

(7)

then one has Q,,/A=4F’/(1+Z,F]2=4Z’/]Z,+Z~2.

(8)

This is suitable for a convincing interpretation. In the variational method it is assumed that the field distribution of the scattered wave on the absorber surface is that of the spur of the incident sound wave (see below). So Z, coincides with the radiation impedance of the scattered wave. Then expression (8) can be represented by the simple analogue circuit of Figure 2. A pressure source with the open-loop pressure 2P,, with P, the amplitude of the incident plane wave, and with the (internal) source impedance Z, acts on the absorber impedance Z as the load impedance. The power which is consumed by the load in this analogue circuit is just the power absorbed by the absorber.

Figure 2. Analogue circuit for finite-size absorbers.

This result is so suggestive of a generalization that Thomasson [ 1l] proposed to represent an absorber rectangle by a strip of an effective strip width a which is determined from the ratio a = 2A/ U of the area A to the circumference U of the rectangle. Hamet [13] presented an analysis for a rectangular absorber which is quite similar to the variational solution, and he showed that his results become identical to the results in references [7, lo] if one of the sides of the rectangle becomes infinitely long. He also obtained a key variable C with the same role in the absorption coefficient or in the absorption cross-section. He, however, did not mention the relation of his variable to the radiation impedance of the scattered field of the rectangle. The above outline of the background in the literature will help to put the present paper into perspective. Equations (6) or (7) are derived in the literature as results of variational solutions for individual absorber geometries. No estimates of the error magnitude of the approximation are available. After a short sketch of the variational method, we shall generalize the result, and conditions for its validity will be given. Some indications concerning the degree of approximation can be derived from these conditions. The key variable is the radiation impedance of the absorber area, with a velocity pattern which is that of the scattered wave. The approximation of the variational method is that this pattern is set equal to the pattern of the trace of the incident plane wave. Next, analytical solutions and algorithms for the computation of the radiation impedances of strips and of rectangles are presented. Finally, numerical results concerning the edge effect are shown.

228

F. P. MECHEL

2. THE VARIATIONAL

SOLUTION

We give only a short outline of the variational method (for details see references [7, S]), in order to see the assumptions which are made there. We apply it to the absorber strip for which the geometry and the co-ordinates are shown in Figure 2. A locally reacting absorber strip with the normalized wall admittance F = F’+ jF” or the normalized wall impedance 2 = Z’+jZ”= l/F and a width a is centered along the y-axis; the z-axis is directed into the upper half-space from where a plane sound wave pi is incident at the angles fii, Cpior 0, @ (see Figure 2); the relations between them are given in equation (4). The x-y plane surrounding the strip is rigid. The area of the strip is symbolized by A, and s denotes the space vector from the origin to a point (x, y, z). A time factor ejw’ is assumed and suppressed. The sound field p(s) is split into the incident plane wave pi(s), Pi(s) = pi e-j(xk+ykV-zk) = ~~p,(~, Y) e+j%,

(9)

into the wave p,(s) reflected from the rigid baffle, pi(s) = P,P,(x, y) e-jZk;,

(IO)

and into the scattered wave p,(s) which is scattered (reflected) from the absorber, P(s)=Pi(s)+P,(s)+P,(s).

(11)

Pi is the (arbitrary) amplitude of the incident wave, and P,(x, y) is its distribution on the plane z = 0, its spur on this plane. The components of the wavenumber vector satisfy kz+ kz+ kz = kz with ko = w/co. They are determined by the angles of sound incidence. The starting point is the Helmholtz integral equation, which in this case (no sources, incident plane wave) is

PCs) =Pits)

JJLG(sb,)

+PrCs)+

$P(SU) 0

-p(so) $

0

Gbbo)

2=0

1

dSo,

(12)

with a Green function G(slso) which can be selected suitably. The integration over the variable so = (x0, yo) is performed over the plane z = 0, the normal unit vector no is directed into the absorber (or wall). If the Green function is chosen so that ~G(s~so)/&ro = 0 on z = 0, then, with the surrounding wall being rigid, one has

PCs) =Pits) +PrCs) +

JJWho) $P(SO)

dso.

(13)

0

A

With the boundary condition V, = Fp at the surface of a locally reacting absorber and with the momentum equation o, = j/(koZo)itp/c3n, one has, at the absorber surface,

pb,

Y,

G(x, Y, 01x0, YO, WP(XO, YO,0) dxo dye.

0) = 2flP,(x, v) -jkoF

(14)

A

A suitable Green function which satisfies this condition is

(1%

FINITE

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RADIATION

IMPEDANCE

229

It is the field of point sources at the source point (x0, yo, zo) and at the mirror point (x0, yo, -zo), with the distances to the field point (x, y, z) given by Ri = (x - xo)2 + (y - yo)2 + (z f z~)~. If both s and so are on the absorber surface, as in equation (14), then Rz+ R2=(~-~o)2+(y-yO)2. Now an approximation is introduced into the analysis of the problem. It is assumed that the field pattern on the absorber surface is that of the trace of the incident wave: P(x, YPO)=

piBpeCx,

.Y),

with a constant factor B which is as yet undetermined. and (lo), that Ps(x~ Y, 0)=(B-2)P,Pe(x9

(16)

This implies, with equations (9) Y):

(17)

that is, the pattern of the scattered field is also proportional to the trace of the incident wave on the absorber surface. The task is to determine B. A variational principle is used for this purpose. It is based on the Helmholtz superposition theorem which says that the cross power (in our case, through the absorber area) of the real sound field with the conjugate field, in which the source point and the field point are interchanged, must be zero if the field description is exact. If the assumption (16) is not exact, the cross power will become a non-zero function of the factor B. The best value for B is determined from the requirement that the cross power of the real field with the conjugate field through the absorber surface shall be a minimum. The result is: B=22/(2+2,),

(18)

with a quantity of the type of a normalized impedance, z

=J&

A

A The

-jCk~(x-x,)+k~(y-y,)l

G(x,Y,~~xo,Yo,~)~

A

dxo dye dx dy.

(19)

A

absorbed sound power, with expression (18), is lIa = A(~P~~2/2Z~)4Z’/~Z+Z~~2,

(20)

and from this the normalized absorption cross-section is Q,/A=4Z’/IZ+ZA12.

(21)

This is the same result as in expression (8) if the quantity ZA is identical with the radiation impedance Z,. Instead of proving ZA = Z, for the individual geometries for which the variational solution can be obtained, we shall generalize equation (8) by an indirect method. For that we need some information concerning radiation impedances. 3. RADIATION

IMPEDANCE

It was shown in [14] that several definitions of the radiation impedance are used in the literature. The most general definition, of which the other definitions are special cases, is by the relation: n=;z*

II A

]b,]* dA,

(22)

230

F. P.

MECHEL

where 17 is the radiated power, 2, is the radiation impedance, and v, is the particle velocity amplitude at the surface A of the radiator, normal to it. In the complex radiation impedance 2, = 2: + jZt, the real component is a measure of the radiated effective power Zl’, while the imaginary component is a measure of the oscillating power F’. If one defines a field impedance ZF at any point of the radiator surface normal to this surface in the direction of v, (assumed to be outwards in the moment) by p(sO) = ZF(s,,)v,(sO), with s0 on the surface A, then: fl=f

-G(~o)bn(~o)l* dSo.

(23)

A

If it is assumed that the field impedance Z,(a) shall be constant over A, then one can take it out of the integral, and a comparison to equation (22) shows immediately Z, = Z, in this case. If the velocity pattern on the surface is such that )v,(s,,)( is constant over A, then one obtains zS=+

Z&J

II

dS,.

(24)

A

The radiation impedance in this case is the linear average of the field impedance. If not only the magnitude /v,,(sO)lis constant, but the phase also, then

n=fv;

p(s,,) dS, = +Z,Av,v:

(25)

A

from which it follows that

zs=&

(26)

p(so)dSo= @)A/+

n A

The latter condition is met with piston-like radiators and with pulsating bodies. Here we present, for later applications, a representation for the radiation impedance which was ,derived by Bouwkamp [15]: a/l+jm

2?7

Z, = -$

k;AZ,,

I0

chp

JD( 8, rp)12sin 6 da, I0

(27)

in which D( 8, cp) is the far field angular distribution of the radiated field normalized to a maximum value of unity, 6 and qc are the polar angle and the azimuthal angle, respectively. The limits of integration indicated in the integral over 8 are a shorthand notation of the path of integration in the complex a-plane, which first goes from 0 to r/2 along the real axis. This part will produce the real component Z: of the radiation impedance. Then the path of integration follows a parallel to the imaginary axis from 7r/2 + j.0 to ~/2 + j.m. The result of this section of the integration path will be the imaginary component Zr. 4. CONDITIONS

FOR THE ABSORPTION

FORMULA

We start again from a decomposition of the sound field according to equation (11) into the incident plane wave pi(S) as in equation (9), into the wave p,(s) reflected from

FINITE

SIZE

ABSORBER

RADIATION

231

IMPEDANCE

the rigid baffle wall according to equation (lo), and into the wave ps(s) which is scattered from the finite absorber. In what follows, we shall make use several times of the fact that the surface distribution function P,(x, y) = exp (-j(xk, + ~4)) has the magnitude I&(x, Y)I = 1. The field must satisfy the boundary conditions at the rigid wall and at the absorber surface. At the baffle wall, the normal velocity u, (all velocities counted positive in the +z-direction) must vanish. Since U,i= -uET, one has u,,(x, y, 0) = 0 outside A. Therefore the “radiator” of the scattered field is surrounded by a rigid wall. Because Vzi= -u,, holds inside the absorber area A also, the total velocity inside A is u,(x, y, 0) = u,,(x, y, 0). The boundary condition on the absorber surface is P(x, Y9 O) = (Pi+Pr+Ps)z=O=

-zzOUz(x9

Y, O),

(28)

and this, because of pi = p,. at z = 0, becomes 2P,p~(x~Y)+px(x~Y~o)=-zz0uz(x~y~O)~

(284

The absorbed sound power which is wanted is

II, =i Re {ZZ,}

lu,(x, y, O)l”dx dy.

(29)

A

If one now assumes, as a sufficient condition, within A, Ps(X, Y, 0) = Z~Zc&(X, Y, 0) = Z,Zo~z(X, y, 0),

with a constant field impedance (normalized)

(30)

Z,, then one obtains, from equation (28a),

2pip~~x~Y~~~zO~z+zF~~~~x~Y~o~~

(31)

This, with u, = uz,, leads to II, = A(lq\*/2Z,)4Z’/(Z+ZF)*,

(32)

which corresponds to equation (20) and to the desired result, if Z, = Z, holds. An interpretation of Z, can be derived from the momentum equation f&s= wkJZcJ)ifp,lfJz = (l/ZFZ,)Ps(X,

Y, 0),

(33)

which, close to the absorber surface, gives I/ZFZO = Cj/kOZO)(Waz) ln (ph,

Y, 0)).

(34)

According to the general formula for the field admittance F, in the direction n [16], F,, =-

1

(35)

k,Z,

for any sound field p(s) = (p(s)1 ej@(‘), equation (34) describes just the normalized field admittance of the scattered wave on the absorber surface. Under the above assumption that Z, shall be constant over A, it is, according to the special case after equation (23), just the radiation impedance of the scattered wave Z,. If one reads expression (33) as a differential equation, the solution near to the absorber surface is ps(x, y, z) =p,(x,

y, 0) e-jcko’ZF)z.

(36)

There exists a plane wave radiation if Z, is purely real. If Z, is purely imaginary-and then necessarily positive, because otherwise the radiation condition would be violated-

232

F. P. MECHEL

the scattered wave is a nonradiating near field. Radiation and near field are combined in the general case of a complex 2, (with a then positive imaginary part). So one can draw a first conclusion: the assumption of a constant field impedance for the scattered wave is sufficient to obtain the absorption formula (8). The second equality in equation (30) makes ps proportional to u,, and then it follows from the boundary condition (28a) that it is proportional to P, on the absorber surface. This is just the approximate assumption of the variational solution. Therefore one can draw a second conclusion: if one assumes ps to be proportional to P,, then the absorption formula (8) will necessarily follow with the meaning of 2, as the radiation impedance of the scattered wave. This is shown here without reference to a special absorber shape; the only condition is that it must be locally reacting. We next assume that the field impedance Z, is not constant in x, y, perhaps in the real sound field or in the analytical description. Then we can derive an estimate of the error in the calculation of the absorbed power from a formula of the type (32) (for ease of writing we assume for the moment that the wall impedance Z and the field impedance , Z, are not normalized). From equation (28a) one obtains 2pipe(x9 Y) = -“z(x~ Y, o)(z+Ps(xv

Yvo)/“zs(x~ Y9O)).

(37)

If therein one sets PAX, Y, O)/%sb,

Y, 0) = Z&l +“fcx, Y)),

(38)

where Z, is still constant, one obtains Z+P,lv,,=(Z+Z,){l+Z,/(Z+Z,lf(x,Y)}.

(39)

The absorbed power assumes the form of equation (32) if the equation

(40)

holds. Thus the error of expression (32), in cases of a non-zero f(x, y) is less than

l&lII

(Mx, y)l’dx dy.

(41)

A

The mean square deviation of p,/u,, from the constant value Z,, therefore, is important for the error. It goes to zero for small absorber areas as well as for large ones. Further, it will be small if the wall impedance Z is large compared to the radiation impedance Z, (these are sufficient conditions). One can also derive from expression (40) a necessary condition for the validity of expression (32). If one puts all terms in expression (40) under one integral, then the requirement is II

[I~,Z+p,(~-/u,Z+v,Z,1~]dxdy=Min=O.

(42)

A

One can see also from this that the value zero will be exactly attained if ps = Z,u,, = ZFu, holds. If one expands the terms of the integrand,

P Re {Z(GP~ -~~l~~,~2~~+~Ip~12-I~F21zs~2~1 dx dy, A

(43)

FINITE

SIZE

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RADIATION

233

IMPEDANCE

then the integration of the term v,,pf gives just the double value of the power radiated by the scattered wave. The integral over the first term will therefore vanish if 2, is the radiation impedance as defined in equation (22). The integral of the second term goes to zero in the same approximation, in which the radiation impedance can be computed from expression (24) (which holds exactly if lu,,(x, y)l = constant). One concludes from these results that it is important for use in practical applications to compute the radiation impedances of waves scattered from finite areas, in a formulation which is independent of the absorber task: i.e., the radiation impedances of finite areas surrounded by rigid baffles, the velocity pattern of which agrees with the trace of an incident plane wave. Analytical and numerical results for strips and for rectangles are presented in the next two sections.

5. RADIATION

IMPEDANCE

OF STRIPS

The geometry is as in Figure 2. Using the angles 0, @ of the incident wave one starts from [ll] c C= C'+jC"= 2, = C/sin Sp, c = koa sin Qi, 1-: cos (x sin 0)Hb2’(x) dx. I(0 ) (44) One writes, for the components of C, c {C’; C”} = Re ej”“{Jo(x); Ye(x)} dx-f [I 0

ocx e’““{Jo(x); Ye(x)} dx . I

I

(45)

Therein J,,(x), Ye(x), Hb2’(x) = J,(x) -jY o( x ) are the Bessel, the Neumann and the Hankel functions of zero order and (Y= sin 0 for abbreviation. The recurrence relations of reference [17, equation 5.1.(5)] can be applied for integrals of the type f,,n(x) = 1’ e-P’t”W,(t)

dt,

(46)

with W,,(r) = {J,,(t), Y,(t)} representing either the Bessel or the Neumann function of order n. In the present case m = 0, 1, n = 0, p = -ja, and hence one finds [x ejax( W,(x)

&l(x) = $ The components

+jaW,(x))-jolf&)l.

(47)

of C then are

{C’; C”) = Re L&,0(x)-(ll~lfi,~(x)l~

So there remains the determination of the integral f&(x). In the real component C’ one obtains from reference [17, equation 10.3.(3), (4)], fo,o(x)[G = 2 ejcsin8 z j”u,(-sin n=O

O)J,+,(c),

(49)

rp

(50)

with the definition U,(cos cp)=sin [(n+l)rp]/sin

234

F. P. MECHEL

from which follows j”U,(-sin

O)=(l/cos

B)[cos((n,+1)0)-jsin((n,+1)0)],

(51)

where the term with even numbers n, = 2,4,6, . . . is taken if n is even, and the term with odd numbers n, = 1,3,5.. . if n is odd. Finally one has, for the real component, C’=Re{~[2(cos2

y)---&

0+j

(II<,=1

x ;

cos n,OJ,O(c)-j

z sin n,0J,(c) n,=2

)

-J,(c)-jsin

OJ,(c)

II.

(52)

This is a development in Bessel functions of ascending order with real arguments. The sequence of Bessel functions can be generated quite easily and the sums converge quite rapidly if the order becomes larger than the argument, so there are no numerical problems. This is quite different in the case of the imaginary component, where W,(t) = Y,(t). Here one takes, from reference [17, equation 10.3.(10)], f,,,(X)l~=%,jcsine

x

K

T j”U,(-sin n=O

0)

y+lnf-6.~,~)J.+,(c)-~(~)nt’~~~m!(m:n+l)’(~)mJ”(c)], (53)

with y = 0.577216 and 6, = 0 for n = 0, 6, = 1 for n > 0. With this one obtains, from expression (48),

(54) This expression not only is rather complicated, but it is also of only restricted value for numerical computations. The terms under the sums become large for c/2 > 1 so that large rounding errors occur. For the range of large c = k,,a sin @ one decomposes in equation (48) as follows: (55)

AJ,o(x)l;;=fo,o(x)l~-so,O(x)l:. The first integral on the right side is a Bessel transform for which one finds ej*‘Jo(t) df=&=&, . 2 arcsinff eja’Yo( t) dt = j L =J m/i7

20 77 cos

(56) 0’

with (Y= sin 0. For the second integral, reference [17, equations 10.5.(6), (7)]] gives asymptotic expansions for large c. With these and after some transformations one obtains,

FINITE

ABSORBER

RADIATION

235

IMPEDANCE

C’,

for the real component C’=-

SIZE

cco~~sRe{c[cos(csin8)+jsin(csin8)][J,(c)+jsin8J,(c)]}

+Re{(l+s)

X

x

1

--- 1 coso

1 X6

cos(c(l+sin [

@)+rr/4)

a&) &+, cos(c(l-sin@)+9r/4)-jsin(c(l-sin8)+~/4) E,(--Ok(p+jCZk+l 1 -sin 8 )+ (2)

XkiO(-l)k$-j*

M

(57)

,

C

and for the imaginary component C”=-

@)+rr/4)+jsin(c(l+sin l+sin 0

C”,

~Re{c[cos(csinB)+jsin(csinB)][Y,(c)+jsinBY~(c)]}+~

-Im{(l+s) cos(c(l+sin

X--

[ 7r cos 28 0+&z 1

[

0)+7r/4)+jsin(c(l+sin l+sin 0

@)+rr/4)

(1) cos(c(l-sin@)+rr/4)-jsin(c(l-sin@)+rr/4) a:‘k?, %+jF x ktO (-l)k l-sin 8 ) ( CZk

x kip) k(!$_j$L)]]}_

(58)

The coefficients a(kv2)are computed from ab’*” = S,( T,,2),

TL2 =-

2 1*ta’

S,(T) = 1,

Sk+,(T)=l+

k+l k+(W)

TS,(T).

Numerical tests show that c must be really large in order to produce convergence of the sums because the a(kV2)can attain large values and are alternating in sign. Therefore one can take other asymptotic expansions of the integrals from reference [ 17, equations lOS.(lO), (ll)] which, however, are applicable for (YC= ha sin 8 sin Cp>>1 only. They lead to

(60)

The corresponding integral with the Neumann function Y0 is obtained if the right side is multiplied by -j and if the minus sign of the second term in the brackets is changed

236

F. P. MECHEL

to a plus sign. After insertion into expressions (55) and (48), one obtains C’=

&

Re {c(cos (c sin O)+j sin O))(J,(c)+j

sin @Jo(c)))

+Re{( I+%)[-&-$$

X

sin O))+j sin (c(l+sin

I+?Y&(cos(c(l+

0)))

[

*-?I

m

’ nz, [2c(l isin

” (l+sin @)I” k:O

4kk!

@)k

+ 1 ‘syi

o (cos (c( 1 -sin

0))

III’

.--n m ” (1 -sin 63)” -j sin (c(l -sin 0))) .r, [2c(sinl o _ 1),n kz, 4kk!

(61)

and C” =

&[~+Re{c(cos(csinB)+jsin(csinO))(Y,(c)+jY~(c))j]

-Jm{(l+S)[& +

I~s~~O(cos(c(l+sin8))+jsin(c(l+sin6))) m

.-n

’ nz, [2c(l isin

n (1 +sin @)k 1 -j @)I” k;O 4kk! -1 -sin 0

’ III(62)

n (1 -sin @)k _ l)1n z

x (cos (~(1 -sin 0)) -j sin (~(1 -sin 0))) “?+ L2c(sinj

k

0

4kk!

Numerical tests show that there remains a gap between expressions (52) and (54) for small c and expressions (61) and (62) for large values of c. The immediate numerical integration of equation (45), therefore, is of interest as another method for the computation of C. Numerical problems arise from the fact that the integrand becomes singular at the lower integration limit for the imaginary component C” because of the singularity of Y,(x) there; this can be overcome by using an algorithm for integration in an open interval. More serious is the fact that the integrands oscillate very rapidly near the upper integration limit if this is large. It is advisable, therefore, to apply an integration scheme, such as the Romberg scheme, which automatically doubles the number of computing points until the error falls short of a prescribed value. A last method of computation of C shall be reported only. There exist polynomial representations with a good precision of Jo(x) and of Ye(x) in x for small arguments and in l/x for large arguments. After putting these polynomials into expression (45), the integrals appearing are integrable. The result is relatively simple for the real component C’ with small values of c, e.g. 0~ c S 4. The Bessel and Neumann functions can be represented by 2”

Yo(x)=~ln~Jo(x)+

i *=0

b,

0

i

,

0GxG4,

(63)

FINITE

SIZE

ABSORBER

RADIATION

IMPEDANCE

237

with coefficients a,, b, for example in reference [17, equations 1.4.7.(3), (4), (6)]. The result for C’ in OGcs4 is

with (Y= sin 0 and k,, k, odd and even integers, respectively. The corresponding formula for C” fills nearly a full page and the integral sine function Si(x) as well as the integral cosine function Ci(x) appear in it. It therefore is not given here. The results are even more complicated in the range x > 4, with Fresnel integrals S(x) and C(x) appearing in it, so they will not be presented here either [18].

c Figure

3. Contour

lines of constant

values of C’(c, 0).

Numerical results for C’ and for C” are plotted in Figure 3 and Figure 4 respectively as contour lines in the plane of c = k,a sin @ and 0. With this, the (normalized) radiation impedance 2, of a strip can be obtained easily from equation (44) (it is an advantage of the selection of the angles 0, @ that the dependence of 2, on the three independent variables koa, 0, @ can be shown with only one contour diagram for each component of 2,). If the strip width (or the frequency) k,a increases, starting from koa = 0, both 2: and 2: start from zero. For k,a sin @ < 0.7 the influence of 0 is small, and the excitation across the strip is conphase. For 0 = 0 (incidence in the plane parallel to the strip axis and normal to the wall), the radiation resistance 2: levels off at a value of about one; the radiation reactance 2: passes a maximum of about 0.5 and then goes down to zero. This is a behaviour which one is used to finding with piston radiators. For wide strips and 0 >>0, the radiation resistance can increase to values higher than one and the reactance can retain large values even for large koa. This is important, if one remembers the role of 2, in the analogue circuit of Figure 1 limiting the maximum power transfer to the load impedance 2. Technical absorbers mostly have large reactances Z”. If their sign is negative

238

F. P. MECHEL

Figure 4. Contour lines of constant values of C”(c, 8).

(spring type reactance), they can come to resonance with the positive (and sometimes also large) reactance 21 of the radiation impedance. One must therefore expect to find maximum absorption values for wall impedances with a negative reactance component, and the maximum will be attained at large angles 0 for ]Z”[ > 1.

6. RADIATION

IMPEDANCE

OF RECTANGLES

Next we consider the radiation impedance of rectangles with area A = ab excited by the trace of an incident plane wave. The co-ordinates are shown in Figure 5. The pattern of the particle velocity normal to the surface (counted positive in the positive z-direction) then is u(x, y) = V, e-j(xkV+yk, 1, with the components

(65)

of the wavenumber being

k, = b sin Si cos Cpi= kop,;

k, = b sin 8; sin pi = bp,,,

a/2

Figure 5. Co-ordinates of the rectangle.

(66)

FINITE

SIZE

ABSORBER

RADIATION

239

IMPEDANCE

where pu,, CL,,have been introduced for abbreviation. The sound pressure field with these velocity sources is e-&R

o(xo,yo)ydS,,

(67)

A

with dS, = dx,, dy, and R* = (x - x0)* + (y - yd2 + z*. One way to the radiation impedance could be via the radiated power:

which, with the definition of 2, (not normalized) ) V,(*A) gives

in equation (22) (where the integral is

e-jkoR _ e-j[k~(x,-x)+k~(y,-Y)l

(69)

dc&

R A

A

Four integrations must be performed here, and the integrand becomes infinite whenever x = x0 or y = yo, which would happen all over the area in a direct numerical integration. Another way to the radiation impedance 2, is via the Fourier transform of the velocity pattern of the rectangle in the otherwise rigid plane z = 0 [19]:

v(xo,yo) e-j(k,xo+kzYo)dx,

V(k, , k,) =

dye

=

Voab

sin ((k, + k,)a/2)

sin ((k2+ k,)b/2)

(k, + kM2

(kz+ k,)b/2

’

A

(70)

with the result:

z

s

=z

sin((Cc,-y2)k+/2) 2

sin((IL -

ydkd2) (/A - Y,uw/2

k&ob O 4lr*

by -

yJkobl2

-m

dy, dy2

w

(71)

and with the abbreviations yI = k,/ k,,, y2 = k2/ ko. Here only two integrations are necessary with an integrand which becomes singular along a curve in the plane of the integration variables. A third starting point is from the Bouwkamp integral (27) with the directivity function

D(fi, cp) =

sin y ( y

(sin &

COScpi-SiIlJfCOSp) >

(sin 6i

COS Cpi -sin

6

COS (9)

sin !$ (sin ai sin (pi- sin r9 sin cp) )* ( +f

(sin Si sin cpi-sin 6 sin cp) (72)

We will start from expression (71) and introduce further abbreviations: a’= koa/2,

b’= k,b/2,

/_&*=1-y:.

(73)

240

F. P. MECHEL

We consider the integral

II

=

JOc?(sin

(~‘(Y~-P.,))

(74)

-m

and apply to it the Parseval theorem which, for pairs of Fourier transforms

Jcc

f(y)=

F(x) ejxy dx,

-m

(75)

says that

2;J 1mfdvlfi(y)

-

m

dy =

J -03

F,(x)F,(-x)

dx.

(76)

We take the first factor of the integrand of equation (74) as F,(x) and the second factor as I$(-x) and obtain, from tables of Fourier transforms, (77) The radiation impedance with this has an intermediate Z,

AZ,b a0 J

r

ken (kOa -bb

shape:

~0s bL,~)Z2(1~0

dy,

(78)

with the integral

I,(I~I) = Ja

-m

(Sinj~~~~~~)))2~S,o dy,.

(79)

Y

This again can be modified by Parseval’s theorem, taking the first factor as F,(y2) and the second factor as F2(--y2): x 2b’bV - 1x1)+;cos UQ)* J0

L(M)=_JK

(~9) dx.

(80)

Inserting this into equation (78) and interchanging the variables x, y (for better readability), gives the radiation impedance as e -jm z = 2j Z, koa kob s

-r

k,,akob Jx=o Jy=o

(81)

(kOn-x)(k,b-y)cos(~~)cos(~,y)~dxdy.

This is the analogue to equation (44) for the strip. A double integral now appears because of the limitation of the area in two directions and a factor (exp (-jR))/R takes over the role of the Hankel function, which is a common substitution when passing from a one-dimensional radiator to a two-dimensional one. This formula could be used for numerical integrations. The “inner” integral now has an integrand which becomes singular at x = y = 0 only. Tbe “outer” integral is regular. By the way, with normal sound incidence, & = 0, one obtains e-jJx2+y2

(koa -x)(kob

-y)

dx dy,

-Jm

x

which is another representation radiator (see reference [ 141).

of the radiation impedance

+Y

of the rectangular

(82) piston

FINITE

SIZE

ABSORBER

We will continue to treat equation performed:

RADIATION

(81) analytically.

x = r cos Q,

241

IMPEDANCE

A substitution

of variables is

y = r sin Q,

(83)

with r and Q explained by Figure 6. The range of integration B(x, y) is subdivided into two subranges:

&(r,

Q)

=

OS Q S arctg (b/a)

&(r, Q) =

OSrSk,a/coscp

arctg (b/u) S Q S 7~/2 OSrSk,b/sinQ

(84

Y t kob

uoa

0

x

Figure 6. The substitution of variables.

One can write equation (81), after this substitution, as

=2j --

z

’

20 7~ koak,,b

arctg( b/ (I )

[I

I( koa/cos

o

Q)

dQ +

I(k&lsin

Q) dQ , I

(85)

where the integral I is of the type I(R)

=

( U + Vr + Wr’) cos (ar) cos (/?r) e-j” dr,

(86)

with the abbreviations. U = k,,abb, W= sin

Q cos Q,

V = -( koa sin (Y = j.L, COS Q,

Q +

kob cos

Q),

/3 = cay sin

Q.

(87)

The task now is to solve for I(R): this will be done by repeated partial integrations. It would be possible to represent the result as an explicit formula for 2, which would contain only simple terms (one can see that at the end of the ensuing analysis). Wowever, it would fill pages and the chance of reproducing it without printing errors is small. Therefore, a more algorithmic presentation is preferred here, which can give useful instructions for programming the numerical task.

242

F. P.

I.0

MECHEL

I

I

I

I

(b)

Figure 7. Radiation resistance of rectangles with different side ratios. 4, =30”, cp,=O”. (a) Z,i/Z,,; 0, from Figure 3; (b) Z:/Z,,; 0, from Figure 4. Values of b/a: . . ., 1, - - - -, 4; - - -, 16; -, 64.

Further abbreviations

are introduced in order to relieve the writing. We introduce:

C(x) =

cos (xr),e-jr dt =

x sin (xr) - j cos(xr)) e-j’ = C’(x) e-j’,

S(x) =

-1 sin (XI) e-j’ dr = (x cos (xr)+j X2-1

sin (xr)) e-j’= S’(x) e+,

(88)

where the primes in the last forms serve to maintain the notation when splitting off the common factor exp (-jr). We further introduce CC(x, y) =

CS(x, y) =

cos (xr) cos (yr) e-j’ dr,

SS(x, y) =

cos (XI) sin (yr) e-j’ dr,

SC(x, y) =

J

J J

sin (xr) sin (yr) e-j’ dr, sin (XT) cos (yr) e-j’ dr.

(8%

These are special cases of the following integrals with a general function f(r): 1,(x

(f( r)) =

J

cos (xr)f(r)

e-j’ dr,

1,(x/f(r))

= j sin (xr)f(r)

e-“d r.

(90)

FINITE

SIZE

I.5

ABSORBER

RADIATION

I

I

IMPEDANCE

I

243

I

(a)

+ I.0 ,/*~

cp.“‘:____*----* .... ,,

R--J *..*

. ..‘.

I’

,A

,/ /.’ /.:* ,,’ /: ,’ _.* ,t

@5-

/ -&$/” ,&

I

Figure

These can be transformed

8. As Figure

I

I

I

7, but 9, = 89”, cpi = 0”.

by partial integration into

After a partial integration of equation (86) one obtains Z(r)=(U+Vr+

Wr’)CC(a,/?)-(V+2Wr)

J

CC(a,P)dr+2W

P) dr2, JJCC(a, (92)

where the use of r in Z(r) indicates that it is the indefinite integral. For the determination of CC( (Y,/3) and of its integrals, one can apply the rule (91) on the functionsf( r) = cos (yr)

244

F. P. MECHEL

1.0.

(b)

b/a= .0.5 . . ..p4--

--.

0.25

0 if

a

Figure 9. As Figure 7, but 19~=45”, (9, =O”, and values of

b/a

as indicated.

and f(r) = sin (yr), and one finds

CCb, y) = C(x) cm (Y’)+-$$

[xSS(x,Y) -jWx, ~11,

X(X, y) = S(x) cos (yr)

-&

Wx, Y) = C(x) sin(yr)

----&CxWx, y)-.iCC(x, ~11,

Wx,

Y)

[xCS(x, y) +jWx, Y)I,

= S(x) sin(yr)+&[xCC(x, v)+jSC(x, Y)I.

(93)

This is an inhomogeneous system of four linear equations in the four unknowns CC, SC, CS and SS. Only CC is needed originally in equation (92), but we shall need the other unknowns too for the integrals over CC. Such systems of linear equations will appear still several times. Therefore we use a symbolic representation [M](z) = (c) for them with the column of the unknowns (z) = (z,, z2, z,, zq) and the column of the right side (c) = (cr , c2, c,, cd). The matrix of the coefficients will always have the form

(94)

FINITE

SIZE

ABSORBER

RADIATION

I.5

.

(b)

L+85/ 2’ ,’

I.0 I’/ ,,

0

Figure

10. Radiation

impedance

245

IMPEDANCE

,/‘_, I’/’ y/

8’ 60” _-----_

-_.

_

IO

of rectangles

20

for different

polar

angles;

b/a = 0.33,

cp, = 0”. (a) Z:/Z,;

(b) Z::/Z,.

with real u and u. Its determinant

is

D=1-2&22u2-2(uv)2+u4+u4.

(95)

The solution can be obtained by applying Kramer’s rule, Zi= Di/D, i = 1, . . . ,4, with the determinants

The system (93) shall be named System a, with the unknowns (2,) = (CC, SC,

cs,SS),

(974

and the right-hand column

Cc) = (C(a) cos (fir), S(a) cos W), C(m) sinW-1, S(a)

sin(Pr)),

(97b)

F. P. MECHEL

246

Ib) 0.85

-

,

-

(

I

I

*

/

095

F 0 &

I

0.8

0.9

SO-

5 4‘

,

\

-

0.7

Ia

0.6

I.05

0.5

I.075

0.4 0.3

30 -

0.2

I.075

A

0.1 I.05 0

I

I

I 30

I

I

I 60

I

I

. 90 p,

I

0

I

I

1

I

I 60

30

1

I 90

(degrees1

Figure 11. Curves of constant values of the radiation resistance (a) and reactance (b); hu = 5.4, k,b = 5.4, b/a=l.

and, finally, the quantities in the coefficient matrix U = P/(c?-

l),

u = c$?/(&‘-

1).

(97c)

All Zai must be determined. The equations (93) are integrated next in order to determine the integral over CC in equation (92). Integrals of the cOiwill be needed in this step. By application of equation (91) and of equation (89) one derives the system of equations, called System b, with the unknowns (%) = (UP I C’(a)),

UP I C’(~)),

m

P’(4),

MP IS’W)),

WW

S(P)S’(~))

(98b)

with the right-hand column (CtJ = (C(P)C’(a),

W)C’(,),

C(P)S’(a),

and the quantities in the coefficient matrix u = (u/(@-

u = czP/(j3*- 1).

l),

All four unknowns must be determined again.

‘p,

(degrees)

Figure 12. As Figure 11, but k,a = 2.7, b/a =2.

(98~)

FINITE

SIZE

ABSORBER

RADIATION

247

IMPEDANCE

-0.35 0

045 I

1

I 30

I

I

I 60

I

I

I

90 ‘pi

1

0

I

I

I

50

I

I

1

60

90

(degrees)

Figure 13. As Figure 11, but k,a = 2.7, k,b = 10.8, b/a ~4.

Then the system of equations for the integrals over CC, SC, CS and SS, which is called System c, has the unknowns ( Z,)=(I

CC(a, P) dr,

I

SC(a, P) dr,

I

CR&, P) dr, / SS(a, P) dr),

(99a)

the right-hand column (cc) = (%I,

zb3,

zb2,

zb4),

WJb)

with the solutions from System b. The matrix contains the terms u=P/(a”-l),

v=c@/(cw”-1).

(99c)

Only z,~ must be computed in this system. The double integral over CC in equation (92) is finally needed. For this System c is integrated. The integration over the unknowns yields the wanted double integral over CC (and also over SC, CS and SS which, however, are not needed). The integrals over the right-hand column of System c are obtained by integration of System b. This leads to an integration of the product terms in (cb). After the products have been multiplied, making use of equation (88), one sees that the integrals over the cbi can be expressed by the CC, SC, CS and SS: i.e., by the solutions of System a. Therefore one has to solve a fourth system of equations, System d, with the unknowns (G) = (1 L(a 1C’(P)) dr, 1 &(a 1C’(P)) dr, 1 I,(cr (S’(B)) dr, [ &((u IS’(p)) dr), (looa) with the elements of the right-hand side cdl

=

(u2_l~~p2_1) [z,,+j~z,2+jPz,3-(rPz,41, -1

cd2= (a’-

l)(@-

cd3= (a’-

1)(/?2-- 1)

1)

-1

[-jPz,,

+ N3z,2+

[-_iazoI +

za2+

za3+jaza41,

4ze3+jfk41,

F. P. MECHEL

248 IO

8

.._

6

4

,.

,.

Figure

14. Curves of constant

b

\

o.,

values of the radiation

5

.,;

resistance

Z_L/Z, (a) and reactance

Z:/Z,

(b). di =O”.

and with the quantities of the coefficient matrix u=

ff/(p-

l),

21= (Y/3/@‘- 1).

(1OOc)

All unknowns zdi are determined. With these, the fifth and last system of equations, System e, can be solved. It has the unknowns Z, ( )=(II

CC(a, P) dr*,

P)dr* PI dr’ , dr*, >, JJWa, JJCS(a,P) JJSC(a, (101a)

FINITE

SIZE

ABSORBER

RADIATION

IMPEDANCE

249

b

to It

I

6

4

2

C

kob Figure 15. As Figure 14, but 9; =45”.

the right-hand side

(ce)= (zdl, zd3, zd2, zd4)>

(1Olb)

and the matrix quantities u = p/((Y’- l),

u= @/(cu’-

1).

(1Olc)

One only needs z,~, which is the double integral over CC. Now all terms in equation (92) are known. The procedure for the solution of these systems of equations must be repeated four times for the computation of 2, from equation (85), twice with the upper integration limits of the ranges B, and B2, and twice with the lower integration limits r = 0. The value of Z(r) for r = 0 could be evaluated analytically.

250

F. P. MECHEL

6

TZ. \_ .,.._. ,”. .._.........

6

._... ,,.

/-cc--

0.6

;

4.

,..

-‘,

,.

._.....

..,

..,,,............,.,.,,....... i..,...,,,. .._

I

-Gz. --

Figure 16. Curves of constant values of the radiation resistance 2:/Z, (p,= 0”.

(a) and reactance 2:/Z,

(b). fii = 45”,

The saving of computing time does not compensate, however, for the larger amount of programming. Kramer’s rule gives explicit solutions of the systems of equations. There is no principal difficulty in writing down an explicit expression for the radiation impedance of the rectangle by insertion of these solutions according to the outline above. This expression would be a finite sum of elementary functions, i.e. trigonometric functions. This is remarkable, because all other known explicit solutions consist of infinite series and/or contain higher transcendental functions. For Si = 0 one should obtain the radiation impedance of the rectangular piston radiator. This can be used as a check; the agreement with results from reference [14] ia perfect.

FINITE

4

2

-..

SIZE

ABSORBER

RADIATION

251

IMPEDANCE

.

. \yj.!$z&t ..- 0.4; 0.5 _....-

I _.._““..,_ ,,.,._.,.._j.._“,,,_._ i . _“._.,.“._. 1 _...I_.

/

0.2 o!s :_ o.,LJ-

jf..._\..jy

1Ii

f..._

t

i

._..t

)._

,t.

.._._:

i

=

.j..._

_.I

6 kob

Figure 17. As Figure 16, but Bi = 45”, cpi= 45”.

In Figures 7(a) and (b) are shown the normalized radiation resistance and reactance, respectively, for field-excited rectangles with different side ratios b/ a with sound incidence under 6i = 30” and vi = 0. Also included in these diagrams are points which were taken from Figures 3 and 4 for the strip. One sees that the radiation impedance of the rectangle can be approximated by that of the strip for side ratios b/a at about b/a > 4. This still holds for nearly grazing incidence, as Figures 8(a) and (b) show for 4, = 89”. This finding supports Thomasson’s proposal to apply for rectangles the results from the strip with an effective strip width. However, the diagrams restrict this method to side ratios of about b/a > 4. In Figures 9(a) and (b) are illustrated the influence of side ratios b/a < 1, i.e., with the plane wave incident normal to the short side.

252

F. P. MECHEL

Figure 18. Curves of constant angles; ha = 1, ~9~= 0”, cpi = 0’.

absorption

coefficient

(~(a,, cp,) for sound

incidence

on a strip at discrete

The polar angle 8i is varied in Figures 10(a) and (b) with the wave vector of the incident plane wave along the x-axis (vi = 0). The character of the curves changes strongly for large koa. A better idea of the influence of the incidence angles ai, vi can be derived from contour diagrams of the components of 2, over the plane of the angles ai, pi. Squares are nearly isotropic in 2, with respect to pi, as can be seen from Figures 1l(a) and (b). This approximately still holds for the real component of 2, of rectangles with not too large side ratios, as can be seen from Figures 12(a) and (b) for b/a = 2. If the side ratio becomes larger, as in Figures 13(a) and (b) with b/a =4, then one finds a pronounced anisotropy.

6

Figure

19. As Figure

18, but fii =75”.

FINITE

SIZE

ABSORBER

Figure

RADIATION

20. As Figure

253

IMPEDANCE

19, but cp, =90”.

The sound absorption coefficient for diffuse sound incidence with discrete angles, (Y(6i, vi), according to

adifl

fOllOWS

from

that

7-r/2

(Y(61,

Qi) COS 6,

Sin

6, d6+2

I

LT(Si)

COS 6,

sin 6, d6i,

0

(102)

where the last relation is an approximation R%peCt

t0 Qie hl

eCpBtiOIl(6)

for

(Y( 6iy Qi) Only

Figure

21. As Figure

if the absorber is (nearly) isotropic with 2, depends on Qi. It makes sense, therefore,

18, but k,a = 6.

F. P. MECHEL

254

Figure 22. As Figure

19, but k,a = 6.

to apply a radiation impedance’which is averaged over (Piwhich we identify by underlining: Z,(ai,

(103)

pi)dqi.

The integration can start from equation (Bl), in which only pu, and CL,,are dependent on vi. The integral can be performed [20, equation 3.876.71, and one obtains e-jJYW

(104)

(%~-x)(~b-y)J~(~~sin9,)~dxd~.

6

,

Figure 23. As Figure 20, but k@ = 6.

I

FINITE

SIZE

ABSORBER

RADIATION

-2

Figure

,“t,

24. As Figure

255

IMPEDANCE

2

4

22, but k,a = 18.

This formula is suited for numerical integration. Contour diagrams of the components of zS over the plane of ha, k,b are shown in Figures 14(a) and (b) for normal incidence, 6, = 0, and in Figures 15(a) and (b) for oblique incidence, ai =45”. If one compares Figure 15 to similar contour diagrams, but now of Zs(Si, vi), as in Figures 16(a) and (b) for cpi= 0 and in Figures 17(a) and (b) for cpi= 45”, then one sees, however, that the method of averaging over ‘pi is restricted to side ratios no larger than about b/a = 2. The best agreement between zs!s(Si) and Zs( 41, vi) is obtained for pi = 45”. It is plausible that the averaged radiation impedance Z of rectangles with not too large side ratios has values similar to those of the radiation impedance of a field-excited

Figure 25. As Figure

23, but k,a = 18.

256

F. P. MECHEL

Figure 26. Curves of constant absorption coefficient ndiB for diffuse sound incidence on a strip; &a = 1.

circular disk with radius a, which is 00

,3 ?r

2mJ:(k,av)

I I_ov2J1-;IUdu du9

(105)

u=o

with V2= U2+sin2 9Yj-2u sin 6, cos ff. 7. NUMERICAL EDGE EFFECT Numerical examples of the absorption coefficient and of the edge effect shall be presented for absorber strips of width a. The co-ordinates are as in Figure 2. First, contour

Figure 27. As Figure 26, but &,a = 3.

FINITE

SIZE

ABSORBER

RADIATION

251

IMPEDANCE

6

-2

Figure

,“,c

28. As Figure

2

4

26, but k,,a = 6.

diagrams with lines of constant absorption coefficient (Y( Cti, pi) according to equation (6) are plotted over the complex plane of the normalized wall impedance 2 = l/F = Z’+ jZ”. Parameters are IQ+ and 6i, pi. In Figure 18 is shown the absorption coefficient for a narrow strip, ha = 1, at normal sound incidence. The maximum of the absorption is shifted towards the side of negative rectances. The influence of the angles is small with such small ha unless the sound incidence approaches grazing incidence. Figure 19 is an example thereof (fii = 75”) with vi = 0 (sound normal to the strip axis) and Figure 20 with (Pi= 90” (sound along the strip axis). The value of the absorption coefficient can exceed unity considerably. Figure 21 for a strip with medium width ( koa = 6) at normal incidence qualitatively resembles what

Figure 29. As Figure

26, but ha = 12.

258

F. P. MECHEL

Figure 30. Curves of constant values of the numerical edge effect KE(B,, sound incidence at discrete angles; k,n = 1, 8, = 0”, cp, = 0”.

cp,) % of an absorber

strip for

is expected: a maximum of about one at the point of impedance matching 2 = 1. For flat sound incidence, however, as in Figure 22 with 6i = 75” and sound normal to the strip axis (cp,= 0), as well as in Figure 23 with the same 6, but sound incidence parallel to the strip axis (vi = 90”), the behaviour becomes quite different from that. The point and the value of maximum absorption are shifted. Even with relatively wide strips, as in the examples of Figures 24 and 25 for k,a = 18, there exist significant differences from the infinite absorbers, at least for nearly grazing incidence, as can be seen from the position and the value of the absorption maximum.

Figure 31. As Figure

30, but 0, =60”.

FINITE

SIZE

ABSORBER

Figure

RADIATION

IMPEDANCE

259

32. As Figure 30, but ~9~= 85”.

It is also of interest to compare the sound absorption with diffuse sound incidence of the infinite absorber to that of an absorber strip (see equation (102)). It is known that the maximum sound absorption of locally reacting infinite absorbers is about (Y,,= 0.96 at the normalized impedance of about Z = l-6. In Figure 26 are shown lines of constant values of the diffuse-field sound absorption of an absorber strip at low frequencies (/~,a = 1). The absorption coefficient can go as high as about 4 and the maximum is at about Z =0*5 -jO*5. As the series of Figures 27-29 shows, with increasing frequency and/or strip width the maximum values of the sound absorption go down and the points of the maxima are shifted upwards in the resistance, up to about Z’ = 1.6; the reactance of the maximum remains, however, at about Z”= -0.5.

Figure

33. As Figure

30, but k,,a = 12.

260

F. P. MECHEL

Figure

34. As Figure

30, but k,a = 12, 6, ~45”.

The diagrams of the sound absorption coefficient do not show the edge effect immediately. In order to make the edge effect clearer, we plot contour diagrams with lines of contant values of the quantity (KE = “Kanten-Effekt”), i.e., KE[koa, ai, ~oi)=[a(ai,

Pi)-am(~ii)llQw(~i),

(106)

in percent, where a( 6i, 4pi) is the sound absorption of the absorber strip and am(8i) is the sound absorption of the infinite absorber with the same normalized wall impedance Z (which is independent of cp,). The corresponding quantity KE(k,a)then will be shown for diffuse sound incidence.

Figure 35. Curves

of constant

values of the numerical edge effect K&,0 sound incidence; k,a = 1.

% of an absorber

strip for diffuse

FINITE

SIZE

Figure

ABSORBER

RADIATION

36. As Figure

IMPEDANCE

35, but k,a = 12.

This numerical edge effect over the plane of the normalized wall impedance for normal sound incidence on a narrow strip ( koa = 1) is shown in Figure 30. One sees from this example that the edge effect must not always be positive. However, the positive edge effect is much larger than the negative one. The maximum is near to the negative reactance axis (there is no absorption immediately on the reactance axis, and therefore no edge effect). The edge effect at low frequency strongly increases with increasing polar angles of sound incidence, as can be seen from Figure 31 (Si = 60”) and Figure 32 (4, = 85’). The edge effect reduces with wider strips. Examples for k,,a = 12 with normal and oblique incidence, respectively, are shown in Figures 33 and 34. The numerical edge effect for diffuse sound incidence is shown in Figure 35 at low frequencies (ha = l), in Figure 36 at k,a = 6 and in Figure 37 at high frequencies

Figure

37. As Figure

35, but &a = 20.

F. P. MECHEL

262

(k,a = 20). It is somewhat surprising that even for rather wide strips (kg = 20 in Figure 37) the edge effect can be as high as 50% for negative reactances and small resistances (which can be found with resonator absorbers below resonance).

8. CONCLUSIONS

The results presented here are from an analysis which is an approximation to the exact solution. Exact results are available for locally reacting absorber strips [4,5]. The comparison shows that the present computations on the basis of the variational solution of the Helmholtz integral equation are quite satisfactory. They are clearly better than the results from the iterative solution of the integral equation [6].

REFERENCES 1. W. Ku~~1960 Acustica 10,264-276.Der Einfluj3 der Kanten auf die Schallabsorption porSser Materialien. 2. F. P. MECHEL 1986 Journal of Sound and Vibration 107, 131-148. Absorption cross section of absorber cylinders. 3. J. R. PELLAM 1940 Journal of the Acoustical Society of America 11, 396-399.Sound diffraction and absorption by a strip of absorbing material. 4. J. ROYAR 1974 Dissertation Math.-Nat. Fakultat, Unioersitat Saarland. Untersuchungen zum akustischen Absorber-Kanteneffekt an einem zweidimensionalen Modell. 5. F. P. MECHEL 1984 Fortschritte der Akustik, DAGA 1984, 31-52. Schallabsorber. 6. F. P. MECHEL 1989 Journal of Sound and Vibration 134, 489-506. Iterative solutions for

finite-size absorbers. 7. A. LEVITAS and M. LAX 1959 Journal of the Acoustical Society of America 23,3 16-322. Scattering

and absorption by an acoustic strip. 8. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. See pp. 156 ff, and 458 ff. 9. T. D. NORTHWOOD, M. T. GRISARU and M. A. MEDCOF 1959 Journal of the Acoustical Society of America 31, 595-599. Absorption of sound by a strip of absorptive material in a diffuse sound field. (It should be noted that the definition of the sound absorption coefficient in this paper differs from the usual definition by a reference to the normal incident power.) 10. T. D. NORTHWOOD 1963 Journal of the Acoustical Society of America 35,1173-l 177. Absorption of diffuse sound by a strip or rectangular patch of absorptive material. 11. S. I. THOMASSON 1982 Report TRITA-TAK-8201, Royal Institute of Technology, Stockholm. Theory and experiments on the sound absorption as function of the area. 12. S. I. THOMASSON 1980 Acustica 44, 265-273. On the absorption coefficient. 13. J. F. HAMET 1984 Reoue d’Acoustique 71, 204-210. Coefficient d’absorption acoustique en champs diffus d’un materiau plan, rectangulaire, de dimensions finies, pos6 sur une surface infinie parfaitement rCfltchissante. 14. F. P. MECHEL 1989 JournalofSound and Vibration (in press). Notes on the radiation impedance, especially of piston-like radiators. 15. C. J. BOUWKAMP 1945 Philips Research Report 1,251. A contribution to the theory of acoustic radiation. 16. F. P. MECHEL 1968 Proceedings of the 6th ICA, Tokyo, H-217-220. New method of impedance measurement. 17. Y. L. LUKE 1962 Integrals of Bessel Functions. New York: McGraw-Hill. 18. The author can send the results in manuscript form on request to interested readers. 19. M. HECKL 1977 Acustica 37, 155-166. Abstrahlung von ebenen Schallquellen. 20. I. S. GRADSHTEYN and I. M. RYZHIK 1980 Table of Integrals, Series and Products. London: Academic Press.