A note on the relation between complex acoustic intensity and specific acoustic impedance

Journal of Sound and Vibration (1987) 118(3), 549-554

A NOTE

ON THE

RELATION BETWEEN COMPLEX ACOUSTIC SPECIFIC ACOUSTIC IMPEDANCE 1.

INTENSITY

AND

INTRODUCTION

In recent years, studies dealing with complex acoustic intensity have received considerable attention [l-4]. In addition to the well known use of the acoustic intensity (the real part of the complex intensity, also sometimes called the active part), analysis of the reactive part of the complex intensity is quite useful in acoustic field descriptions [l-5]. This note presents a general relation between the complex acoustic intensity and the complex specific acoustic impedancle. Examples of the general relation are given for some simple acoustic fields, and also applications of the analysis to intensity measurements are briefly discussed. 2. The instantaneous

(physical)

FORMULATION

acoustic

intensity

I =p(x, For fields of periodic time dependence intensity is (where T is the period)

is

r)u(x, t). the mean

(1) time

average

of the instantaneous

T I(x)

=+

-

I0

P(X, ~Mx,

t) dt.

(2)

Thus only the instantaneous intensity is in the direction of u(x, t). If complex quantities are used to represent a field of simple harmonic one has the identity

time dependence,

I(x) = 4 Re [ i(x)i

(3)

Here and in what follows a tilde (“) denotes a complex quantity and an asterisk (*) denotes a complex conjugate. Thus fez acoustic fields of simple harmonic time dependence, a mean complex acoustic intensity, C,can be defined as E=+[p'fi*]=i+jQ.

(4)

Here p’ is the acoustic pressure, i is the particle velocity, i is the real acoustic, active, mean intensity (as given in equation (3)) and G is an acoustic reactive mean intensity. It can be seen from equations (l)-(4) that the geometrical directions of the various intensities thus defined-instantaneous (l), periodic mean (2), simple harmonic mean (3), and complex mean (4)-are all, in general, different. In particular, the geometrical direction of the simple harmonic mean intensity is not that of the complex mean intensity but rather that of only its real part. The directions of the two components of the complex mean intensity, i and Q, may in general, be different from each other, and different again from that of i*, which is of course the direction of the total mean complex intensity C. For any given direction specified by a unit vector n, the component of the mean complex intensity in that direction can be expressed, from equation (4), as E.n=f[p’fi*

.n]=!J~~~]=I;,+jQj,.

(5)

549 0022-460X/87/210549+06

$03.00/O

@ 1987 Academic

Press Limited

LETTERS

550 3.

complex impedance The define it in a given arbitrary velocity is

TO THE

THE COMPLEX

where uR,, and u,,, are the components and hence

= uf7,fu+ju,,,u,

n(uR,,

2, =i/tZ, R, being the resistance

Equation

+h,,)

=

uR(x) and ul(x)

(7)

THE

(8) can be written,

COMPLEX

Hence the

(8)

for the n direction. IMPEDANCE

by using equation

i,, = R,+jX,,

parts, respectively.)

= R,+jX,,,

and X,, the reactance

BETWEEN

in that direction,

nun.

R and I denote real and imaginary in the n direction is

(Here the subscripts complex impedance

RELATION

(6)

of the vectors

u”(x) =

THE

IMPEDANCE

is not a vector; for the complex impedance one can only and real direction. In the direction defined by n the particle

u,(x)

4.

EDITOR

AND THE COMPLEX

INTENSITY

(5), as

=f(~u’~)/tu”,~~=(2/1~n12)[ln+jQ,].

(9)

(Here, and in what follows, the overbars indicating mean quantities are omitted for brevity.) Thus, the real and imaginary parts of the mean complex intensity in the n direction are related to the real and imaginary parts of the impedance in that direction by I, =$&12Rn,

Qn =~~lq2Xn.

(1% b)

From equations (10a) and (lob) the following relation among complex impedance parts for the n direction can be derived:

the complex

intensity

(11)

QnlL = WK,. Some examples of the explicit forms of this general ratio relationship in the next section, for simple representative acoustic fields. 5.

APPLICATION

TO SIMPLE

and

ACOUSTIC

(11) are given

FIELDS

5.1. Harmonic plane waves The acoustic pressure p’ and the particle velocity ue in a direction x-direction are, for a simple harmonic plane wave field, p’= A fie

Using equations

=

[(COS

,j(wr-k-x)

e)lpc][A

+

B

ej(wr-kx)-

(12) and (13) in the intensity

at an angle 0 to the

ej(wf+kx),

B

(12)

(13)

ej(wr+kr)].

and impedance

equations

6, = ZO+jQB = [(cos B)/2pc][(A*-B2)+j2AB

gives

sin (2kx)],

(14)

~,=R,+jX,={pc[(A2-B2)+j2ABsin(2kx)]}/{cos~[A2+B2-2AB~~s(2kX)]}. (15) From equations

(14) and (15), QB/ IH = X,/R,

Note that relationship

= 2AB sin (2kx)/(A’-

(16) does not depend

B2),

upon the direction

angle

J16) 0.

LETTERS

If the field is urn-directional,

TO THE

551

EDITOR

i.e., A = 0 or B = 0, one has Qe/Ie=XJRe=O.

(17)

This result corresponds to the fact that the complex impedance and complex mean intensity for a plane traveling wave are purely real. It should be noted then when 6 is ~12, i.e., in a direction at right angles to the propagation, e0 =0 and &+co, from equations (14) and (15). The ratios (16) and (17) are then, of course, both indeterminate. 5.2. Spherical wave jield The acoustic pressure p’and the particle velocity I& in a direction at an angle 0 to any given radius vector ii in the case of a finite monopole (i.e., a pulsating sphere) can be written as j(wt-krtko)

,

(1819) where a is the radius of the finite monopole and Q. is the source strength. The actual complex velocity G is of course in the radial, r, direction. By using equations (18) and (19), the intensity and acoustic impedance can be written as

ee=Ze+jQe

=

Q;kpc cos 8 327r2r2(l+ k’a2)

’ [ 1 k+L

(20)

r ’

(21) From equations (20) and (21), Qe/Ie = X,/R,

= l/kr.

(22)

As in plane waves, the ratio relationship (22) is not dependent on the direction angle 6. For both cases, of course, the fields are one-dimensional, being made up of forward or backward traveling waves along x and r, respectively. 5.3. Two harmonic plane wavejields perpendicular to each other The acoustic pressure p’and the particle velocity r& for two harmonic plane waves which are perpendicular to each other are given by $=A ej(wr-kx)+B ej(wr+kx)+c eJ(wl-k,vl+D ej(mr+kv), t-23)

6,

=y

[A

ej(wr-k.x) _

B

ej(wf+k.xl]

+

‘zce[c

ejtwf-kv’

_

D

ej(wr+ky)].

Here, as is evident, 0 is the angle between the selected direction in the x-y plane and the x-axis. By using equations (23) and (24), the intensity and acoustic impedance can be written as follows, for a simple case where B = 0 and D = 0: ~~=ZIe+jQe=(1/2pc)[cos~{A2+ACcos(kx-ky)}+sin~{C2+AC~~~(kX-ky)}]

+j(l/2pc)[cos8{ACsin(kx-ky)}+sin8{-ACsin(kx-ky)}],

_ -~-(R’~~xe)=[cos

@{A’+AC

cos (kx-ky)}+sin

8{C2+AC

(25) cos(kx-ky)}

+ j{AC sin (kx - ky)(cos 0 -sin ~)}]/[cos’ 0 A2 + sin’ 13C2 + 2 cos 8 sin 8 AC cos (kx - ky)].

(26)

552

LEJTERS

TO THE

EDITOR

Hence Qe/ IO = X0/ RO= AC sin (kx - ky)(cos

8 -sin

O)/ O)].

(27)

Unlike the plane wave and spherical wave cases the ratio between reactive intensity active intensity here is dependent on the direction n; however, the ratio relation holds for a given (arbitrary) direction.

and (27)

BC2)+AC

[(cos OA’+sin

5.4. Two harmonic

spherical

cos(h--ky)(cos

O+sin

wave jields

The acoustic pressure p’ and particle spherical waves, from point monopole located at two different points, are

velocity sources

u”,, in a direction of equal strength

n for two harmonic but different phase,

(28) (29) By using equations

(28) and (29), the intensity

and acoustic

impedance

can be written

as

En = I,, +jQ,,

x{kr,

&=kpc

cos (k/3)-sin

(kp)}++

I1

f z{kr2

f~+$~+f~~{cos(kp)+kr,sin(k/3)} 1 2 [ I 2

+~~~{cos(k~)-kr2sin(k@)} I 2

cos (kp)+sin

(kp)}

1,

(30)

1

$$+f$+$!?{kr,cos(k@)-sin(kp)) I 2 2 1 (Ifk2rf)+l

+f$z{kr,cos(k/3)+sin(kp)} I 2

12 1

+2~$$~{(l+k”r,r,)cos(k@)+k(r,--r2)sin(k/3)} where kp = kr, - kr2+ a. Hence the ratio relationship

Qn Xn _=-= 4, R

-$ 2+$3+$

,

2

1 2

${kr,

+-$z{krzcos(k/3)-sin(k@)}

cos (kp)-sin I/F

ar, ‘(ltk’ri) rz (an)

1,

(31)

is (kp)}

--+-;jz rf1 ar, an

1z ar,

12 +$--${cos(kp)+kr,sin(kp)}+-$~{cos(k~)-kr,sin(k/3)} 1 2

1 2

As for the crossed plane waves case of section 5.3, equation relationship depends on the direction, n. Also, the relationship valid in any given direction.

I

.

(32)

(32) shows that the ratio Qn/ I, = X,,/ R, remains

LETTERS

TO THE EDITOR

553

The reason why the ratio relationship depends on direction in more complicated fields such as these crossed plane wave and spherical wave fields is connected with the fact that the mean intensities (real or complex) are well known to be solenoidal vectors, so that intensity flow is analogous to an incompressible Buid flow, or magnetic flux. This property of solenoidality is satisfied if the intensity flux lines in any arbitrary region in the field either enter and leave the region from points on the boundary or describe closed circuits in the interior of the region. In a general field, both types of intensity flux line patterns may be present. The closed circuit types clearly can be regarded as “reactive”, in the sense that they are associated with zero energy transfer across the boundary, but those that do enter and leave at boundary points are not necessarily “active” in the opposite sense. At a wall of a room, for example, flux lines of this type could occur, entering from one point on the panel and leaving from another, which would be associated with the actually reactive near field of the panel, a passive element being driven by the acoustic field in the room in a higher order mode vibration pattern. Also, flux lines entering and leaving at the boundary in general will not lie along straight lines. It is evident from the examples presented here that the general ratio relationship of equation (11) can be expected to be strictly independent of direction only for cases of straight flux lines. There are many cases of practical interest, of course, in which the flux lines will be approximately straight, at least over limited regions. 6.

DISCUSSION

It can be seen that the ratio relationship (11) is of general validity and may be useful in deriving complex impedance values from complex intensity values and vice versa. In the case of travelling plane waves, it is known that the reactance is zero and correspondingly the reactive intensity becomes zero with only the active or propagating part of the intensity given by equation (17). In the case of the single spherical wave field, the term kr which is useful in describing the near to far field transition also appears in equation (22). As would be expected, its inverse is the ratio between the reactive and active parts to the complex intensity and impedance, respectively. The ratio relation between impedance and intensity components can evidently be used in determining the impedance of an acoustical field in a given direction from measured values of the active and reactive intensity components in a given acoustical field, by using an estimated or measured acoustic velocity and the measured complex intensity, the resistive and reactive parts of impedance can be evaluated. This method of evaluation of the impedance could have practical applications, e.g.. in obtaining the impedance at a boundary, at two points of interest in the field itself. ACKNOWLEDGMENT

The authors wish to thank Professor P. E. Doak for his useful suggestions. The authors wish to acknowledge IBM Corporation, Poughkeepsie, New York for their support through a research contract. Also, the authors thank Mr W. Kim, Stevens Institute of Technology for his assistance. Noise and Vibration Control Laboratory, Department of Mechanical Engineering, Stevens Institute of Technology, Castle Point, Hoboken, New Jersey 07030, U.S.A. (Received t Present

24 August address:

1985,

and in revisedform

Zonic Corporation,

Milford,

13 March 1987)

OH 45150, U.S.A.

M. G.

PRASAD

s. Y.

HAMt

554

LETTERSTOTHE

EDITOR

REFERENCES 1. G. W. ELKO 1984 Ph.D. Thesis, The Pennsyloania State Unioersity. Frequency domain estimation

of the complex intensity and acoustic energy density. 2. J. TICHY 1984 Proceedings of the International Conference on Noise Control Engineering, Hawaii, 1149- 1154. Basic intensity flow relationships in acoustic fields. 3. G. W. ELKO and J. TICHY 1984 Proceedings of the International Conference on Noise Control Engineering, Hawaii, 1061-1064. Measurement of the complex acoustic intensity and energy density. 4. M. G. PRASAD and S. Y. HAM 1985 Second International Congress on Acoustic Intensity, Senlis, France. Interference of the acoustic intensity and pressure fields of a primary source due to a secondary source. 5. B. FORSSEN and M. J. CROCKER 1983 Journal of the Acoustical Society of America 73 1047- 1053. Estimation of acoustic velocity, surface velocity, and radiation efficiency by use of the twomicrophone technique.