Measurement of acoustic flow resistance: Further comments

Journal

of Sound

nnd Vibrufion (1987) 118(3), 543-544

LETTERS TO EDITOR MEASUREMENT

OF ACOUSTIC

FLOW RESISTANCE:

FURTHER

COMMENTS

Ingard and Dear [I] have discussed my comments [2] on their method for measuring the acoustic flow resistance of porous materials [3], and I should like to add a few final points to the discussion, for the sake of clarification. Concerning equation (3) in reference [3] I agree that, strictly speaking, this is correct as it stands. But it is perhaps more common (without being universal practice), when as Z/PC = ,9 -ix, using “-i” theory (as Ingard and Dear have done), to write impedance rather than as 0 +ix, as in equation (1) of reference [3]. This has the advantage of preserving a positive sign on the mass reactance (for example), and thus maintaining a degree of consistency between “-i” and “+i” usage. Dr. Ingard has, indeed, used this notation in the past [4,5] (though not consistently-reference [6] is an exception) and this fact led me to suspect that equation (1) in reference [3] should possibly have read Z/PC = 8 -ix, in which case equation (3) would have acquired the negative sign that I referred to in my comments [2]. In any case, this is a minor point that I mentioned [2] in passing. As far as the flow resistivity of the material is concerned, I may not have made myself sufficiently clear; at all events, Drs Ingard and Dear seem not to have understood the point that I was attempting to make. Certainly, Ingard and Dear’s “flow impedance” z is not the same quantity as the acoustic flow resistivity per unit thickness of material a(w), and (as I pointed out in equations (4a, b) of reference [2]) there is a relationship between z and a(w) = a, +ioi, which is z/I = a, + icr, + iq’op,/n

(1)

(the +i sign convention is being used here), where I is the material thickness, q and R are the geometrical tortuosity and volume porosity (respectively) of the material, w is radian frequency and p. is fluid density. Equation (1) is valid for a rigid porous medium. My argument was principally that Ingard and Dear did not sujiciently stress the di$erence between z and o(o), and I also wished to point out the relationship (1). The quantity (+(o) is used in many rigid frame models of porous materials and is a fundamental characterizing parameter of the bulk acoustic properties. It seemed that a cursory reading of their paper might lead the reader to confuse z/I and o(w), notwithstanding the statement in the Introduction of reference [3] that “the reactive part of the flow impedance is due to the effective inertial mass density of the fluid in the material”. My comments [2] on the accuracy of measurement of cri still hold. Ingard and Dear [l] assert that I was concerned that the “error can become considerable, as the thickness of the material increases” (sic), but one can see from my comments [2] that this is not the case. In fact, I merely pointed out that since the sample thickness must necessarily be small, materials having low flow resistivities would have correspondingly low values of z, and the percent accuracy in the measurement of z would thus be limited. A further possibility of confusion in reference [3] arises in connection with the structure factor of the material. This, as discussed in section 3 of [3], would seem to include not only purely geometrical effects (i.e., q2) but also the inertial effects in the viscous acoustic boundary layer adjacent to the solid frame of the material, as represented by oi (see equation (1)). Thus, since Ingard and Dear evidently derive empirical values of the structure factor as the ratio between x/l and w/c, (c, being the acoustic speed in the fluid), their “structure factor” is equal to m I,D = c+i/‘pow+ q2/L?. The momentum equation 543 0022-460X/87/210543+02

$03.00/O

0

1987 Academic

Press Limited

LETTERS TO THE EDITOR

544

for the fluid in the pores of the material

is then

Vp + orV + iwp,m,,DV = 0, rather than the more common

version

(2a)

(see the review paper by Attenborough

Vp+c~(w)V+iwp,q*V/Ll

=O,

[7]) (2b)

(where p is sound pressure and V is acoustic particle velocity) or its equivalent. Since gi is a positive quantity, according to the available microstructure models for porous media, m,,, would be greater than q2 (the “geometrical” structure factor), possibly by as much as a factor of two. Department of Mechanical and Aerospace Engineering, University of Missouri, Rolla, Missouri 65401, U.S.A.

A. CUMMINGS?

(Received 18 November 1986) REFERENCES 1. K. U. INGARD and T. A. DEAR 1986 Journal of Sound and Vibration 109,51.5-518.

Measurement of acoustic flow resistance: discussion. 2. A. CUMMINGS 1986 Journal of Sound and Vibration 109,512-514. Comments on “Measurement of acoustic flow resistance”. 3. K. U. INGARD and T. A. DEAR 1985 Journal of Sound and Vibration 103,567-572. Measurement of acoustic flow resistance. 4. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. 5. U. INGARD and V. K. SINGHAL 1975 Journal of the Acoustical Society of America 58,788-793. Effect of flow on the acoustic resonances of an open-ended duct. 6. K. U. INGARD 1981 American Society of Mechanical Engineers Journal of Engineeringfor Industry 103, 302-313. Locally and nonlocally reacting flexible porous layers; a comparison of acoustical

properties. 7. K. AITENBOROUCH materials.

1982 Physics Reports 82, 179-227.

t Now at the Departmentof Engineering

Design and Manufacture,

Acoustical characteristics

University

of porour

of Hull, Hull HU6 7RX, England.