Received 8 August 1966 5.3

Measurement of Specific Acoustic Impedance*

JOSEPH WOOD ROGERS

Bucknell University, Lewisburg, Pennsylvania 17837

The theory of an experimental procedure that can be used to measure specific acoustic impedance is pre- sented in this paper. No moving parts are required. Only two measurements need be made. A sequence of simple calculations is presented for reduction of these data.

HE theory of a simple experimental procedure for the measurement of specific acoustic impedance

is given in this paper. There are no moving parts re- quired, nor is there any need to calibrate the transfer coefficients of a microphone and coupling either. The only equipment required consists of a wave analyzer, an electronic counter, a source (usually a headphone), two microphones with amplifiers, a piece of transmission tube, and a means of coupling the microphones to the tube by way of small holes in the side of the tube.

The success of the measurement depends on the tube. It must have uniform cross section. The exact center of

the tube along its axial dimension is determined and two small holes are drilled exactly the same distance either side of the center. Number 60 drill holes are

adequate. The microphones are coupled to the tube at these points and detect the pressures there. The sample whose specific acoustic impedance is to be found is affixed to one end of the tube and a source whose fre-

quency can be controlled is attached to the other. The source is adjusted until a relative minimum in

the pressure standing-wave pattern is located at the microphone closer to the termination. The two micro- phones' voltages are recorded and their ratio is deter- mined. Then the source and termination are exchanged and the process is repeated. The product of the two voltage ratios is exactly the same as the square of the pressure ratio. From this information and the knowledge of the distances that the holes are from the termination, we can calculate the specific acoustic impedance of the

* This work is a part oi the author's dissertation submitted for partial fulfillment of the degree of Doctor of Philosophy in Elec- trical Engineering at The University of Michigan. The author was supported by a Cooperative Graduate Fellowship granted by the National Science Foundation. The author thanks Professor R. K.

Brown, chairman of the dissertation committee, for his help in this work.

sample. The following discussion verifies this statement and also gives formulas for the calculations.

We first define two kinds of transfer coefficients that

are independent of intensities. The first kind relates the source velocity with the pressure at either microphone, whereas, the second relates the pressure at the micro- phones with their output voltages. Both kinds are constant for linear systems.

Let z• be the ratio of the pressure at the microphone closer to the termination to the velocity of the source and let zs be the ratio of pressure at the other micro- phone to the velocity of the source. We note that, when the tests are made, the microphone closer to the termination in the first test becomes the microphone further from the termination after the source and

termination are exchanged. z• and z2 do not change since they are determined by the termination alone in a linear system. They are not microphone properties; they depend only on the locations of the microphones. Let t• be the ratio of the output voltage to the pressure of the first microphone and ts be the same for the second microphone. For convenience, the first microphone will be the one closer to the termination for the first test. t•

and ts are properties of the microphones, their couplings, and amplifiers; they are not dependent on the locations of the microphones. Let u• and us be the respective source velocities during the first and second tests. Let en and e•s be the output voltage of the first microphone during the first and second tests, respectively, and let es• and ess be the same voltages for the second micro- phone.

The ratio of the pressure at the first microphone to that at the second is given by

The Journal of the Acoustical Society of America 1431

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j. w. ROGERS

This ratio is independent of the source velocity (as it would be expected to be).

We next form (½n/½sl)' (½ss/½1s), which is the product of the ratios of output voltages for the first and second tests:

e•ess Ulz•t•usz•ts (z•) s esters u•zstsuszsl• x /' (2) We observe that this ratio is also independent of the source velocity and that the transfer coefficients of the microphones do not affect it. For experimental purposes, then, how the microphones are coupled to the acoustic tube or how the signal is amplified (as long as linearity is maintained) has no effect on the tests. In other words, the shape of the block coupling the microphone to the tube is unimportant. The important result is that this product of ratios is the square of the pressure ratio R,

R s= eness/es•e•s. (3)

Therefore, the knowledge of four voltages is sufficient to find the pressure ratio. This ratio is not the usual pressure standing-wave ratio unless the two micro- phones are located exactly an odd number of quarter- wavelengths apart. This situation would be unusual.

The next problem is how to use this ratio. We may assume that the distance from the termination to the

first microphone is d. For all z greater than d, we could consider that an equivalent termination is located at d having reflection coefficient kd. The reflection coefficient k of the actual termination must then be related to k• by the following equation in which -• is the complex propagation constant of the tube:

k=kae s*a (4)

Since at the point z= d there is a relative minimum in the pressure standing-wave pattern, ka is a negative real number. This fact is important and is used later. The complex pressure at any point z greater than d is given by

p= Poe•(z-a)-+- kae-•(•-a), (5)

in which P0 can be considered as a scaling factor de- pendent on the source velocity. If the distance between the microphones is s, then the complex pressure at the second microphone is given by

Ps= Po(e•'•-i-kae-•'•). (6)

We can relate the ratio of pressures at the two micro-

phones to the ratio R, which is measureable'

R= [ Po(e*•+kae-*•)/Po(l+ka) l . (7)

We now make use of the fact that ka is real. By expanding e *• and e-* • (for •=a-t-il•) and by letting be the ratio (1-ka)/(l+ka), we obtain

R•= N•(cosh•as sinSBs+sinh•'as cos•'/•s) h- 2N coshas sinhas

+ (coshSas cosS/•s + sinhSas sinS/•s). (8)

This equation is a quadratic in N, which can be easily solved. Every term in this equation can be obtained from experiment. In a lossless tube, N would be the pressure standing-wave ratio. In many cases, very little error is introduced by assuming that the tube is lossless. Next, ka is computed from the equation

ka= (1--N)/(1-i-N), (9)

in which N is the positive solution to Eq. 8. The other solution has no significance. Then k can be calculated from Eq. 4. Finally, the specific acoustic impedance of the real termination is found from the following equation:

z=

While this procedure can be used at least in theory for every case except when a is zero and/•s is an integral multiple of ,r, it should not be used unless the reflection coefficient is fairly large (magnitude at least 0.5 and preferably higher). Difficulty is encountered if the holes are not drilled the correct distance apart. The distances depend on the frequencies at which the tests are to be run and on some knowledge of the termination. The whole process can become a trial and error procedure and, hence, quite expensive unless there is a general idea of the specific acoustic impedance to be measured.

This method has been used to measure the reflection

coefficient of an unflanged circular pipe terminated in the open air. The results have agreed quite well with the values expected theoretically • for reflection coefficients whose magnitudes exceeded 0.6. This experiment and some of the difficulties encountered in its performance are discussed elsewhere?

• H. Levine and J. Schwinger, "On the Radiation of Sound from an Unflanged Circular Pipe," Phys. Rev. 73, 383-406 (1948).

•' J. W. Rogers, "The Effects of Slotting Organ Pipes," PhD dissertation, Univ. Mich. (1965) (Univ. Microfilms, Ann Arbor, Mich.), pp. 122-123.

1432 Volume 40 Number 6 1966

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