Absorption Coefficients and Impedance

In this laboratory exercise you will measure the absorption coefficients and acoustic impedance of several acoustic samples using the Brüel & Kjær Standing Wave Apparatus Type 4002, along with the B&K Real-Time Frequency Analyzer Type 2133 and the B&K Frequency Analyzer Type 2107. Absorbing materials play an important role in architectural acoustics, the design of recording studios and listening rooms, and automobile interiors (seat material is responsible for almost 50% of sound absorption inside an automobile).

The reverberation time of a room is a very important acoustic quantity of concern to both architects and musicians. The theory explaining reverberation time, and the acoustical behavior of large and small rooms, was developed by Sabine. This theory requires a knowledge of the absorbing properties of the materials which cover the surfaces of the room under analysis. The Standing Wave Apparatus Type 4002, one of B&K's first products (almost 50 years ago), was developed primarily to measure the sound absorbing properties of building materials, and is still widely used today. The standing wave tube (also called an impedance tube) method allows one to make quick and easy, yet perfectly reproducible, measurements of absorption coefficients. The impedance tube also allows for accurate measurement of the normally incident acoustic impedance, and requires only small samples of the absorbing material.

The B&K Standing Wave Apparatus is shown in the figure below. A loudspeaker produces an acoustic wave which travels down the pipe and reflects from the test sample. The phase interference between the waves in the pipe which are incident upon and reflected from the test sample will result in the formation of a standing wave pattern in the pipe. If 100% of the incident wave is reflected, then the incident and reflected waves have the same amplitude; the nodes in the pipe have zero pressure and the antinodes have double the pressure. If some of the incident sound energy is absorbed by the sample, then the incident and reflected waves have different amplitudes; the nodes in the pipe no longer have zero pressure. The pressure amplitudes at nodes and antinodes are measured with a microphone probe attached to a car which slides along a graduated ruler. The ratio of the pressure maximum (antinode) to the pressure minimum (node) is called the standing wave ratio SWR. This ratio, which always has a value greater than or equal to unity, is used to determine the sample's reflection coefficient amplitude R, its absorption coefficient , and its impedance Z.

An alternate method for determining the absorbing properties of a material, also in wide use today, involves placing a unit-area piece of material (say one meter squared) in a special reverberation room. The difference in the reverberation time with and without the material yields the absorbing properties of the material. This method is generally more expensive, requiring precisely calibrated sensors and a specially designed reverberation chamber, and is much less convenient. Although this method does not yield normally incident acoustic impedance data, it is superior for measuring absorption characteristics for randomly incident sound waves, and is preferable for determination of absorbing properties that depend on the size of the material.

On a historical note, water-filled impedance tubes became critically important to the development of absorptive coating for submarines during World War II. Such coatings, developed first by Germany, were used (and still are) to protect submarines form sonar detection.

II. Theory

Assume that a pipe of cross-sectional area S and length L is driven by a piston at x=0. The pipe is terminated at x=L by a mechanical impedance Z_mL. If the piston vibrates harmonically at a frequency sufficiently low that only plane waves propagate [NOTE: For a circular waveguide (pipe) filled with air, the highest frequency at which only plane waves will propagate is given by f_max ≈101 / a where a is the radius of the waveguide. For the 10 cm diameter pipe used in this experiment f_max≈2000Hz.] then the pressure wave in the pipe will be of the form

(1)

where A and B are determined by the boundary conditions at x=0 and x=L. Using Euler's Equation , one may obtain the particle velocity in the pipe,

(2)

The acoustic impedance of the plane waves in the pipe may be expressed as

(3)

The mechanical impedance load at x=L may be written in terms of this acoustic impedance as

(4)

If we choose to write

(5)

then

(6)

Thus, given the ratio of incident to reflected amplitudes, and the phase angle theta, the mechanical impedance of the sample may be determined. Substitution of (5) into (1) and solving for the pressure amplitude P=| p| of the wave, one obtains

(7)

This pressure amplitude is shown below. The figure on the left shows the pressure amplitude in the pipe with a rigid termination at $x=L$. All of the sound energy incident upon the termination is reflected with the same amplitude. However, there may be some absorption along the walls as the waves travel back and forth along the pipe. The figure on the right represents the case when the pipe is terminated at $x=L$ with some acoustic absorbing material. Now some of the incident sound energy is absorbed by the material so that the reflected waves do not have the same amplitude as incident waves. In addition the absorbing material introduces a phase shift upon reflection. The amplitude at a pressure antinode (maximum pressure) is A+B, and the amplitude at a pressure node (minimum pressure) is A-B. It is not possible to measure A or B directly. However, we can measure A+B and A-B using the standing wave tube.

We define the ratio of pressure maximum to pressure minimum as the standing wave ratio

(8)

which may be arranged to provide the sound power reflection coefficient

(9)

A pressure minimum occurs when

(10)

which requires that

(11)

or

(12)

where the quantity (L-x) equals the distance from the test sample to the first pressure minimum (n=1) as shown in the figure. The B&K Standing Wave Tube Apparatus allows for measurement of the maximum pressure, (A+B), the minimum pressure, (A-B), and the distance from the sample of the first minimum, (L-x).

The complex mechanical impedance of the test sample is then obtained by substituting (12) and (9) into (6). The mechanical impedance of the test sample may be a complicated function of frequency, and it may be necessary to repeat the above measurements over the range of frequencies of interest. When a large number of frequencies are measured, it is inconvenient and unnecessary to carry out all the calculations as outlined above. Instead, use of a Smith calculator, (see reference [5]) a nomographic chart, enables rapid determination of the real and imaginary parts of the impedance directly from measurements of the standing wave ratio and the position of the node nearest the sample.

The sound power absorption coefficient for the test sample at a given frequency is given by

(13)

As was the case for the impedance, the absorption coefficient may be a function of frequency, and measurements over the frequency range of interest may be required.

Example Calculation

(13)

and

(13)

which means that 89% of the incident sound power is absorbed by the sample.

III. Equipment

• Brüel & Kjær Standing Wave Apparatus Type 4002
• B&K Real-Time Frequency Analyzer Type 2133 (or equivalent 1/3 Octave Band Analyzer)
• B&K Frequency Analyzer Type 2107.
• Function/Waveform Generator (Hewlet Packard 33120 )
• Audio amplifier
• Several samples of acoustic absorbing material

• Place a sample of acoustic absorbing material in the cap of the impedance tube, and clamp the cap down tight. Be sure to record the type of absorbing material.
• Bring up the B&K Real-Time Frequency Analyzer Type 2133 to its default state.
• Set the Bandwidth to 1/3 Octave from 200 Hz to 2.5 kHz. Set the Averaging to hfilbreak Exp T:1/4s tau: 1/8s Fast.
• Set the Channel A input to: Ch. A.Direct.
• Connect the microphone jack to the Direct input on the Frequency Analyzer. You will have to use alligator clips and banana plugs to make this connection. Connect the HP function generator through the amplifier to the loudspeaker. Turn on the function generator and set the amplifier volume to the specified level.
• Press the [Input Autorange] key on the Frequency Analyzer keyboard to autorange the input signal. You may have to do this again, whenever the analyzer indicates an overload in channel A.
• Set the function generator to produce a 200 Hz sine wave.
• Press the [Start] key on the frequency analyzer to begin taking a measurement. You may have to press the [Auto Scale] key to see the data.
• Set the cursor on the Frequency Analyzer to the 200 Hz band. Move the microphone car until the cursor reading indicates a maximum level for the 200 Hz band. Record the maximum level (in dB) for the 200 Hz band.
• Move the microphone car until the cursor reading indicates a minimum level for the 200 Hz band. Record the minimum level (in dB) for the 200 Hz band.
• Locate and record (using the ruled scale under the microphone care) the position of the first minimum from the sample end of the tube.
• Return to step ¤ and repeat for the next 1/3 octave band (250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500 Hz).

• Calculate the maximum and minimum rms pressures from the recorded decibel levels using

  	LeveldB = 20 log prms          and         prms =10(LeveldB / 20) 

• Calculate the standing wave ratio SWR using

  SWR = p_rms,max / p_rms,min

• Calculate the sound power reflection coefficient (B / A) using (9).
• Calculate the sound power absorption coefficient α using (13).
• Plot the sound power absorption coefficient versus frequency.
• Calculate the phase θ from (12) with n=1 and with the quantity (L-x) equal to the position of the first minimum.
• Calculate the complex mechanical impedance Z_mL of the test sample using (6) or a Smith Chart. (Use ρ = 1.21 kg/m³ and c=343m/s for air.)

The B&K Frequency Analyzer Type 2107 is designed especially for use with the Standing Wave Apparatus Type 4002, so that the sound power absorption coefficients may be measured directly without calculation.

• Keep the same test sample in the impedance tube.
• Disconnect the microphone from the B\&K Real-Time Frequency Analyzer Type 2133 and turn the analyzer off.
• Connect the microphone to the "Amplifier Input" jack on the Frequency Analyzer Type 2107.
• Set the function generator to produce a 125 Hz sine wave.
• Using the "Frequency Tuning" dial and the "Frequency Range" knob, set the frequency of the analyzer to 125 Hz.
• Set the "Input Potentiometer" knob to about 5, and adjust the "Meter Range" and "Range Multiplier" knobs until the needle on the "Indicating Meter" shows a response somewhere in the middle of the scale.
• While watching the needle, move the microphone car until you find a pressure maximum (don't worry about the exact value indicated by the needle on the scale).
• Adjust the "Input Potentiometer" knob so that the needle reads 100\% on the top line of the bottom scale.
• Move the microphone car until you find the pressure minimum closest to the sample.
• ⇒ Read off the percent from the top line of the bottom scale as indicated by the needle. This percent (converted to a decimal: 86\%$\rightarrow$0.86) is the sound power absorption coefficient.
• ⇒ For this sample, compare these measured values of the sound power absorption coefficient to the calculations previously obtained. Except for the lowest frequency band (125 Hz), the measured and calculated values should agree closely.
• ⇒ Return to step ¤ and repeat for the 250 Hz, 500 Hz, 1000 Hz and 2000 Hz octave bands..
• Repeat for TWO additional samples. Be sure to record a description of the samples you are measuring.

• Compare calculated and measured values for the absorption coefficient of your first sample.
• Plot the absorption coefficient as a function of frequency for each of the three different samples. You can plot results of all three measurements on the same graph for comparison.
• Since the standing wave tube measures the absorption coefficient only for normal incidence, the measured coefficients are generally somewhat smaller than those measured using the reverberation room method. Using the curve(s) below, correct your measured values for α to the statistical values α_st which would apply for random incidence when placed on a wall in a room.
• Some suggestions for analysis of your results: How does the absorption vary with frequency? Which samples provide the greatest amount of absorption? Over what frequency ranges are the samples most effective?

1. "Instructions and Applications" for Standing Wave Apparatus Type 4002 and Frequency Analyzer Type 2107, (Brüel & Kjær, 1967).

2. Reynolds, Engineering Principles of Acoustics: Noise and Vibration Control, (Allyn & Bacon, 1981).

3. Kinsler, Frey, Coppens, and Sanders, Fundamentals of Acoustics, Third Edition, (John Wiley & Sons, 1982).

4. Bies and Hansen, Engineering Noise Control, Second Edition, (Chapman & Hall, 1996).

5. Elmore and Heald, Physics of Waves, Appendix B, ( Dover reprint, 1985).