Acoustic High-Pass, Low-Pass, and Band-Stop Filters

Many interesting problems in acoustics involve the propagation of sound in ducts. Furthermore, many applications of acoustics to ventilation / exhaust systems in buildings and automobiles use mufflers and acoustic filters to reduce the level of noise propagating down a duct or radiating from the end of a duct. In this laboratory exercise you will explore the behavior of acoustic waves in a duct with changes in cross-sectional area, side branches, and resonators. In the process you will observe the behavior of acoustic low-pass, high-pass, and band-stop filters as they are used in a duct system.

II. The Theory of Acoustic Transmission Lines

A rather complete theoretical development of acoustic waveguides, transmission lines, and filters may be found in Chapters 9 & 10 of Fundamentals of Acoustics, 3rd Ed., Kinsler, Frey, Coppens, and Sanders, (J. Wiley & Sons, 1982).

A. Waveguides and transmission lines

A waveguide is a structure which forces wave propagation along a path parallel to its longest dimension. Acoustic wavequides are structures with constant cross-sectional area and shape. Simple examples of such structures include hoses, tubes, and pipes, referred to hereafter as ducts. If a duct is excited by a pressure disturbance with a wavelength larger than twice the duct's largest cross-sectional dimension, then only plane waves will propagate down the duct. For a circular duct containing air at room temperature, the highest frequency at which only plane waves will propagate is given by f = 100/a where a is the radius of the duct cross-section. Once plane waves are generated inside the duct, they will propagate down the duct, even if the duct has bends or turns in it. A propagating plane wave may encounter a change in the acoustic impedance of the duct when the duct (i) opens into free space, (ii) is connected to another section of duct with a different cross-section, (iii) branches off into two ducts, or (iv) is terminated in some other way. This impedance change causes partial reflection and partial transmission of the incident plane waves.

Assume that a duct of cross-sectional area S and length L is driven by a piston at x=0. The pipe is terminated at x=L by an acoustic impedance Z_L. The input acoustic impedance as seen by the driver (looking into the duct at x=0) may be written in terms of the terminal impedance,

(1)

Equation (1) is called a transmission line equation; similar equations are used for electromagnetic waveguides (transmission lines), as well as for comparing the input mechanical impedance to the termination impedance for transverse waves on a string or longitudinal waves in a beam.

B. Duct driven at x=0 and open at x=L

Let the duct be driven by a rigid piston at x=0 and open-ended at x=L. At first guess one might think that the termination impedance for an open end would be Z_L =0, which would reduce Eq. (1) to Z₀ = (p c/S) i tan (kL) resulting in resonance frequencies occurring at f_n = nc/2L, for n=1,2,3, . . . This is the assumption made in most elementary physics books; it is not, however, correct. The boundary condition is not zero at the open end, because the open end of the duct radiates sound into the surrounding medium. The proper value for the terminating impedance is then Z_L= Z_r where Z_r is the radiation impedance of the open end of the pipe. The radiation impedance is complex; the real part (radiation resistance) represents the energy radiated away from the open end in the form of sound waves, and the imaginary part (radiation reactance) represents the mass loading of the air just outside the open end. For unflanged and flanged open ends, the radiation impedance is

	unflanged;	(2a)
	flanged.	(2b)

The input acoustic impedance for an unflanged, open-ended duct may be obtained by substituting Eq. (2) into the transmission equation line equation Eq. (1). Resonance occurs when the input impedance becomes a minimum; or when 1/Z₀ becomes a maximum. Figure 1 shows the input acoustic impedance calculated from Eqs. (1) and (2a) for a duct of length L=1.1 m and radius a= 2.0 cm. The first two resonances occur at approximately f₁ = 170 Hz and f₂=340 Hz.

Figure 1: Input admittance (inverse of input impedance) for an unflanged, open-ended duct of length L=1.01 m and radius a= 2.0 cm.

The resonance frequencies for an unflanged, open duct may be approximated by

(3)

where n=1,2,3, . . . , c is the speed of sound (343 m/s for air), and the length of the duct includes an "end correction" for the open end.

III. Acoustic Filters

An acoustic low-pass filter may be constructed by inserting an expansion chamber in the duct. An expansion chamber serves as a simple model of a muffler, and also has applications in architectural acoustics (the plenum chamber in a building's HVAC system is an expansion chamber). Keeping track of all incident and reflected waves from both junctions, one can derive the sound power transmission coefficient as

(5)

Figure 3(a) shows this coefficient for an expansion chamber with dimensions similar to what you will use in this lab. Notice that there are frequencies at which all incident power passes right through the filter, and there are other frequencies where a minimum of power is allowed to pass by. Frequencies for complete transmission correspond to standing waves being set up within the muffler chamber.

In a low frequency limit (kl << 1) the expansion chamber may be treated as a side branch of acoustic compliance C = V/ p c² where the volume V = S₂ L₂. In this low frequency limit, the side branch approximation of the sound power transmission coefficient in Eq. (5) becomes

(6)

This low frequency approximation is shown in Figure 3(b). The expansion chamber appears to pass low frequencies, and block high frequencies. Thus, it is called a "low-pass" filter.

Figure 3: (a) Sound power transmission coefficient for an expansion chamber; (b) low frequency approximation acting as a low-pass filter.

An acoustic high-pass filter may be constructed simply by inserting a "T" junction, or a short side branch into the duct. If both the radius and the length of the side branch are smaller than a wavelengths of the plane waves in the duct then the acoustic impedance of the side branch opening becomes The first term represents the radiation of sound from the side branch opening, and the second term represents mass loading of the fluid at the side branch opening. For low frequencies the sound power transmission coefficient (which describes how much sound energy makes it past the filter) may be derived as,

(4)

The transmitted power is shown in Figure 2 for a side branch you may encounter in this lab. This acoustic filter blocks low frequencies and passes high frequencies, thus it is called a "high-pass" filter. It is important to realize that low frequency energy is not radiated out of the side branch, but is reflected back towards the source.

Figure 2: Sound power transmission coefficient for a side branch in the low frequency approximation of a high-pass filter.

If a cavity is attached to the side branch, as shown at right, then the side branch has both mass (inertia) and compliance. Such an acoustic system is called a Helmholtz resonator --- it behaves very much like a simple mass-spring system (have you ever blown into a pop bottle to make a sound?). This resonator has a neck with radius a and area S_b, an effective neck length of L_eff = L + 1.7a, and a cavity volume V. This cavity resonates at a frequency

(7)

and in the process of resonating it absorbs energy at this frequency. All the energy absorbed by the resonator during one part of the acoustic cycle is returned to the pipe later in the cycle. The phase relationship is such that all the absorber energy is returned back towards the source -- it does not get sent on down the duct. Since no energy is removed from the system, just returned, then the real part of the branch impedance R_b = 0. The imaginary part of the impedance may be expressed in terms of the compliance and inertia of the resonator, X_b = p (w L_eff/S_b - c² / wV), so that the sound power transmission coefficient may be written

(8)

The transmitted power is zero when w = w₀ in Eq. (7), which is the resonance frequency of the resonator, whence all energy is reflected back towards the source. This transmitted power is shown in Figure 4 for a Helmholtz resonator tuned to 172 Hz. These filters block any sound within a band around the resonance frequency, and pass all other frequencies. Thus they are called "band-stop" filters. The bandwidth of the filter may be increased by adding some porous absorbing material (steel wool) in the cavity of the resonator.

Figure 4: Sound power transmission for a band-stop filter tuned to 172 Hz.

• PVC pipe:
  • a 1 m length (2-inch diameter) to serve as the unfiltered duct
  • several varying lengths of the 2-inch diameter pipe (instructor can pre-cut these lengths, or have students cut their own pieces)
  • several varying lengths (less than 40 cm) of 3-inch or 4-inch diameter pipe for building expansion chambers and Helmholtz resonators
  • T-joints, couplings between 2-inch, 3-inch, and 4-inch diameter pipe, end caps for all diameter sizes
• loudspeaker (good low-frequency response preferable) with connection to pipe (a horn loudspeaker driver fits rather nicely into a 2-inch diameter pipe)
• Frequency analyzer with averaging, multiplot display, and noise source, and floppy disk drive (Hewlet Packard 35670A)
• Audio amplifier to amplify noise input to speaker.
• Two small diameter (1/4") microphones and necessary preamplifiers

You will need a frequency analyzer to observe the frequency response of the duct as measured by the microphone. Set up the analyzer to display power spectrum of the microphone input over a frequency range of 0-800 Hz. If your analyzer has memory for saving data and the ability to display more than one plot or measurement at a time, you can readily observe the effects of each filter and compare the filtered sound measurement with the unfiltered duct. Save the unfiltered measurement to a data register and set up the analyzer to display the saved measurement and each new filtered measurement simultaneously (two separate displays or two measurements on the same plot). If your analyzer has a floppy disk drive you can save data to disk to plot later using a PC. You will also need a noise source to drive the loudspeaker. Using a noise source from the analyzer has the advantage that the frequency range of the noise source can be set to match the frequency range of analysis. For wave propagation down a duct this will ensure that only plane waves will be generated at the driver end of the duct. The noise source should be white noise (equal energy at all frequencies) or pink noise (equal power in all frequency bands).

B. Measurement of the duct resonances

• Locate the long duct (approx 1 m in length). Be sure to record all pertinent dimensions of the duct.
• {Connect the 1 m duct to the loudspeaker and position the microphone about 1 cm from the center of the open end as shown below.
• Turn the source on and start a measurement. When the measurement is done, save the data to disk for plotting later. In addition, if you save this measurement to a data register you can display this saved data simultaneously with future measurements of the duct with various filters to see how the filters behave.
• Plot this data (or save to floppy disk for later plotting).
• Locate and record the values of any resonance frequencies of the duct. How do these frequencies compare to what you might calculate from Eq. (3)?

• Set up the duct with expansion chamber as shown below, but keep the total length of the duct/filter combination equal to the length of the original duct in subsection B.
• Be sure to record all dimensions for this system.
• Turn the source on and start a measurement. If you have set up the analyzer correctly, you should see this measurement superposed on top of the duct measurement.
• Did the measured response of the duct change? If so how?
• Plot this data (or save to floppy disk for later plotting)..
• Try a different size expansion chamber, but keep the total length of the duct/filter system the same.

• Set up the duct with side branch as shown below, but keep the total length of the duct/filter combination equal to the length of the original duct in subsection B.
• Be sure to record all dimensions for this system.
• Turn the source on and start a measurement. If you have set up the analyzer correctly, you should see this measurement superposed on top of the original duct measurement.
• Did the measured response of the duct change? If so how?
• Plot this data (or save to floppy disk for later plotting)..
• Try changing the length of the side branch and describe what you hear. What length seems to be the best at cutting out low frequencies?

• Set up the duct with Helmholtz resonator as shown below, but keep the total length of the duct/filter combination equal to the length of the original duct in subsection B.
• Be sure to record all dimensions for this system.
• Turn the source on and start a measurement. If you have set up the analyzer correctly, you should see this measurement superposed on top of the original duct measurement.
• Plot this data (or save to floppy disk for later plotting).
• Did the measured response of the duct change? If so how? How does the measured value of the band-stop frequency match a calculated value of the resonance frequency of the Helmholtz resonator?
• Try to find resonators that will knock out each of the first two resonances of the original duct.

• Compare plots of the response of the duct alone to the duct with the various filters. Do the filters behave as the theory predicts? How so? or How not so?
• Comment on how such filters might be used in automobiles, air-conditioning systems, etc.

1. Kinsler, Frey, Coppens, and Sanders, Fundamentals of Acoustics, Third Edition, (John Wiley & Sons, 1982) Chapters 9&10.
2. Reynolds, Engineering Principles of Acoustics: Noise and Vibration Control, (Allyn & Bacon, 1981), Chapter 9.
3. Dowling and Ffowcs Williams, Sound and Sources of Sound, (Ellis Horwood, 1983), Chapter 6.

This laboratory exercise has been used very successfully in a rather popular senior level course, "PHYS-580 / ME-530, Acoustics, Noise, and Vibration," which serves as an elective for Mechanical Engineering, Electrical Engineering, and Applied Physics majors at GMI Engineering & Management Institute. Students really enjoyed putting the duct systems together, and most experimented with "monster" filter networks to see what effect multiple filters have. The data shown below and on the following pages was obtained for a 1 m length of PVC pipe, 2-inch diameter with appropriate T-joints, 3-inch and 4-inch pieces, and necessary connectors. The duct was driven with a horn loudspeaker which fit nicely into one end of the PVC pipe. The general effect of high-pass, low-pass, and band-stop filters is rather clearly demonstrated for this system.

Measured response for duct, length L=1.01 m, radius a=1.95 cm

Resonance frequencies at approximately f₁=170 Hz and f₂=340 Hz

On the following plots, the dotted curve is the unfiltered duct response, shown above.

Duct with expansion chamber --- Low-pass filter

Duct with side branch --- High-pass filter

Duct with Helmholtz resonator --- Band-stop filter

Duct with Helmholtz resonator tuned to f₁=172 Hz