Acoustic radiation patterns are measured over a 320 deg arc for a point source in a finite length, hard walled, unflanged circular duct. The measured results are compared with computed results which are based on the Wiener-Hopf solution for radiation from a semi-infinite unflanged duct. Measurements and computations are presented for frequencies slightly below and slightly above each of the first four higher order radial mode cutoff frequencies. It is found that the computed and measured patterns show better agreement below the mode cut-off frequencies than above and that the agreement is better at lower frequencies that at higher frequencies. The computed radiation  patterns do not show fine lobes which are caused by diffraction from the back end of the duct

Holm, R. G.

English

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N A S A  TECHNICAL NOTE NASA TN D-7360 
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ACOUSTIC RADIATION PATTERNS 
FOR A SOURCE IN A HARD-WALLED 
UNFLANGED CIRCULAR DUCT 
by Raymond G. Holm and WiZZiam E. Zornmski 
Langley Research Center 
Humpton, Va, 23 665 
N A T I O N A L  AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  @ W A S H I N G T O N ,  D. C. 0 NOVEMBER 1973 
https://ntrs.nasa.gov/search.jsp?R=19740003410 2020-03-23T13:59:21+00:00Z
A SOURCE IN 
A HARD-WALLED UNFLANGED CIRCULAR DUCT 
Raymond G. Holm and William E. Zorumski 
501-24-01-01 
NASA Langley Research Center 
Hampton, Va. 23665 
Technical Note 
National Aeronautics and Space Administration 
Washington, D. C. 20546 
Acoustic radiation patterns a r e  measured over a 320' a r c  for a point source in a 
finite-length, hard-walled, unflanged circular duct. The measured results a r e  compared 
with computed results which a r e  based on the Wiener-Hopf solution for radiation from a 
semi-infinite unflanged duct. Measurements and computations a r e  presented for frequen- 
cies slightly below and slightly above each of the first four higher-order radial mode cut- 
off frequencies. It is found that the computed and measured patterns show better agree- 
ment below the mode cut-off frequencies than above and that the agreement is better at 
lower frequencies than at higher frequencies. The computed radiation patterns do not 
show fine lobes which a r e  caused by diffraction from the back end of the duct. 
Acoustic radiation Unclassified - Unlimited 
Duct radiation 
I 
For sale by the National Technical Information Service, Springfield, Virginia 22151 
ACOUSTIC RADIATION PATTERNS FOR A SOURCE IN A 
HARD-WALLED UNFLANGED CIRCULAR DUCT 
By Raymond G. Holm and William E. Zorumski 
Langley Research Center 
SUMMARY 
Acoustic radiation patterns a r e  measured for a source in a hard-walled unflanged 
circular duct. The duct is of the form of a short barre l  (length = diameter) with a point 
source of sound in the center of its closed end. Acoustic radiation patterns a r e  measured 
in the angular range 00 to *160° from the duct axis a t  frequencies just above and just below 
the cut-off frequencies of the f irs t  four higher-order radial duct modes. The measured 
radiation patterns a r e  compared with computed patterns which a r e  based on the Wiener- 
Hopf solution for  radiation from a semi-infinite unflanged duct. Measurements a r e  also 
made of the far-field sound pressure level a s  a function of frequency a t  several angles. 
The measurements of sound pressure level a s  a function of frequency show sudden 
changes of up to 20 dB a t  the duct mode cut-off frequencies. The measured radiation 
patterns above and below the cut-off frequencies a r e  also significantly different. The 
computed radiation patterns a r e  in fa i r  agreement with the measured patterns. In general, 
the agreement is better below the mode cut-off frequencies than above and is better a t  
lower frequencies than a t  higher frequencies. The measured radiation patterns show many 
fine lobes which a r e  caused by diffraction from the back end of the duct, whereas the com- 
puted patterns do not have lobes in this angular range. 
'. 
INTRODUCTION 
The radiation of sound from a circular waveguide is an acoustical problem which 
must be considered in a number of practical situations. Two mathematical models have 
been used in the study of this problem. These models a r e  (1) the duct with an infinite 
rigid flange and (2) the unflanged duct. 
Morse (ref. 1) has derived equations for  the radiation impedance of a hard-walled 
circular duct with an infinite rigid flange and has made a short table of these radiation 
impedances. Morse shows how the method of successive approximations may be used, 
once the mode radiation impedances a r e  known, to calculate radiation from the flanged 
circular duct. In reference 2, Morse's results a r e  extended to the case of annular ducts 
with acoustically treated walls and it is shown that the radiation impedances may be used 
(in a se t  of linear equations) to find the sound field inside the duct (due to a source inside 
the duct) and the radiated sound field. 
The mathematical problem of radiation from a semi-infinite unflanged circular duct 
is a mixed boundary-value problem which can be solved by the Wiener-Hopf technique 
which is discussed in reference 3. This solution for  radiation from an unflanged circular 
duct with higher-order radial and circumferential acoustic modes is given in reference 4. 
The purpose of this paper is to compare measured radiation patterns with theo- 
retical patterns for  an unflanged circular duct. A point acoustic source is used inside 
the duct since it is easy to represent this source mathematically and to create it 
experimentally. 
The method of determining the sound field for  this point source is given in refer -  
ence 2 for  the case of a flanged duct and the same analytical procedure is valid for the 
unflanged duct if unflanged duct reflection coefficients and radiation equations a r e  used. 
The unflanged duct reflection coefficients and radiation directivity factors used in calcu - 
lations in the present paper were obtained from reference 5. 
APPARATUS AND PROCEDURES 
A schematic illustration of the experiment is shown in figure 1. A source of sound 
originates inside the duct at the closed end, is transmitted to the front of the duct, and is 
then radiated outward where it is measured by the movable microphone. The boom micro- 
phone moves on an a r c  about the center of the duct opening and traverses a total angle of 
320°. 
The duct and i t s  mathematical model a r e  shown in figure 2. The duct (fig. 2(a)) is 
made of a double-walled reinforced steel  shell which is packed with fiber glass to elim- 
inate structural radiation. One end of the duct is closed with a heavy aluminum plate. A 
speaker is threaded into a small hole in the center of this plate to provide a point source 
of sound. The mathematical model of the duct (fig. 2(b)) is a semi-infinite circular duct 
with a rigid plate which is recessed a distance of 38.1 cm from the open end of the duct. 
The duct assembly was mounted in an anechoic chamber with the assembly center 
line about 2 m above the floor grating so  that grating reflections would be minimized. The 
temperature in the chamber was recorded so that the local sound speed could be calculated. 
A knowledge of this sound speed permitted selecting theoretically pertinent operating f re -  
quencies near the duct cut-off frequencies. Once a frequency was selected, the rotating 
boom was swept from o0 to +160° and then from o0 to -160' a t  a rate of 1 .2g0/sec. An 
attempt was made to synchronize the boom sweep rate with the sweep rate on a graphic 
level recorder used to polar-plot the output of the sweeping microphone. The microphone 
signal was passed through a 50-Hz constant-bandwidth filter centered on the frequency 
recorded by the digital frequency meter.  The microphone output was monitored on an 
oscilloscope to insure that the speaker was not being overdriven. Also, a t  specific polar 
angles (i.e., 00, 200, 40°, etc.), linear records were obtained of the sound-pressure-level 
(SPL) variation a s  the beat oscillator frequency was swept from 1000 Hz to 6000 Hz. 
These sweeps were made to demonstrate the sudden variations of radiated sound near the 
duct cut-off frequencies. 
RESULTS AND DISCUSSION 
The duct radial mode cut-off frequencies in t e rms  of the product of wave number k 
times duct radius b a r e  given in the following table: 
These cut-off frequencies a r e  calculated from the hard-walled duct boundary condition 
that the acoustic velocity normal to the duct wall is zero. This condition is expressed 
mathematically by the equation ~ b ( k b )  = 0, that is, the derivative of the Bessel function 
of the first kind of order  zero is zero. The values in this table a r e  based on a sound 
speed of 347.27 m/sec and a temperature of 26.90 C. 
Mode 
0 
1 
2 
3 
4 
The directivity results obtained a r e  presented in figures 3 to 1.0. The dotted theo- 
retical curves accompany the experimental results with tick marks exterior to the polar 
grids to indicate the experimental 20° angular increments. Since the boom microphone 
and chart sweep rates were not precisely synchronized, there is distortion in the experi- 
mental angular scale. 
The calculated sound pressure levels were matched to experimentally measured 
SPL's on axis (0°) in figures 3 to 10 by shifting the calculated values. This procedure 
was used because the actual sound power could not be controlled. The actual acoustic 
power radiated by the speaker could be determined analytically only from an extensive 
study of the speaker's dynamics, which is beyond the scope of the present paper. 
kb 
0 
3.83 
7.02 
10.17 
13.32 
A comparison of figures 3 and 4 shows that the radiation directivity pattern changes 
from below to above the cut-off frequency of the first higher-order radial duct mode. 
Cut-off 
frequency, 
Hz 
0 
1111 
2037 
2951 
3865 
Below the cut-off frequency, only the zeroth-order mode (plane wave) is propagating with 
the higher-order modes being exponentially attenuated. Above the cut-off frequency, the 
first higher-order mode propagates and, as predicted, dramatically enhances the axial 
and side-line sound radiation. Comparing figures 5 and 6, figures 7 and 8, and figures 9 
and 10 shows similar features for the second, third, and fourth higher-order radial modes. 
In general, the agreement is better below the cut-off frequencies than above and better a t  
lower frequencies than at  higher frequencies. Many small lobes, which the analysis does 
not predict, a re  measured behind the plane of the duct opening. These small lobes a re  
due to diffraction about the source end of the duct. 
The sound-pressure-level variations at  specific polar angles, as the speaker fre-  
quency was swept from 1000 Hz to 6000 Hz while maintaining the voltage to the speaker 
constant, a r e  presented in figure 11. These graphs, particularly at  sweeps of 20' and 80°, 
show a sudden increase in radiated sound pressure level as the frequency passes through a 
specific duct mode cut-off frequency. These results agree with the theoretical prediction 
that the power radiated from a constant volumetric strength source is infinite just above 
the cut-off frequencies of an infinite hard-walled duct. 
CONCLUDING REMARKS 
Acoustic radiation patterns have been measured and calculated for a source in a 
hard-walled circular duct without a flange. The measurements and calculations a r e  in 
fa i r  agreement; the agreement is better at lower frequencies. Theoretical predictions 
of sudden increases in side-line radiation a s  the frequency increases through a cut-off 
frequency a re  confirmed. The measured radiation patterns show a number of fine lobes 
behind the plane of the duct opening which a r e  not predicted by theory. These lobes a re  
caused by diffraction from the back end of the duct. 
Langley Research Center, 
National Aeronautics and Space Administration, 
Hampton, Va., September 5, 1973. 
REFE rnNCES 
1. Morse, Philip M. : Vibration and Sound. Second ed., McGraw -Hill Book Co., Inc., 1948. 
2. Zorumski, William E . : Generalized Radiation Impedances and Reflection Coefficients 
of Circular and Annular Ducts. J. Acoust. Soc. Amer., vol. 54, no. 6, Dec. 1973. 
3. Noble, B. : Methods Based on the Wiener-Hopf Technique for the Solution of Partial 
Differential Equations. Pergamon Press ,  Inc., c. 1958. 
4. Vajnshtejn, L. A. (J. Shmoys, transl.): The Theory of Sound Waves in Open Tubes. 
Propagation in Semi-Infinite Waveguides, Res. Rep. No. EM-63 (Contract 
No. AF-19(122)-42), Inst. Math. Sci., New York Univ., Jan. 1954, pp. 87-116. 
5. Lansing, D. L.; Drischler, J. A.; and Pusey, C. G.: Radiation of Sound From an 
Unflanged Circular Duct With Flow. NASA paper presented a t  the 79th Meeting 
of the Acoustical Society of America (Atlantic City, N.J.), Apr. 21-24, 1970. 
Figure 1. - Experiment schematic plan view. 
(a) Experimental duct and speaker assembly. 
I & 1.27 cm Aluminum P l a t e  
In f in i t es imal ly  Thin Rigid Wall t o  - 
alp[ I 
E! €I 
0 0 
a d  
&G 
Ln M 
Rigid P l a t e  I I 
0 
Point Acoustic 
lCIl 
Source 
/ / / / / / / / I /  / A 
Fiber-Glass Packing 
/ 1) / 1 1 1  1 / 1 
\speaker  lush ~ o u n t e d )  
(b) Mathematical model of duct with acoustic point source. 
Figure 2. - Experimental duct and mathematical model. 
Experiment 
......-... Theory 
Figure 3. - Radiation patterns a t  1082.5 Hz. 
Figure 4. - Radiation patterns a t  1133.1 Hz. 
Figure 5. - Radiation patterns at 1998.2 Hz. 
Experiment 
. . . .. . . . . - . Theory 
Figure 6. - Radiation patterns a t  2058.9 Hz. 
Figure 7 .  - Radiation patterns at 2883.1 Hz. 
I \ 
Experiment 
........ ... Theory 
Figure 8.- Ftadiation patterns at  2977.8 Hz. 
Experiment 
.- - - - Theory 
Figure 9. - Radiation patterns at 3776.4 Hz. 
Experiment 
e m e s e e e - o a e  Theory 
Figure 10.- Radiation patterns a t  3897.3 Hz. 





Acoustic radiation patterns for a source in a hard-walled unflanged circular duct

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