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On the use of acoustical holography to locate sound sources on complex structures
Journal ofSoundand
TO
Vibration (1976) 48(2),
157-168
ON THE USE OF ACOUSTICAL HOLOGRAPHY LOCATE SOUND SOURCES ON COMPLEX STRUCTURES E. E. WATSON? Applied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A. AND
W. F. KING III Deutsche Forschungs- und Versuchsanstalt fiir Luft- und Raumfahrt E. V., Institut fiir Turbulenzforschung, 1 Berlin 12, Germany (Received 5 April 1915, and in revisedform 20 April 1976)
Until recently, it was extremely difficult to detect and locate far-field sound sources on complex structures. An experimental technique is presented in this paper for the identification of such sound sources on complex vibrating structures. Results are given for a preliminary investigation designed to determine the feasibility of applying a modified form of acoustical holography to the location of sound sources on ventilator fans. Such sources are identified on a three-bladed ventilator fan. Optical and computer sound source reconstructions are compared and analyzed. Results of this preliminary study are very encouraging and indicate the versatility and usefulness of this technique for modifying and controlling far-field noise radiation from complex vibrating structures.
1. INTRODUCTION The scientific study of any natural phenomenon involves quantitative measurements of its influence on observables under known circumstances. Until recently, however, it was very difficult to obtain quantitative information from far-field sound sources that would allow identification of their origins. This deficiency was remedied in 1969 by Graham [I], who first applied long wavelength acoustical holography to the detection of radiated sound sources. Graham’s technique was a new application of holography, which had previously been used only for the imaging of objects. Improvements in the technique of holographic sound source location were made shortly thereafter by Watson [2, 31 and Hannon and Watson [4], who have identified far-field sound sources on a variety of vibrating two-dimensional structures, including mass-loaded and ribbed plates. In this paper, we report on our preliminary efforts to use long wavelength acoustical holography to locate and identify radiated sound sources on complex three-dimensional vibrating structures. We investigated the radiated sound field from a three-bladed ventilator fan driven at one of its natural frequencies. To fully exploit the three-dimensional information stored in the holograms, we reconstructed the wavefronts both optically and numerically. Since this use of holographic techniques for sound source location is neither widely appreciated nor understood as being different from the more conventional imaging applications, a brief historical review of acoustical holography is given in section 2. A theoretical development t Present address: Environmental Decatur, Alabama, U.S.A.
Acoustics Division, Universal Oil Products, Wolverine Tube Division, 157
E. E. WATSON
1%
AND W. F. KING 111
appropriate to sound source detection studies is given in section 3. Our experimental results, including both optical and computer reconstructions of the wavefronts, are presented in section 4 together with a discussion of the method.
2. HISTORICAL
REVIEW
OF
HOLOGRAPHY
The field of optical holography began in 1948 when Gabor [5] proposed a new lensless imaging technique to improve the electron microscope. In the late 1940’s, this instrument was plagued by serious problems of spherical aberration. To avoid these distortions, Gabor’s suggestion was to record the electron diffraction pattern of the object and to then use this pattern to reconstruct the image. The essential steps in this method are illustrated in Figure I.
FILM PLATE
PLANE ’ REFERENCE WAVE
Figure
I. The reference and scattered
I/ I
I \
object
/,---. ,,‘\ ‘1’ \
\ \
\\._/j/i! I ,;;
VIRTUAL IMAGE (TRUE
Figu1.e 2. A schematic image\. and observer.
wave in a semi-transparent
,,/
\
I
,I (CONJUGATE
IMAGE)
representation
showing
the relative
positions
IMAGES
of the hologram,
real and
vi!-tual
USE OF ACOUSTICAL HOLOGRAPHY
159
A plane monochromatic wave irradiates a semi-transparent object. The scattered and incident waves combine to form an interference pattern on photographic film. When the developed film is itself irradiated with a plane monochromatic wave, the interference pattern becomes a diffraction grating that scatters the wave to create two wavefields. As indicated in Figure 2, one field converges to produce a real image, while the other diverges and results in a virtual image. This phenomenon of two overlapping images in on-axis holography is called the twin image effect. Gabor never actually succeeded in creating satisfactory images for the electron microscope with this technique. We should emphasize that since this imaging method relies on the interference of two waves, coherent sources of radiation are required for its successful application. Very little progress was made in holography until 1962 when Leith and Upatnicks [6] noticed the similarity between Gabor’s suggestion for lensless imaging and the synthetic aperture antenna. They modified Gabor’s suggestion by irradiating the object from the side to create an off-axis interference pattern. When this pattern or hologram is used to reconstruct the wavefront, the twin images are separated and do not overlap. This modification to the technique, combined with the development of the gas laser, which provides a coherent source of radiation, made possible the dramatic progress in optical holography that took place during the past decade. As we have seen, the development of optical holography was greatly hindered for at least ten years after Gabor’s paper by the lack of coherent sources. No such obstacle stood in the path of acoustical holography, however, since almost all acoustic sources that can be used for imaging are coherent. In spite of this availability of both ideas and equipment, it was not until 1965 that Greguss [7] recorded acoustical holograms on photographic film. Shortly thereafter, Thurstone [7] demonstrated that visible images could be produced from acoustical holograms. Since that time, the major thrust of research in acoustical holography has been towards imaging acoustically opaque objects at frequencies in the MHz region. An important exception is the work of Graham [I] and Watson [2, 31 on the location of far-field sound sources with long wavelength acoustical holography. It is this use of holography that is exploited in the work reported on in this paper. 3. THEORY The detection and location of far-field sound sources requires an experimental procedure that differs somewhat from that used in conventional imaging acoustical holography. This, in turn, leads to modifications in the theoretical analysis of the process. To make clear this difference, we will outline the theory of acoustical holography and indicate where our approach differs from the usual treatment. The following analysis is adapted from an excellent discussion of conventional acoustical holography by Aoki [9]. Consider the three planes shown in Figure 3. Let a discrete frequency sound source located at the intersection of the x1 and y, axes irradiate an object in the x,-y, plane. In this discussion, we will label all acoustic fields by their velocity potentials @*,where the subscript will identify the particular field. The illuminating field @, in conventional acoustical holography passes through the semitransparent structure to be imaged and forms a diffraction field QD, in addition to the background illuminating field. These fields interact with one another to generate an interference pattern that is recorded as a sound hologram in the x,-y, plane. In our experiments to locate far-field sound sources on vibrating structures, we do not actually irradiate the object under investigation with sound waves. Instead, we mix a plane wave with the sound field Gs generated by sound sources on a vibrating structure. As in conventional acoustical holography, this interaction of sound wave fields produces an interference pattern that is detected and recorded as a sound wave hologram in the x,-y, plane. The effect of
E. E. WATSON AND W. F. KING 111
160 XI
FICTITIOUS SOURCE t
Figure 3. Co-ordinate system for the construction
of acoustical
holograms
mixing the plane wave with QS is equivalent to illuminating the structure with a fictitious sound source located at s1 = 0~ (see Figure 3). In our analysis, we will let the distance from the fictitious source to the vibrating structure be very large but finite. The effect of this approximation is to illuminate the structure with a spherical rather than a plane wave. As the source distance s1 becomes very large, however, the practical difference between a spherical and plane wave becomes insignificant and our analysis should not be affected by this choice. With square law detection of the wave fields, the pattern recorded in the x-,-y, plane is (1)
5=(~,+~S)(~,+~S)*=/~,/Z+/~DSi2+f~~~+~,~S*. where the star denotes complex conjugate. By using the paraxial express the illuminating wave from the fictitious source as ~,(x,,?,,j=(A/s,)exp(-ik,s,[l
+(xt+lf.2.$},
ray approximation,
we can
(2)
where (x,,y,) are points in the plane of vibration, A is a constant, and X, is the wave number of the sound wave with wavelength i.,. If we treat the wave field GS issuing from the radiating sources on the structure in the same way that Aoki [9] treats the diffraction wave, our mathematical results will be completely analogous to those of conventional acoustical holography. For paraxial rays, @S(~3r~3) can be described in the plane of the hologram by the FresnelKirchhoff integral [lo] as
where s2 is the distance from the vibrating structure to the hologram plane and T(x,,yJ is a function representing the effective transmission of the structure. The wavefront may be reconstructed by irradiating the reduced hologram with a coherent laser beam of wavelength &. In any wavefront reconstruction process, images can be formed with little or no distortions in directions transverse to the z-axis in Figure 4, but distortions in depth are more difficult to avoid [ 111. If we change the linear scale of the hologram by R-l, then lateral dimensions will also change by the same factor: i.e., x’ = R-lx, y’ = R-‘J. From the optical imaging relation [6], however, the depth dimension wit1 become z’ = (&/R212)z.
(4)
161
USE OF ACOUSTICAL HOLOGRAPHY x4
REAL IMAGE t
Figure 4. Co-ordinate
system for the reconstruction
of acoustical holograms.
To obtain an undistorted image, we should, therefore, choose &/A2 = R. For acoustic wavelengths of interest in our investigations, the ratio I,/& lies between lo4 and 105. Reduction factors of order R are a practical impossibility with our present optical reconstruction techniques. Since we are interested in locating far-field sound sources, rather than imaging objects, this distortion or stretching is, in fact, very desirable. If R were to approach the ratio A,/&, the twin image effect would present serious problems for the reconstruction. With the stretching of the z-co-ordinate, these images are well separated, the focal points are not too close together, and background noise is reduced. On the other hand, some reduction in the hologram is necessary to ensure that the focal plane of the hologram is within the laboratory. In the image plane shown in Figure 4, the optical field can be written as [9]
JI(x4,y4) =
&- eeikzs3~ j S(Rx3,Ry,) ew 2
3
[
+2s
{(-x4
-
~3)’
+
(~4
-
~3)~)
dx3
G3,
(5)
I
.
where k2 = 271/Azand s3 is the distance from the hologram to the images. Note that @: Qs and C?~@z in equation (1) reconstruct the real and virtual images, respectively. In discussions of the reconstructed images, we can replace the hologram function 4 in equation (1) by @:(Rx3, Ry3) @,(Rx,, Ry,) and @,(Rx3, Ry,) @S*(Rx3, Ry,). When equations (2) and (3) are substituted into equation (5), an expression is obtained for the optical field $(x4,y4). The distance from the hologram to the image planes is [ 121 ~3 =
+(A/~,
R2)s2h
+
~2>h
(6)
where the upper and lower signs correspond to the virtual and real images, respectively. These will have the same meaning in the remainder of this discussion. When this expression for s3 is substituted into the expression for $(x4, y4), the result is IC/(X~,Y~) = *WI
AW2/s2(sl
+ s~)~I e-ikzs3 T(+ctx,, +a~,),
(7)
where 0:= si R/(s, + s2).
(8)
From equation (7) we see that the images of the original sound source are reconstructed with magnitude (sl + s,)/s, R times that of the original source.
162
E. E. WATSON AND W. F. KING 111
4. EXPERIMENTAL
RESULTS
AND
DISCUSSION
The experimental arrangement used in these investigations is shown schematically in Figure 5. The structure to be studied is suspended by wires and vibrated at one of its natural
-PROPELLER DIFFRACTION NEARFIELD ACOUSTIC FARFIELD
I--
DISTANCE FROM SOURCE TO SCANNER
Figure
5. Experimental
arrangement
for the construction
of acoustical
holograms
frequencies by an electromagnetic shaker. Any far-field sound sources on the vibrating structure will generate an acoustic field which can be detected by four microphones attached to the scanning mechanism shown in Figure 5. In operation. the scanner frame moves stepwise horizontally while the microphones traverse vertically. thereby sweeping over the entire area enclosed by the frame. To effectively irradiate the object under investigation with an infinitely distant source, a sine wave at the driving frequency of the vibrator is mixed with the signals detected by the microphones. This produces an interference pattern in the scanning plane, similar to that which would obtain from the interference of a true plane wave with the sound field. Light emitting diodes (LED’s), fastened to the microphones, are wired to respond to this interaction ofthe sound field and sine wave. Since the LED’s move with the microphones, they trace out the interference pattern in that plane. For the present series of experiments, we produced acoustical holograms in the plane of the scanner by using a camera with an open shutter to record the light patterns on 35 mm film. The dimensions of the aperture swept out by the microphones are 203 cm high and 224 cm long, as shown in Figure 5. For the experimental conditions used in these investigations, the vertical dimension of the aperture recorded on the 35 mm film is 20 mm, or a reduction of about 100 from the original aperture. To reconstruct the far-field sound sources with as little longitudinal distortion as possible, the hologram should be reduced in size by the factor R-l. where R = i.,/l.,. As discussed in section 3, however, we are not using holographic techniques to image objects; in our experiments to locate sound sources, we actually benefit from the co-ordinate stretching that occurs when R is less than the ratio i,/i2. On the other hand, we need some reduction of the hologram to locate the focal plane within the laboratory. With both of these limitations in mind, we reduce the vertical dimension of the hologram to 4 mm and obtain a total reduction factor of about R = 500.
USE OF ACOUSTICAL
HOLOGRAPHY
Figure 6. Drawing of three-bladed
163
fan.
The first complex structure on which we attempted to locate sound sources was a threebladed ventilator fan with a diameter of approximately 45 cm (see Figure 6). We suspended the fan by wires and attached it along the centerline of its hub to the shaker. After the natural vibrations were measured, we excited the fan at frequency fR, corresponding to one of its higher natural modes. This generated radiated sound of non-dimensional acoustic wavelength I = /II/D = O-076,
(9)
where D is the fan diameter. We chose one of the higher natural modes because, as I + 0.5, it becomes increasingly more difficult to localize sound sources on the blades. At 1= O-076 the SPL was approximately 90 dB re 00IO2 pbar at a distance of 15 cm from the fan. The
Figure 7. Acoustical hologram for the three-bladed
ventilator fan vibrated on the hub with I, = O-076.
164
Figure
E. E. WATSON
8.
:a1 wavefront
reconstruction
AND
of far-field
W. E. KING
111
sound sources on the three. ?laded
LelGilator
fan.
hologram obtained with this procedure is shown in Figure 7 and its optical wavefront reconstruction for sound sources is shown in Figure 8. To clearly identify the relative locations of the sound sources, we outlmed the fan on all of the on-axis reconstructions. Bright areas on these reconstructions indicate regions that act as far-field sound sources. As can be seen in Figure 8, various areas ofthe blades are acting as far-field sound sources with apparent sources outside the fan. These extraneous sources will be discussed in detail below. Sound emanating from the bright region around the fan hub demonstrates the well-known piston effect. In addition to the optical reconstructions of the wavefronts, we have also written a computer program to numerically reconstruct the sound sources [9, 13-151. The technique is to convolve a digitized hologram with a so-called propagation function, given by [IO]. f(x,
-
x3,
y,
-
y3) = is, exp (2, I-)/+ i,,
(10)
with r=[s~+x:+_b$“2.
(11)
This is done by a computer evaluation of the integral in equation (5), which is the desired convolution. The hologram can be regarded as a complex aperture that transmits a monochromatic reference wave, with either zero or x phase shift. The hologram, convolved with the propagation function, gives a description of the amplitude distribution on any desired plane behind the hologram. We have found that the intensity of a reconstructed sound source is dependent upon the source (a) amplitude, (b) location, and (c) phase, as well as on the number and position of other sources. In the following numerical reconstructions, the sound sources are plotted on-axis, in perspective, and in relief, with relative intensity, in the latter two cases, represented by the heights of the curves. The intensity scale is linear, with zero intensity represented by zero height. All values are scaled in proportion to the greatest intensity, which
165
USE OF ACOUSTICAL HOLOGRAPHY
--
__ = -
=
Ix= -
-I=L -
Figure 9. Computer reconstruction view with noise cut-off level at 0.1.
z
--
of far-field sound sources on the three-bladed
ventilator fan, on-axis
is chosen as unity. With this normalization procedure, the reconstruction of a single point source will have a peak intensity of unity, regardless of the amplitude, location, or phase of the source. We have not yet determined how these parameters affect the ratio of the peak intensity to either the first lobe or background noise level. Consequently, our results are presented more to demonstrate the feasibility of locating far-field sound sources on complex structures than to analyze in detail the radiation from any particular structure. Computer wavefront reconstructions for the various viewing angles are shown in Figures 9-14. Background noise and speckle can be serious problems, particularly in optical reconstructions, because of the practical difficulties involved in suppressing them without affecting
Figure 10. Computer wavefront reconstruction fan, partially rotated with noise cut-off level at 0.1.
of far-field sound sources on the three-bladed
ventilator-
166
E. E. WATSON AND W. F. KING 111
Figure 11. Computer wavefront reconstruction of far-field sound sources on the three-bladed
ventilator
fan, rotated 90 degrees with noise cut-off level at 0.1.
the true sound sources of interest. In the numerical reconstructions, we can more easily control the signal to noise ratio and display any desired signal level. For example, in Figures 9-11, all source intensities with amplitudes less than loo/ of the peak value are suppressed, while in Figures 12-14, all values less than 20”/) are suppressed. The on-axis views in Figures 9 and 12 correspond to the optical reconstruction in Figure 8. As can be seen, the level of sound displayed in the computer reconstructions is different. Figures 12-14 indicate that one blade is acting as a stronger source of sound than is apparent in Figures 8-l 1. Such
Figure 12. Computer wavefront reconstruction on-axis view with noise cut-off level at 0.2.
of far-field sound sources on three-bladed
ventilator fan,
USE OF ACOUSTICAL HOLOGRAPHY
167
F
Figure 13. Computer reconstruction of far-field sound sources on the three-bladed ventilator fan, partially rotated with noise cut-off level at 0.2.
asymmetrical behavior is probably caused by imperfections in either the blade material or its construction. Note also that Figure 13 clearly shows the relative intensities of the sources. Noise caused by constructive interference of radiated sound with wall and equipment reflections can also result in pseudo-sound sources located outside the structure. Such sources are evident in Figures 8 and 9, and, somewhat less, in Figure 12. In most cases, the pseudosound source intensities are less than those of true sound sources, and increasing the cut-off
Figure 14. Computer wavefront reconstruction of far-field sound source on the three-bladed ventilator fan, rotated 90 degrees with noise cut-off level at 0.2.
E. E. WATSON AND W. F.
168
KING
111
level on computer reconstructions eliminates them. The walls, floor, and ceiling of the room used for these experiments were covered with foam rubber. While this reduced unwanted sound sources somewhat, better provisions would have to be made if further reductions are desired. One obvious advantage of the computer reconstructions is that the most intense far-field sound sources can easily be located simply by raising the background cut-off level until only the desired stronger sources are displayed. In an actual study of ventilator fan radiated noise, the mode geometries would be plotted and correlations sought between acoustic sources and mode shapes. These results indicate that it is possible to use holographic techniques to locate and identify far-field sound sources on complex structures such as ventilator fans. With this diagnostic tool, the effects of structural modifications on radiated noise can be studied directly and the rources of such noise identified. ACKNOWLEDGMENTS We would like to thank Robert Cohen, Mark Lang, Kent Eschenberg, Wilden Nuss, and Donald Thompson for their contributions to the work presented in this paper. Special thanks are due to Fred Stocker at The Pennsylvania State University Computation Center for his valuable assistance with the computer reconstruction. This work was supported in part by the Naval Sea Systems Command under contract with the Applied Research Laboratory at The Pennsylvania State University.
REFERENCES 1. T. S. GRAHAM 1969 Ph.D. acoustic
Dissertation,
The Pennsylvania
State
Unicersity.
Long wavelength
holography.
2. E. E. WATSON 1971 Ph.D. Dissertation, The Pennsylcaniu State Unicersity. Detection of sound radiation from plates using long wavelength acoustical holography. qf’America54,685-691. Detection ofacoustic 3. E. E. WATSON 1973 Journalofthe AcousticalSociety sources using long wavelength acoustical holography. 4. R. J. HANNON and E. E. WATSON 1974 Journal qfAppliedPhysics 45, 1951-1953. Visualization of vibration and sound using scanning and holographic methods. 5. D. GABOR 1948 A’uture 161, 777-778. A new microscope principle. 6. N. LEITH and J. UPATNICKS 1962 JournaI of the Optical Society qf’ America 52, 1123-l 129. Reconstructed wavefronts and communication theory. 7. P. GREGUSS 1965 Research Film 5, 330-337. Ultraschallhologramme. 8. F. L. THURSTONE I966 Proceedings of Symposium on Biomedical Engineering, Marquette Unioersity, Milwaukee, Wisconsin 1. 12-l 5. Ultrasound holography and visual reconstruction. 9. Y. AOKI 1970 Institute of Electricul and Eelctronic Engineering Transactions on Audio and Electroacoustics AU-18, 258-267. Optical and numerical reconstruction of images from sound-wave holograms. IO. J. W. GOODMAN 1968 Introduction to Fourier Optics. New York: McGraw-Hill Book Company, Inc. See p. 41. 1I. R. W. MEIER 1965 Journal qf’the Optical Society of America 55, 987-992. Magnification and third-order aberration in holography. 12. M. BORN and E. WOLF 1959 Principles of Optics. New York : Pergamon Press. 13. R. K. MUELLER 1971 Proceedings of the Institute of Electrical and Electronic Engineers 59, 13 19-l 335. Acoustical holography. 14. T. S. HUANG I97 I Proceedings qf the Institute ofElectricalandElectronic Engineers 59,1335-l 348. Digital holography. 15. Y. AOKI and A. BOIVIN 1970 Proceedings of the institute of Electrical and Electronic Engineers 58, 821-822. Computer reconstruction of images from a microwave hologram.
TO
Vibration (1976) 48(2),
157-168
ON THE USE OF ACOUSTICAL HOLOGRAPHY LOCATE SOUND SOURCES ON COMPLEX STRUCTURES E. E. WATSON? Applied Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A. AND
W. F. KING III Deutsche Forschungs- und Versuchsanstalt fiir Luft- und Raumfahrt E. V., Institut fiir Turbulenzforschung, 1 Berlin 12, Germany (Received 5 April 1915, and in revisedform 20 April 1976)
Until recently, it was extremely difficult to detect and locate far-field sound sources on complex structures. An experimental technique is presented in this paper for the identification of such sound sources on complex vibrating structures. Results are given for a preliminary investigation designed to determine the feasibility of applying a modified form of acoustical holography to the location of sound sources on ventilator fans. Such sources are identified on a three-bladed ventilator fan. Optical and computer sound source reconstructions are compared and analyzed. Results of this preliminary study are very encouraging and indicate the versatility and usefulness of this technique for modifying and controlling far-field noise radiation from complex vibrating structures.
1. INTRODUCTION The scientific study of any natural phenomenon involves quantitative measurements of its influence on observables under known circumstances. Until recently, however, it was very difficult to obtain quantitative information from far-field sound sources that would allow identification of their origins. This deficiency was remedied in 1969 by Graham [I], who first applied long wavelength acoustical holography to the detection of radiated sound sources. Graham’s technique was a new application of holography, which had previously been used only for the imaging of objects. Improvements in the technique of holographic sound source location were made shortly thereafter by Watson [2, 31 and Hannon and Watson [4], who have identified far-field sound sources on a variety of vibrating two-dimensional structures, including mass-loaded and ribbed plates. In this paper, we report on our preliminary efforts to use long wavelength acoustical holography to locate and identify radiated sound sources on complex three-dimensional vibrating structures. We investigated the radiated sound field from a three-bladed ventilator fan driven at one of its natural frequencies. To fully exploit the three-dimensional information stored in the holograms, we reconstructed the wavefronts both optically and numerically. Since this use of holographic techniques for sound source location is neither widely appreciated nor understood as being different from the more conventional imaging applications, a brief historical review of acoustical holography is given in section 2. A theoretical development t Present address: Environmental Decatur, Alabama, U.S.A.
Acoustics Division, Universal Oil Products, Wolverine Tube Division, 157
E. E. WATSON
1%
AND W. F. KING 111
appropriate to sound source detection studies is given in section 3. Our experimental results, including both optical and computer reconstructions of the wavefronts, are presented in section 4 together with a discussion of the method.
2. HISTORICAL
REVIEW
OF
HOLOGRAPHY
The field of optical holography began in 1948 when Gabor [5] proposed a new lensless imaging technique to improve the electron microscope. In the late 1940’s, this instrument was plagued by serious problems of spherical aberration. To avoid these distortions, Gabor’s suggestion was to record the electron diffraction pattern of the object and to then use this pattern to reconstruct the image. The essential steps in this method are illustrated in Figure I.
FILM PLATE
PLANE ’ REFERENCE WAVE
Figure
I. The reference and scattered
I/ I
I \
object
/,---. ,,‘\ ‘1’ \
\ \
\\._/j/i! I ,;;
VIRTUAL IMAGE (TRUE
Figu1.e 2. A schematic image\. and observer.
wave in a semi-transparent
,,/
\
I
,I (CONJUGATE
IMAGE)
representation
showing
the relative
positions
IMAGES
of the hologram,
real and
vi!-tual
USE OF ACOUSTICAL HOLOGRAPHY
159
A plane monochromatic wave irradiates a semi-transparent object. The scattered and incident waves combine to form an interference pattern on photographic film. When the developed film is itself irradiated with a plane monochromatic wave, the interference pattern becomes a diffraction grating that scatters the wave to create two wavefields. As indicated in Figure 2, one field converges to produce a real image, while the other diverges and results in a virtual image. This phenomenon of two overlapping images in on-axis holography is called the twin image effect. Gabor never actually succeeded in creating satisfactory images for the electron microscope with this technique. We should emphasize that since this imaging method relies on the interference of two waves, coherent sources of radiation are required for its successful application. Very little progress was made in holography until 1962 when Leith and Upatnicks [6] noticed the similarity between Gabor’s suggestion for lensless imaging and the synthetic aperture antenna. They modified Gabor’s suggestion by irradiating the object from the side to create an off-axis interference pattern. When this pattern or hologram is used to reconstruct the wavefront, the twin images are separated and do not overlap. This modification to the technique, combined with the development of the gas laser, which provides a coherent source of radiation, made possible the dramatic progress in optical holography that took place during the past decade. As we have seen, the development of optical holography was greatly hindered for at least ten years after Gabor’s paper by the lack of coherent sources. No such obstacle stood in the path of acoustical holography, however, since almost all acoustic sources that can be used for imaging are coherent. In spite of this availability of both ideas and equipment, it was not until 1965 that Greguss [7] recorded acoustical holograms on photographic film. Shortly thereafter, Thurstone [7] demonstrated that visible images could be produced from acoustical holograms. Since that time, the major thrust of research in acoustical holography has been towards imaging acoustically opaque objects at frequencies in the MHz region. An important exception is the work of Graham [I] and Watson [2, 31 on the location of far-field sound sources with long wavelength acoustical holography. It is this use of holography that is exploited in the work reported on in this paper. 3. THEORY The detection and location of far-field sound sources requires an experimental procedure that differs somewhat from that used in conventional imaging acoustical holography. This, in turn, leads to modifications in the theoretical analysis of the process. To make clear this difference, we will outline the theory of acoustical holography and indicate where our approach differs from the usual treatment. The following analysis is adapted from an excellent discussion of conventional acoustical holography by Aoki [9]. Consider the three planes shown in Figure 3. Let a discrete frequency sound source located at the intersection of the x1 and y, axes irradiate an object in the x,-y, plane. In this discussion, we will label all acoustic fields by their velocity potentials @*,where the subscript will identify the particular field. The illuminating field @, in conventional acoustical holography passes through the semitransparent structure to be imaged and forms a diffraction field QD, in addition to the background illuminating field. These fields interact with one another to generate an interference pattern that is recorded as a sound hologram in the x,-y, plane. In our experiments to locate far-field sound sources on vibrating structures, we do not actually irradiate the object under investigation with sound waves. Instead, we mix a plane wave with the sound field Gs generated by sound sources on a vibrating structure. As in conventional acoustical holography, this interaction of sound wave fields produces an interference pattern that is detected and recorded as a sound wave hologram in the x,-y, plane. The effect of
E. E. WATSON AND W. F. KING 111
160 XI
FICTITIOUS SOURCE t
Figure 3. Co-ordinate system for the construction
of acoustical
holograms
mixing the plane wave with QS is equivalent to illuminating the structure with a fictitious sound source located at s1 = 0~ (see Figure 3). In our analysis, we will let the distance from the fictitious source to the vibrating structure be very large but finite. The effect of this approximation is to illuminate the structure with a spherical rather than a plane wave. As the source distance s1 becomes very large, however, the practical difference between a spherical and plane wave becomes insignificant and our analysis should not be affected by this choice. With square law detection of the wave fields, the pattern recorded in the x-,-y, plane is (1)
5=(~,+~S)(~,+~S)*=/~,/Z+/~DSi2+f~~~+~,~S*. where the star denotes complex conjugate. By using the paraxial express the illuminating wave from the fictitious source as ~,(x,,?,,j=(A/s,)exp(-ik,s,[l
+(xt+lf.2.$},
ray approximation,
we can
(2)
where (x,,y,) are points in the plane of vibration, A is a constant, and X, is the wave number of the sound wave with wavelength i.,. If we treat the wave field GS issuing from the radiating sources on the structure in the same way that Aoki [9] treats the diffraction wave, our mathematical results will be completely analogous to those of conventional acoustical holography. For paraxial rays, @S(~3r~3) can be described in the plane of the hologram by the FresnelKirchhoff integral [lo] as
where s2 is the distance from the vibrating structure to the hologram plane and T(x,,yJ is a function representing the effective transmission of the structure. The wavefront may be reconstructed by irradiating the reduced hologram with a coherent laser beam of wavelength &. In any wavefront reconstruction process, images can be formed with little or no distortions in directions transverse to the z-axis in Figure 4, but distortions in depth are more difficult to avoid [ 111. If we change the linear scale of the hologram by R-l, then lateral dimensions will also change by the same factor: i.e., x’ = R-lx, y’ = R-‘J. From the optical imaging relation [6], however, the depth dimension wit1 become z’ = (&/R212)z.
(4)
161
USE OF ACOUSTICAL HOLOGRAPHY x4
REAL IMAGE t
Figure 4. Co-ordinate
system for the reconstruction
of acoustical holograms.
To obtain an undistorted image, we should, therefore, choose &/A2 = R. For acoustic wavelengths of interest in our investigations, the ratio I,/& lies between lo4 and 105. Reduction factors of order R are a practical impossibility with our present optical reconstruction techniques. Since we are interested in locating far-field sound sources, rather than imaging objects, this distortion or stretching is, in fact, very desirable. If R were to approach the ratio A,/&, the twin image effect would present serious problems for the reconstruction. With the stretching of the z-co-ordinate, these images are well separated, the focal points are not too close together, and background noise is reduced. On the other hand, some reduction in the hologram is necessary to ensure that the focal plane of the hologram is within the laboratory. In the image plane shown in Figure 4, the optical field can be written as [9]
JI(x4,y4) =
&- eeikzs3~ j S(Rx3,Ry,) ew 2
3
[
+2s
{(-x4
-
~3)’
+
(~4
-
~3)~)
dx3
G3,
(5)
I
.
where k2 = 271/Azand s3 is the distance from the hologram to the images. Note that @: Qs and C?~@z in equation (1) reconstruct the real and virtual images, respectively. In discussions of the reconstructed images, we can replace the hologram function 4 in equation (1) by @:(Rx3, Ry3) @,(Rx,, Ry,) and @,(Rx3, Ry,) @S*(Rx3, Ry,). When equations (2) and (3) are substituted into equation (5), an expression is obtained for the optical field $(x4,y4). The distance from the hologram to the image planes is [ 121 ~3 =
+(A/~,
R2)s2h
+
~2>h
(6)
where the upper and lower signs correspond to the virtual and real images, respectively. These will have the same meaning in the remainder of this discussion. When this expression for s3 is substituted into the expression for $(x4, y4), the result is IC/(X~,Y~) = *WI
AW2/s2(sl
+ s~)~I e-ikzs3 T(+ctx,, +a~,),
(7)
where 0:= si R/(s, + s2).
(8)
From equation (7) we see that the images of the original sound source are reconstructed with magnitude (sl + s,)/s, R times that of the original source.
162
E. E. WATSON AND W. F. KING 111
4. EXPERIMENTAL
RESULTS
AND
DISCUSSION
The experimental arrangement used in these investigations is shown schematically in Figure 5. The structure to be studied is suspended by wires and vibrated at one of its natural
-PROPELLER DIFFRACTION NEARFIELD ACOUSTIC FARFIELD
I--
DISTANCE FROM SOURCE TO SCANNER
Figure
5. Experimental
arrangement
for the construction
of acoustical
holograms
frequencies by an electromagnetic shaker. Any far-field sound sources on the vibrating structure will generate an acoustic field which can be detected by four microphones attached to the scanning mechanism shown in Figure 5. In operation. the scanner frame moves stepwise horizontally while the microphones traverse vertically. thereby sweeping over the entire area enclosed by the frame. To effectively irradiate the object under investigation with an infinitely distant source, a sine wave at the driving frequency of the vibrator is mixed with the signals detected by the microphones. This produces an interference pattern in the scanning plane, similar to that which would obtain from the interference of a true plane wave with the sound field. Light emitting diodes (LED’s), fastened to the microphones, are wired to respond to this interaction ofthe sound field and sine wave. Since the LED’s move with the microphones, they trace out the interference pattern in that plane. For the present series of experiments, we produced acoustical holograms in the plane of the scanner by using a camera with an open shutter to record the light patterns on 35 mm film. The dimensions of the aperture swept out by the microphones are 203 cm high and 224 cm long, as shown in Figure 5. For the experimental conditions used in these investigations, the vertical dimension of the aperture recorded on the 35 mm film is 20 mm, or a reduction of about 100 from the original aperture. To reconstruct the far-field sound sources with as little longitudinal distortion as possible, the hologram should be reduced in size by the factor R-l. where R = i.,/l.,. As discussed in section 3, however, we are not using holographic techniques to image objects; in our experiments to locate sound sources, we actually benefit from the co-ordinate stretching that occurs when R is less than the ratio i,/i2. On the other hand, we need some reduction of the hologram to locate the focal plane within the laboratory. With both of these limitations in mind, we reduce the vertical dimension of the hologram to 4 mm and obtain a total reduction factor of about R = 500.
USE OF ACOUSTICAL
HOLOGRAPHY
Figure 6. Drawing of three-bladed
163
fan.
The first complex structure on which we attempted to locate sound sources was a threebladed ventilator fan with a diameter of approximately 45 cm (see Figure 6). We suspended the fan by wires and attached it along the centerline of its hub to the shaker. After the natural vibrations were measured, we excited the fan at frequency fR, corresponding to one of its higher natural modes. This generated radiated sound of non-dimensional acoustic wavelength I = /II/D = O-076,
(9)
where D is the fan diameter. We chose one of the higher natural modes because, as I + 0.5, it becomes increasingly more difficult to localize sound sources on the blades. At 1= O-076 the SPL was approximately 90 dB re 00IO2 pbar at a distance of 15 cm from the fan. The
Figure 7. Acoustical hologram for the three-bladed
ventilator fan vibrated on the hub with I, = O-076.
164
Figure
E. E. WATSON
8.
:a1 wavefront
reconstruction
AND
of far-field
W. E. KING
111
sound sources on the three. ?laded
LelGilator
fan.
hologram obtained with this procedure is shown in Figure 7 and its optical wavefront reconstruction for sound sources is shown in Figure 8. To clearly identify the relative locations of the sound sources, we outlmed the fan on all of the on-axis reconstructions. Bright areas on these reconstructions indicate regions that act as far-field sound sources. As can be seen in Figure 8, various areas ofthe blades are acting as far-field sound sources with apparent sources outside the fan. These extraneous sources will be discussed in detail below. Sound emanating from the bright region around the fan hub demonstrates the well-known piston effect. In addition to the optical reconstructions of the wavefronts, we have also written a computer program to numerically reconstruct the sound sources [9, 13-151. The technique is to convolve a digitized hologram with a so-called propagation function, given by [IO]. f(x,
-
x3,
y,
-
y3) = is, exp (2, I-)/+ i,,
(10)
with r=[s~+x:+_b$“2.
(11)
This is done by a computer evaluation of the integral in equation (5), which is the desired convolution. The hologram can be regarded as a complex aperture that transmits a monochromatic reference wave, with either zero or x phase shift. The hologram, convolved with the propagation function, gives a description of the amplitude distribution on any desired plane behind the hologram. We have found that the intensity of a reconstructed sound source is dependent upon the source (a) amplitude, (b) location, and (c) phase, as well as on the number and position of other sources. In the following numerical reconstructions, the sound sources are plotted on-axis, in perspective, and in relief, with relative intensity, in the latter two cases, represented by the heights of the curves. The intensity scale is linear, with zero intensity represented by zero height. All values are scaled in proportion to the greatest intensity, which
165
USE OF ACOUSTICAL HOLOGRAPHY
--
__ = -
=
Ix= -
-I=L -
Figure 9. Computer reconstruction view with noise cut-off level at 0.1.
z
--
of far-field sound sources on the three-bladed
ventilator fan, on-axis
is chosen as unity. With this normalization procedure, the reconstruction of a single point source will have a peak intensity of unity, regardless of the amplitude, location, or phase of the source. We have not yet determined how these parameters affect the ratio of the peak intensity to either the first lobe or background noise level. Consequently, our results are presented more to demonstrate the feasibility of locating far-field sound sources on complex structures than to analyze in detail the radiation from any particular structure. Computer wavefront reconstructions for the various viewing angles are shown in Figures 9-14. Background noise and speckle can be serious problems, particularly in optical reconstructions, because of the practical difficulties involved in suppressing them without affecting
Figure 10. Computer wavefront reconstruction fan, partially rotated with noise cut-off level at 0.1.
of far-field sound sources on the three-bladed
ventilator-
166
E. E. WATSON AND W. F. KING 111
Figure 11. Computer wavefront reconstruction of far-field sound sources on the three-bladed
ventilator
fan, rotated 90 degrees with noise cut-off level at 0.1.
the true sound sources of interest. In the numerical reconstructions, we can more easily control the signal to noise ratio and display any desired signal level. For example, in Figures 9-11, all source intensities with amplitudes less than loo/ of the peak value are suppressed, while in Figures 12-14, all values less than 20”/) are suppressed. The on-axis views in Figures 9 and 12 correspond to the optical reconstruction in Figure 8. As can be seen, the level of sound displayed in the computer reconstructions is different. Figures 12-14 indicate that one blade is acting as a stronger source of sound than is apparent in Figures 8-l 1. Such
Figure 12. Computer wavefront reconstruction on-axis view with noise cut-off level at 0.2.
of far-field sound sources on three-bladed
ventilator fan,
USE OF ACOUSTICAL HOLOGRAPHY
167
F
Figure 13. Computer reconstruction of far-field sound sources on the three-bladed ventilator fan, partially rotated with noise cut-off level at 0.2.
asymmetrical behavior is probably caused by imperfections in either the blade material or its construction. Note also that Figure 13 clearly shows the relative intensities of the sources. Noise caused by constructive interference of radiated sound with wall and equipment reflections can also result in pseudo-sound sources located outside the structure. Such sources are evident in Figures 8 and 9, and, somewhat less, in Figure 12. In most cases, the pseudosound source intensities are less than those of true sound sources, and increasing the cut-off
Figure 14. Computer wavefront reconstruction of far-field sound source on the three-bladed ventilator fan, rotated 90 degrees with noise cut-off level at 0.2.
E. E. WATSON AND W. F.
168
KING
111
level on computer reconstructions eliminates them. The walls, floor, and ceiling of the room used for these experiments were covered with foam rubber. While this reduced unwanted sound sources somewhat, better provisions would have to be made if further reductions are desired. One obvious advantage of the computer reconstructions is that the most intense far-field sound sources can easily be located simply by raising the background cut-off level until only the desired stronger sources are displayed. In an actual study of ventilator fan radiated noise, the mode geometries would be plotted and correlations sought between acoustic sources and mode shapes. These results indicate that it is possible to use holographic techniques to locate and identify far-field sound sources on complex structures such as ventilator fans. With this diagnostic tool, the effects of structural modifications on radiated noise can be studied directly and the rources of such noise identified. ACKNOWLEDGMENTS We would like to thank Robert Cohen, Mark Lang, Kent Eschenberg, Wilden Nuss, and Donald Thompson for their contributions to the work presented in this paper. Special thanks are due to Fred Stocker at The Pennsylvania State University Computation Center for his valuable assistance with the computer reconstruction. This work was supported in part by the Naval Sea Systems Command under contract with the Applied Research Laboratory at The Pennsylvania State University.
REFERENCES 1. T. S. GRAHAM 1969 Ph.D. acoustic
Dissertation,
The Pennsylvania
State
Unicersity.
Long wavelength
holography.
2. E. E. WATSON 1971 Ph.D. Dissertation, The Pennsylcaniu State Unicersity. Detection of sound radiation from plates using long wavelength acoustical holography. qf’America54,685-691. Detection ofacoustic 3. E. E. WATSON 1973 Journalofthe AcousticalSociety sources using long wavelength acoustical holography. 4. R. J. HANNON and E. E. WATSON 1974 Journal qfAppliedPhysics 45, 1951-1953. Visualization of vibration and sound using scanning and holographic methods. 5. D. GABOR 1948 A’uture 161, 777-778. A new microscope principle. 6. N. LEITH and J. UPATNICKS 1962 JournaI of the Optical Society qf’ America 52, 1123-l 129. Reconstructed wavefronts and communication theory. 7. P. GREGUSS 1965 Research Film 5, 330-337. Ultraschallhologramme. 8. F. L. THURSTONE I966 Proceedings of Symposium on Biomedical Engineering, Marquette Unioersity, Milwaukee, Wisconsin 1. 12-l 5. Ultrasound holography and visual reconstruction. 9. Y. AOKI 1970 Institute of Electricul and Eelctronic Engineering Transactions on Audio and Electroacoustics AU-18, 258-267. Optical and numerical reconstruction of images from sound-wave holograms. IO. J. W. GOODMAN 1968 Introduction to Fourier Optics. New York: McGraw-Hill Book Company, Inc. See p. 41. 1I. R. W. MEIER 1965 Journal qf’the Optical Society of America 55, 987-992. Magnification and third-order aberration in holography. 12. M. BORN and E. WOLF 1959 Principles of Optics. New York : Pergamon Press. 13. R. K. MUELLER 1971 Proceedings of the Institute of Electrical and Electronic Engineers 59, 13 19-l 335. Acoustical holography. 14. T. S. HUANG I97 I Proceedings qf the Institute ofElectricalandElectronic Engineers 59,1335-l 348. Digital holography. 15. Y. AOKI and A. BOIVIN 1970 Proceedings of the institute of Electrical and Electronic Engineers 58, 821-822. Computer reconstruction of images from a microwave hologram.