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ACOUSTIC PROPERTIES OF PARABOLIC REFLECTORS

 

 

Randolph Scott Little

Laboratory of Ornithology

Cornell University

Ithaca, New York*

 

 

 

Since sound power being radiated from a point source varies inversely with the square of distance, i.e., drops 6dB each time distance is doubled, and since the apparent area of the radiating surface also decreases to one-fourth its initial value whenever distance is doubled, it is often desirable that a sound pickup device have both large gain and high directivity.  Aluminum or Fiberglas® parabolic reflectors have been used in natural sound recording since May, 1932 (Kellogg, 1962), but without accurate knowledge of their acoustical properties, beyond the fact that they seem to work.  Recent interest in collapsible or folding sound reflectors points to need for further theoretical and practical study of the characteristics of these reflectors.

The giant reflector telescope at Mt. Palomar Observatory contains a 200-inch parabolic mirror to collect and focus the weak light from outer space.  The antenna at Andover, Maine, which received signals from the Telstar satellites is a section of an even larger paraboloid.  Although the reflectors of these examples are reflecting transverse electromagnetic radiation, several principles will apply to concentration of longitudinal sound waves as well.  The larger the aperture of the reflector, the higher the gain; the truer the reflecting surface, the sharper the definition of focus; and the larger the diameter of the reflector, the greater the directivity.

Each of these statements is, of course, a generalization.  The first statement need not be expanded as it holds well enough.  The second will be of no consequence, except perhaps in the case of a collapsible reflector where irregularities in shape are significant.  The third statement, however, varies in importance with frequency.  At high audio frequencies, reflectors of the type generally used in the Laboratory of Ornithology (spun aluminum, 36-inch diameter, 12-inch focal length) are quite directional.  At low frequencies, on the other hand, they are omnidirectional.  At intermediate frequencies, gain and directivity are irregular.  It is this intermediate range which is of particular interest from the standpoint of analysis.

 

FREQUENCY RESPONSE

Numerous experiments have demonstrated that a body does not reflect radiation unless the dimensions of the obstacle are several wavelengths or more.  Hence, swells from the ocean are unaffected by pilings of a pier, but are reflected from a long breakwater.  Since sound in air travels at about 1000 feet per second, the wavelength of a 1000 Hertz (1 KHz) signal is about one foot.  Thus, objects smaller than a few feet across cannot be expected to reflect sounds of frequencies below 1 KHz.  At frequencies above 10 KHz, the same object may reflect quite uniformly.  Somewhere between these limits resonances will produce irregular response.

For the particular geometric form of the parabolic reflector, the sound pressure at the focal region is given by the integral (Little, 1963):

               j(ωt - ky²/a²)   ⌠ y = y₀         -jk[{(x_f - y²/a²)² + y²}^½  - y²/a²]

P = 2πAe                       │             ye                                                   dy  ;

                                      ⌡ y = 0

 

where A is the acoustic pressure amplitude of the normally incident plane sound wave of radian frequency ω and propagation constant k  ( k = ω/c , where c is the velocity of sound propagation).  The coordinate system is as shown in Figure 1.

Such an integral of the form  e^Ky²dy  is indeterminate by ordinary means, and must be computed and tabulated, as is the similar Gaussian probability integral.  However, an important qualitative conclusion can be drawn without actually evaluating the integral at each of the various frequencies of interest.  This conclusion is that the solution can be expressed in approximate form by Bessel functions (Kinsler and Frey, 1961).  This suggests that the gain will be low at low frequencies, high at high frequencies, and will experience large variations at intermediate frequencies.  These variations would appear as a damped pseudo-sinusoid as depicted in Figure 2.

The actual response of a 36-inch/12-inch aluminum reflector at 0º incidence is given in Figure 3a.  Notice the large peak at 200 Hz.  This is due to cavity resonance of the dished reflector, much as an organ pipe, and is omnidirectional.  Effective reflection begins near 1 KHz, and is quite variable to beyond 10 KHz.  Although accurate measurement at higher frequencies was beyond the capabilities of the instruments, it is suggested that these irregularities in response should decrease at higher frequencies, making the reflector well suited to ultrasonic recording.

 

DIRECTIVITY

If the parabolic surface of the reflector is approximated by a series of small flat facets, each of these facets must be oriented so as to direct reflections through the focal point.  In order that reflections from this hypothetical surface not differ significantly from those from the true parabolic surface, each flat element must not deviate from the true surface by more than about one percent of a wavelength so that phase shift will be less than ten degrees.  Solving the geometry, the radius of a facet cannot exceed  0.1aL^½  if the deviation is to be less than one percent, where a is the parabolic constant and L is the wavelength of the sound.  For a value of  a = 10 (typical for the reflectors used), this means that the radius of a facet cannot exceed the square root of the wavelength.

Considering a plane wave incident upon the reflector, the wavefront may be subdivided into areas corresponding to each facet of the hypothetical reflector.  Each facet will reflect its cylinder of sound toward the focal point.  However, the convergence of all these cylinders of sound will be to a focal region rather than to a discrete point, the cross-section of which will correspond to that of the facets.  Hence, the size of the focal region is proportional to the square root of the sound wavelength, making directivity proportional to the square root of frequency.  The limit of directional resolution for a 36-inch/12-inch aluminum reflector using a microphone with a one-inch diaphragm seems to be about 10º, occurring at frequencies of 5 KHz and higher.  The limit as determined by the  L^½  formula would be about 8º at 8.5 KHz.  The slight loss of resolution is probably due to inaccurate placement of the microphone at the focal region.

 

FOLDING REFLECTOR

Experiments conducted on a collapsible umbrella reflector, recently made available to the Laboratory of Ornithology by Mr. Rodman Ward, gave very encouraging results.  The device, sold under the trade name “Reflectal,” is a nylon fabric umbrella with roughly parabolic shape when set.  The reflecting surface is aluminized Mylar bonded to the nylon, with a fine quilted texture to diffuse light rays.  The reflector was intended for use in floodlighting photographic subjects, with no provision for mounting a microphone at the focal point.  At the time of this writing, suitable fittings are being prepared; but preliminary measurements were conducted with the microphone held in place by a separate tripod.  The focal length appears to be about 12 inches, the diameter about 40 inches.

The results are quite comparable to those for the conventional aluminum reflectors, as shown in Figure 3b.  Overall gain averages several decibels less for the fabric reflector because of less efficient reflection.  Due to the imperfect parabolic shape, the limit of directional resolution is somewhat poorer than the 10º found for the conventional aluminum reflector.

Once the microphone mount has been prepared and the “Reflectal” has been field tested, a more accurate evaluation of its practicality will be possible.  However, based on laboratory tests, the collapsible umbrella reflector promises to give performance comparable to that of a rigid reflector of equal size, at one-quarter the weight and one-twentieth the (folded) volume.  This could amount to considerable advantage from the standpoint of transportation, especially where shipping is a factor.  The main features yet to be proven are the durability and convenience of use in the field.

 

CONCLUSION

These results indicate that, for the size of reflector used in the Laboratory of Ornithology, the device may be thought of as a 20 dB amplifier with a high-pass filter network which cuts off at 1 KHz.  For recordings of bat sounds and other ultrasonics, smaller diameter reflectors could be used without sacrificing uniformity of frequency response.  In order to avoid deep cavity resonances, as indicated by the 200 Hz peak, the microphone should lie outside the plane of the edge of the reflector.  On the other hand, in order to suppress sounds coming from directions behind the reflector, the microphone should be well inside the rim.  A suitable compromise is to place the microphone at the plane of the rim.  This means that the focal length must be one-quarter of the diameter in order to satisfy the formula for a parabolic curve.  In the analysis of sound recordings made with a reflector, the analyst should be aware of the irregularities of frequency response brought to light in this paper, especially the low frequency cutoff, which might otherwise be attributed to acoustical filtering in the voice-producing system of the animal.

 

LITERATURE CITED

Kellogg, P. P.

1962.  “Bird Sound Studies at Cornell.”  Living Bird.  1:37-48.

Kinsler, L. E., and A. R. Frey

1961.  Fundamentals of Acoustics.  Wiley, New York.

Little, R. S.

1963.  “An Investigation of the Acoustic Properties of Parabolic Reflectors.”  School of Electrical Engineering, Cornell University.  Unpublished senior project report.

 

FIGURES

 

 

 

 

This paper originally published in Bio-Acoustics Bulletin, 4(1):1-3, 1964.