Cornell University

College of Engineering

School of Electrical Engineering

EE 4091-EE 4092

Senior Project Report

An Investigation of the

ACOUSTIC PROPERTIES OF PARABOLIC REFLECTORS

Randolph Scott Little

4 June 1963

ABSTRACT

Spun aluminum and molded fiberglass parabolic dishes, ranging in diameter from one foot to four feet, have been used for many years as sound concentrators without any accurate knowledge of their acoustic properties. Most of these reflectors were manufactured for microwave transmission applications, for which theoretical treatments and practical characteristics are readily available. However, it is incorrect to assume that short electromagnetic waves and long compression waves behave alike. Because of increased application in the field of natural sound recording, it is imperative that the acoustic properties of such reflectors be understood and taken into account.

An integration technique for determining the mean acoustic pressure amplitude in a small volume around the focal point, such as the space occupied by the microphone diaphragm, is developed; and a relationship for gain as a function of frequency is determined. An extension for this integration method is developed for expressing gain vs. direction of the incident sound. Experimental observations using an aluminum reflector supplied by the Cornell Laboratory of Ornithology are used to verify theoretical predictions.

The results of this investigation indicate that: non-uniform frequency response should be taken into account over the applicable frequency range; a larger diameter reflector is generally desirable, although in certain instances smaller diameters may be sufficient; and for best results focal length should approximate one-quarter of the diameter.

INTRODUCTION

Since the sound power being radiated from a point source varies inversely as the square of distance, i.e., drops 6 dB each time distance is doubled, and since the apparent size of a radiating surface also decreases to one-quarter of its initial value whenever distance is doubled, it is often desirable that a sound pickup device have both large gain and high directivity. In radio applications, electromagnetic reflecting and focusing arrays provide the necessary qualities.

Since good conductors are good electrical reflectors, metals are widely used in the manufacture of radio reflectors. One class of reflectors, parabolic reflectors, has the property of focusing parallel rays at normal incidence to a single point. At micro-wavelengths aluminum parabolic dishes with diameters from one to many feet are used, giving gain directly proportional to the square of the diameter-to-wavelength ratio. Thus, gain cannot be considered to be uniform over any appreciable bandwidth.

This theoretical relationship has been derived for electromagnetic waves, and is only applicable at wavelengths small compared to the diameter. Hence, it is imperative that we develop an independent theory for the acoustic properties of parabolic reflectors. That discussion is presented in the following section.

THEORETICAL ANALYSIS

Assumptions:

1) that the medium (air) is isotropic and homogeneous;

2) that the medium is perfectly elastic;

3) that gravitational forces may be neglected;

4) that pressure and velocity variation amplitudes are small;

5) that compressions and expansions occur adiabatically; and,

6) that any departure from these assumptions will be so stated.

Definitions:

1) A particle is any element of volume small enough that the acoustic variables may be considered constant throughout the element.

2) Acoustic impedance is the vector ratio of pressure to velocity.

Symbols:

x, y, z Coordinates of a particle of the medium

u, v, w Corresponding component particle velocities

ρ′ Instantaneous density at any point

ρ Constant mean density at any point

s Condensation at any point, defined by: s = (ρ′ – ρ)/ρ

P′ Instantaneous pressure at any point

P_oConstant mean pressure at any point

P Acoustic pressure at any point, defined by: P = P′ - P_o

Φ Velocity potential function

c Velocity of propagation of sound wave.

GENERAL PLANE WAVE EQUATION

By the principle of continuity, the time rate of increase of mass of a volume element must equal the net rate of influx of mass into that volume. Consider the volume element dx·dy·dz shown in Figure 1.

The net influx from flow in the x direction is given by:

(ρ′u - {ρ′u + [δ(ρ′u)/δx] dx}) dy dz = - [δ(ρ′u)/δx] dx dy dz.

Hence, the net influx of mass into the volume is:

- ([δ(ρ′u)/δx] + [δ(ρ′v)/δy] + [δ(ρ′w)/δz] dx dy dz.

The rate of mass increase is:

(δρ′/δt) dx dy dz.

Hence, the equation of continuity is:

(δρ′/δt) + [δ(ρ′u)/δx] + [δ(ρ′v)/δy] + [δ(ρ′w)/δz] = 0.

Since ρ′ = ρ(1 + s) , we can expand the equation of continuity as:

ρ(δs/δt) + ρ(1 + s)([δu/δx] + [δv/δy] + [δw/δz]) + ρ(u[δs/δx] + v[δs/δy] + w[δs/δz]) = 0.

For acoustic waves of normal intensity, we may neglect s with respect to unity. Moreover, at audible frequencies the wavelengths are so long that u, v, w and s change very slowly with x, y and z. Therefore, products such as u(δs/δx) involving two very small terms may be neglected with respect to terms such as (δu/δx). With these approximations the equation of continuity becomes:

(δs/δt) + (δu/δx) + (δv/δy) + (δw/δz) = 0.

The velocity vector of a particle in an irrotational field may be expressed as the gradient of a scalar potential function, or:

V = ▼Φ = (δΦ/δx) + (δΦ/δy) + (δΦ/δz) ; where Φ(x, y, z, t) is the velocity potential.

The equation of continuity, written in terms of this potential function becomes:

(δs/δt) + ▼²Φ = 0.

We must now find a relation between the condensation, s, and the potential function in order to obtain an acoustic wave equation that may be solved for Φ. Considering the forces acting upon the elemental volume dx dy dz, the net pressure force acting in the positive x direction is: dF_x = (- [δρ′/δx] dx) dy dz. This force changes the x component of momentum of the elemental volume, and can be equated to the time rate of change of momentum, or:

dF_x = δ(ρ′u dx dy dz)/δt = (δ[ρ′u]/δt) dx dy dz.

Equating these two expressions for dF_x gives:

(δρ′/δx) dx + (δ[ρ′u]/δt) dx = 0.

Again, we can substitute ρ′ = ρ(1 + s) ≈ ρ , and reduce the equation to:

(δρ′/δx) dx + ρ(δu/δt) dx = 0, where the dx is retained for later use.

Similar expressions can be derived for the other coordinate directions, and the sum of these three analogous equations must be zero. Therefore:

(δρ′/δx) dx + (δρ′/δy) dy + (δρ′/δz) dz + ρ(δ[u dx + v dy + w dz]/δt) = 0.

But, since u = (δΦ/δx), v = (δΦ/δy) and w = (δΦ/δz), we can write:

dP′ + ρ(δ[dΦ]/δt) = 0.

Integrating at some particular time gives: P′ + ρ(δΦ/δt) = constant. When no acoustic waves are present, (δΦ/δt) = 0 and P′ = P₀. Therefore, constant = P₀. By substitution of this and of P = P′ – P₀ , we obtain the equation of motion:

P = - ρ(δΦ/δt).

The bulk modulus of elasticity, B, of a fluid is defined as: B = - (dP/dV)/V = P/s. Letting c² = B/ρ, then P = ρc²s, which may be substituted into the equation of motion to give:

S = - c^-2(δΦ/δt).

This may be substituted into the equation of continuity to give:

(δ²Φ/δt²) = c²▼² Φ, the acoustic wave equation.

Once the acoustic wave equation has been solved for Φ, all of the acoustic variables can be obtained by relations such as P = -ρ(δΦ/δt), s = P/ρc², u = (δΦ/δx), etc.

The constant c represents the velocity of the acoustic wave; and, assuming that volume changes are essentially adiabatic so that the bulk modulus B = P₀, then the expression for c becomes:

c = (B/ρ)^0.5 = (P₀/ρ)^0.5 = (RT)^0.5, where R is the acoustic resistance and T is absolute temperature of the medium.

At 0˚C and one atmosphere pressure in air, c = 33,160 cm/sec or about 1,000 ft/sec.

For an acoustic plane wave traveling in the x direction, the general wave equation may be reduced to: (δ²Φ/δt²) = c²(δ²Φ/δt²). A solution of this plane wave equation expressing the motion as a function of harmonic waves is:

Φ = Ae^{j(ωt – kx)} + Be^{j(ωt + kx)}, where A is the complex amplitude of a plane wave of frequency ω traveling in the positive x direction with a velocity c = ω/k, and B is the complex amplitude of a similar wave traveling in the negative x direction with the same velocity.

SOUND TRANSMISSION AND REFLECTION

Relationships quite analogous to those found in transmission and reflection of light waves can be derived for sound waves, although it is not safe to assume blindly that the two cases may be analogous. For example, quite obviously the sum of reflected and transmitted energies must equal the incident energy. Also, an equation similar to Snell’s Law gives the ratio of sines of the angle of incidence to transmission as being the ratio of the velocities of propagation. A critical angle of incidence exists beyond which no power transmission can occur. This angle equals the arcsine of the ratio of velocities.

For the purposes of this investigation the reflecting surface will be considered to be a perfect reflector. This will be a valid assumption for typical reflectors (aluminum and Fiberglas®) at audio frequencies. An incident sound wave will therefore undergo a phase reversal upon reflection, but this reversal may be neglected so long as all waves eventually impinging upon the microphone diaphragm undergo the same reversal.

The reflecting surface of a paraboloid may be thought of as a many-faceted area composed of small elements which are each planar and each directed so that incoming parallel waves at normal incidence will be focused at the common focal point of the reflector. Each element of area must be small enough so that appreciable error does not result from this approximation. Generally this requires that the surface of each flat element may not deviate from that of the true paraboloid by more than about 1% of a wavelength so that phase shift will be less than about ten degrees. It is apparent that for accurate definition at the focal point, the criterion that each element of area be directed just right is much more critical than the actual deviation between the surface of the paraboloid and of the elements. This is an important point to note in practice; i.e., that whenever a reflector becomes dented it is more important to correct for irregularities in curvature than for departures from exact shape. Although both criteria amount to the same thing in the long run, the correction for proper orientation gives a better first approximation to the true paraboloid.

Actually the integration of reflections from each of these facets poses a stiffer limit on the dimensions of an element than does deviation from the true paraboloid surface. This limitation is dictated as follows. The distances that components of the wave must travel from this element to the focal point must not vary by more than about 1/36 wavelength in order to avoid phase distortion. The simplest way to be sure that this does not occur is to limit the area of each element such that no dimension is large compared to 1/36 wavelength. This restriction is unnecessarily stringent for shallow paraboloids, however, and may be relaxed somewhat for practical reflector dimensions.

Since the problem is being approached from the standpoint of a summation of reflections from plane facets, the reflected elements of the incident plane wave will each be tubular elements of cross-sectional area equal to the projected area of the reflecting facet; hence, the focal point becomes a focal volume approximately equal in cross-section to the projected area of a facet. This should coincide with the diaphragm of the microphone.

PARABOLIC GEOMETRY

In order to perform integration of reflections from differential facets of the parabolic surface, certain relations must be known, such as path length as a function of point of reflection. These relations will be derived here. Since radial symmetry exists, only a radial section will be treated. Figure 2 shows this section, and defines size and focal length of the reflector. Figure 3 shows reflection from a given facet, and defines path length.

The portion of the incident wave reflected from each facet is in the form of a cylinder whose cross-sectional area equals the projected area of the facet. Notice that this area is the same whether projected at the focal point or at the incident wave. Hence, all such cylinders of reflected wave components intersect at the focal point, but the focus occurs in a sphere of cross-sectional area equal to that of the cylinders rather than at a single point. Under actual circumstances, roughness and deformities of the surface of the reflector would cause similar reduction of definition at the focus. So long as the space occupied by the microphone diaphragm coincides with the focal region, the following derivation should be valid.

Since all components of a normally incident wave must travel equal distances to reach the plane of the edge of the paraboloid, all components will be in phase at this plane, and any differences in path lengths will occur beyond this plane. Therefore, all distance measurements will be made commencing with the plane of the edge of the reflector. Hence, in Figure 3 the “distance” traveled by the component being reflected from the facet at coordinates (x, y) will be D = D_i + D_r.

The mean pressure amplitude in the focal region can be found by integrating over all reflected components, taking into account the differing path lengths and resulting phase shift. Since radial symmetry exists, a factor of 2πy dy will give the projected area of an annulus on the surface of the reflector at radius y. Therefore, the mean focal pressure amplitude is given by:

where P is the pressure amplitude of the component being reflected from radius y at the time it reaches the focal point. This amplitude is simply:

P = Ae^{j(ωt
- kD)}

where D is the “distance” traveled by that component as defined above. Since all components undergo the same phase reversal upon reflection, this can also be absorbed and removed from the problem as was the initial distance traveled from the sound source to the plane of the edge of the reflector.

The distance D_i is simply: D_i = x₀ – x. The distance D_r is: D_r = ([x_f – x]² + y²)^0.5. Since the equation of definition for a parabola is y = ax^0.5, the distances can be expressed in terms of y and the constants a, x_f and y₀. The resulting expression for D is:

D = (y₀²/a²) – (y²/a²) + ([{y₀/x_f} – {y²/a²}]² + y²)^0.5.

Therefore, the mean acoustic pressure amplitude in the focal region is:

, where k = ω/c.

This integral does not lend itself to solution by ordinary means of integral calculus, because it is of the form ye^f(y)dy, where f(y) is not a simple function of the variable. Two approaches are available. This equation could be programmed for solution by a computer, or suitable approximations and expansions could be made so that the reduced equation may be solved by the methods of calculus. The particular reductions which must be made lie in the expression for D. Unfortunately, none of the terms of D may be neglected, making reduction impossible. However, the constant y₀²/a² can be taken ahead of the integral to give:

An integral of the form ye^by2 dy is indeterminant by ordinary means, and must be computed and tabulated, as is the similar probability integral. However, a few qualitative conclusions can be drawn without actually evaluating the integral at each of the various frequencies of interest. One of these conclusions is that the solution can be expressed in an approximate form by Bessel functions. This suggests that gain will be low at low frequencies, high at high frequencies, and will experience large variations at mid frequencies. These variations will appear as a damped pseudo-sinusoid, as depicted in Figure 4.

The other possible conclusion is that gain will increase with frequency until it is limited by: 1) departure from the assumption of adiabatic volume changes, with resulting heat generation and power loss; or, 2) scattering of reflected waves due to the irregular microstructure of the surface of the reflector. Figure 4 depicts the result of these qualitative conclusions. Computer programming of the integral equation could provide quantitative results.

Directivity of the reflector will be inversely proportional to wavelength. This can be observed from the discussion earlier where the paraboloid was approximated by a series of small flat facets.

The definition of focus, and hence the portion of the reflected power concentrated upon the microphone, is directly proportional to the diameter of the cylinders of the reflected wave components coming from the various facets. It has been shown that the effective size of each facet is determined by the wavelength; i.e., that the greatest dimension should not exceed 1/36 wavelength. Therefore, the definition of focus is inversely proportional to wavelength for a given degree of accuracy in the surface of the reflector. This means that the focal region will be larger at lower frequencies, and the reflector can be directed further away from the direction of the sound source while the microphone remains within the focal region. Conversely, at high frequencies the reflector can be very directional; i.e., slight misalignment will cause the focal region to shift entirely away from the microphone. Of course, since the parabolic curve does not focus parallel rays if they are not directed along the axis of the parabola, there will be a first order reduction in gain as the source is shifted away from the reflector axis, and a less significant decrease in directivity.

EXPERIMENTAL RESULTS

Measurements were made using a spun aluminum paraboloid supplied by the Cornell Laboratory of Ornithology. The reflective surface was painted with camouflaging colors for use in natural sound recording, and the back surface was coated with automotive body deadener. The reflector had a diameter of 36” and a focal length of 12”. The reflector was mounted on a tripod in such a manner that its direction of orientation with respect to the sound source could be gauged on a protractor. An AKG 200-ohm Studio Dynamic Microphone was mounted at the focal point with a special jig.

The sound source was a Western Electric loudspeaker system, located 15 meters from the reflector. Both reflector and speaker were about 4 feet above the terrain, which was flat and covered with grass. The speaker was driven by an HP 200C Audio Oscillator and a McIntosh 30 Amplifier. Microphone voltages were monitored with an HP 400C VTVM.

Temperature was 76˚ F, with relative humidity 80%. There was a slight irregular breeze which caused large meter fluctuations at frequencies above 4 KHz. The measured voltages are presented in Table 1, and calculated gains are tabulated in Table 2. Gain as a function of frequency is given in Curve Sheet 1, and directional gain in Curve Sheet 2.

Table 1
Microphone rms voltage in millivolts as a function of direction of incidence and of frequency. Background levels: 27 mv mike alone, and 32 mv with the reflector.
Freq.	Mike	0˚	2.5˚	5˚	10˚	20˚	30˚	45˚	90˚	180˚
80 Hz	0.49	0.58	0.56	0.56	0.56	0.50	0.56	0.54	0.50	0.50
100	0.60	0.80	0.80	0.75	0.80	0.76	0.77	0.68	0.64	0.50
200	0.42	1.40	1.46	1.20	1.15	1.15	1.10	1.00	0.56	0.79
240	0.78	2.00	2.10	2.05	2.00	1.90	1.80	1.40	0.70	1.10
300	0.88	1.65	1.60	1.50	1.50	1.55	1.30	0.88	0.51	0.73
350	0.96	1.40	1.26	1.25	1.15	1.10	0.90	0.64	0.53	0.74
400	0.62	0.88	0.80	0.81	0.67	0.60	0.53	0.32	0.38	0.51
450	0.56	0.60	0.64	0.86	0.64	0.46	0.40	0.32	0.35	0.50
500	0.51	0.75	0.82	0.85	0.90	0.68	0.52	0.32	0.32	0.44
550	0.55	1.32	1.25	1.35	1.10	0.90	0.65	0.32	0.32	0.48
600	0.56	1.25	1.50	1.40	1.30	0.91	0.70	0.32	0.32	0.42
650	0.44	1.05	1.10	1.00	1.15	1.10	0.73	0.32	0.32	0.36
700	0.40	0.90	1.05	0.60	1.00	0.64	0.43	0.32	0.32	0.34
750	0.52	1.00	1.30	0.80	1.20	1.00	0.41	0.32	0.32	0.36
800	0.52	0.95	0.40	0.68	0.60	0.75	0.36	0.32	0.32	0.34
1 KHz	0.33	0.90	0.80	1.30	0.85	0.40	0.33	0.32	0.32	0.33
1.5	0.35	4.50	6.00	5.70	3.10	0.45	0.30	0.36	0.50	0.42
2.0	0.75	9.50	8.00	8.00	5.50	1.00	1.05	0.60	0.50	0.42
4.0	0.34	2.30	2.00	2.00	0.47	0.32	0.32	0.32	0.32	0.32
8.0	0.48	5.00	2.80	2.30	0.35	0.32	0.32	0.32	0.32	0.32
10.0	0.28	1.40	1.00	1.10	0.32	0.32	0.32	0.32	0.32	0.32
15.0	0.27	1.60	0.45	0.40	0.32	0.32	0.32	0.32	0.32	0.32

Table 2
Microphone sound level in decibels above that level which was measured with the microphone alone without the reflector. A blank indicates that the level was not above the noise level.
Freq.	0˚	2.5˚	5˚	10˚	20˚	30˚	45˚	90˚	180˚
80 Hz	1.50	1.18	1.18	1.18	0.20	1.18	0.84	0.20	0.20
100	2.54	2.54	2.00	2.54	2.10	2.22	1.14	1.58	0.06
200	10.49	10.85	9.15	8.75	8.75	8.39	7.57	2.53	5.50
240	8.17	10.76	8.42	8.17	7.76	7.28	5.10	-0.91	3.00
300	5.58	5.30	4.76	4.76	5.04	3.50	0.08	-4.62	-1.50
350	3.30	2.36	2.28	1.56	1.20	-0.54	-3.54	-5.14	-2.36
400	3.06	2.24	2.38	0.68	-0.24	-1.64		-4.22	-1.64
450	0.63	1.15	1.71	1.15	-1.69	-2.89		-4.05	-0.95
500	3.38	4.16	4.46	4.96	2.52	0.19			-1.28
550	7.62	7.12	7.84	6.04	4.30	1.47			-1.16
600	6.97	8.61	7.99	7.35	4.25	1.98			-2.47
650	6.55	7.99	7.17	8.35	7.99	3.43			-1.71
700	7.06	8.36	3.54	7.98	4.06	0.64			-1.40
750	5.67	7.95	3.71	7.25	5.67	-2.07			-3.21
800	5.23	8.59	2.31	1.23	3.17	-3.21			-3.71
1 KHz	8.74	7.70	11.94	8.24	1.70	0.04			0.04
1.5	22.20	24.72	24.26	19.95	2.22	7.18	0.26	3.12	1.60
2.0	22.10	20.58	20.58	-2.68	1.54	2.92	-1.90	-3.48	-5.00
4.0	16.66	15.41	15.41	2.86
8.0	20.38	15.35	13.66	-2.70
10.0	13.99	11.08	11.90
15.0	24.48	4.50	3.46

DISCUSSION OF RESULTS

Although it would have been more desirable to make sound level measurements at higher intensities in order to suppress the background noise more, this was not possible with the available equipment. Measurements were made on a relatively quiet, still day at the Radio Astronomy Observatory on Zeman Road northeast of the Cornell Campus. This provided the greatest possible signal-to-noise ratio under the circumstances. Even so, the results in Table 2 show that many readings were not significantly above the ambient level. Considerable improvement could have been achieved by decreasing the source-reflector separation, but the large separation was used to ensure plane waves at the edge of the reflector. As a matter of fact, the 15 meter distance is quite representative of distances actually encountered between the sound source and parabola in the field of natural sound recording.

Some indication of the Bessel function type variation in gain can be gathered from Curve Sheet 1, although the variations in gain are not highly significant in light of variations encountered in making individual readings. The peak near 200 Hz may be due to resonance of the entire mechanical structure of the reflector. The wavelength at this frequency is about 5 feet, or about twice the reflector diameter. Gain becomes appreciable at frequencies above 1.5 KHz, or at wavelengths less than one-quarter of the reflector diameter. High frequency gain of the 36”/12” parabolic reflector is about 20 dB. The front-to-back ratio is about 30 dB; and the beam width is less than 20˚. Below about 1 KHz the reflector is more nearly omnidirectional and useless as a sound concentrator. These limits would be shifted according to the dimensions of the reflector, were it of a different size.

CONCLUSIONS

These results indicate that for the size of reflector used in the Library of Natural Sounds, 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 resonance at 200 Hz, the microphone should lie outside the plane of the edge of the reflector. On the other hand, in order to reduce the pickup of sounds from the back of the reflector around the rim, the microphone should be located within the plane of the edge of the reflector. A suitable compromise would be to place the microphone at the plane of the edge. In other words, the focal length should be about one-quarter of the reflector diameter. In the analysis of recordings made with the aid of a parabolic 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 mechanism of the animal.

BIBLIOGRAPHY

Material for the theoretical discussion of this paper was prepared from notes made in Acoustics 4541-4090, taught by Prof. Ingalls in the Fall of 1962 and Spring of 1963. The text for these courses was:

Kinsler, L. E., and A. R. Frey, Fundamentals of Acoustics, 1961, Wiley, New York.

Helpful suggestions were obtained from: Prof. True McLean, Dr. P. P. Kellogg, and Prof. R. P. Agnew, all of Cornell University.

None of the $100.00 project allocation was used, as equipment was supplied by the Laboratory of Ornithology.

SOUND TRANSMISSION AND REFLECTION

PARABOLIC GEOMETRY

EXPERIMENTAL RESULTS

Table 1

Mike

Table 2

DISCUSSION OF RESULTS