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NBSIR 78-154

Techniques for the Measurement

of Acoustic Impedance of

Asphalt

P. A. Mansbach and C. I. Holmer

Acoustical Engineering Division

Center for Mechanical Engineering and

Process Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

October 1978

Prepared for U.S. Environmental Protection Agency

Office of Noise Abatement and Control

.use

* 78-1541

BBraai of Standard

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NBSIR 78-1541

6^ Cl \oo

TECHNIQUES FOR THE MEASUREMENT

OF ACOUSTIC IMPEDANCE OF

ASPHALT

P. A. Mansbach and C. I. Holmer

Acoustical Engineering Division

Center for Mechanical Engineering and

Process Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D C. 20234

October 1978

Prepared for U.S. Environmental Protection Agency

Office of Noise Abatement and Control

The Environmental Protection Agency has contributed to the development of

the data included in this report but the conclusions and recommendations

are solely the responsibility of the National Bureau of Standards.

U.S. DEPARTMENT OF COMMERCE. Juanita M. Kreps. Secretary

Dr. Sidney Harman. Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

ABSTRACT

Five techniques were used for the measurement of the very high specific

acoustic impedance of an asphalt surface. These techniques are: Impedance

Tube, Pure-Tone Traverse, Pulse-Echo, Broad-Band Cross-Correlation, and Direct

Accelerometer Measurement. These technqiues, as used in the present investigation, are described and evaluated in some detail, and the results of the

measurements are presented. Of the five techniques, the broad-band cross- correlation proved to be the most effective, and is readily capable of even

further improvement. The value of the specific acoustic admittance ratio,

pc/z, (averaged over frequencies) of the sealed asphalt surface obtained with

this technique is .007. The effects of atmospheric wind and temperature

gradients on ray propagation, and spherical wavefront corrections to planewave reflection, are derived theoretically. These refinements are necessary

to realize the full potential of the broad-band measurement technique. Effects

of the finite test surface impedance on source emission measurements are discussed. Variations in measured sound levels of the order of 1-2 dB due to

differing impedances at different test site surfaces are considered likely,

given the limited data on asphalt impedance presently available.

1
ACKNOWLEDGMENTS

The authors wish to acknowledge the contributions of Donald S. Blomquist

and Charles 0. Shoemaker for their assistance in the Sandusky field measurements, Dennis D. Toth for conducting the direct accelerometer measurements, Carl

R. Voorhees for assistance in data reduction, and other members of the Acoustical

Engineering Division for assistance both in the laboratory and through discussions

We wish to thank Dr. Peter Baade* for discussion and for the suggestion of

making the direct accelerometer measurement.

The impedance tube measurements were conducted under the direction of

Y. M. Chang,** in cooperation with International Harvester, Inc.

k

Research Division, Carrier Corp,, Syracuse,

•kit U.S. EPA Enforcement Div., Sandusky, OH

NY

ii
CONTENTS

1. INTRODUCTION ... ............ .1

1.1. Rationale

1 1.2. Definitions and Assumptions .2

1.3. Scope of Experiment and Analysis .4

1.3.1. Experiment

1.3.2. Further Analysis ... .5

2. IMPEDANCE TUBE MEASUREMENTS .6

2.1. Description of Technique 6

2.2. Advantages and Limitations .6

2.3. Experiment

2.4. Results . .7

3. PURE-TONE TRAVERSE 9

3.1. Glancing Angle Geometry • ...... .9

3.2. Description of Traverse Technique • 10

3.3. Advantages and Limitations 11

3.4. Experiment X2

3.5. Results 12

4. PULSE-ECHO MEASUREMENTS 16

4.1. Description of Experiment 16

4.2. Advantages and Limitations 16

4.3. Experiment

4.4. Results 19

5. BROAD-BAND CROSS -CORRELATION 22

5.1. Description of Technique* 22

5.1.1. Data Collection 22

5.1.2. Correlogram Features • 22

5.1.3. Extraction of Reflection Coefficient . 24

5.2. Advantages and Limitations * 24

5.3. Experiment 26

5.3.1. Configuration 26

5.3.2. Locations of Source and Microphones 30

5.4. Initial Data Reduction 30

5.5. Final Data Reduction and Results 32

5.6. Analysis of Delay Times * * 32

5.7. Sources of Error 34

5.8. Corrections for Wind Gradient 37

5.9. Corrections for Temperature Gradient 38

iii
6. DIRECT ACCELEROMETER MEASUREMENT 39

6.1. Description of Technique 39

6.2. Advantages and Limitations 39

6.3. Experiment

6 . A. Results

6.5. Validation

7. EFFECTS OF WIND AND TEMPERATURE GRADIENTS 45

7.1. Description of Effects

7.2. "Non-Effect" of Uniform Wind 45

7.3. Fermat's Principle 48

7.4. Time-of-Flight Lemma 48

7.5. Effect on Propagation Time Difference 50

7.6. Effect on Angle of Incidence

7.7. Focusing Effect

7.8. Wind Profile

8 . SPHERICAL WAVEFRONT CORRECTIONS 61

8.1. Correction Expressions 61

8.2. Application to Impedance Measurement 62

9. EFFECT OF SURFACE IMPEDANCE ON SOURCE EMISSION MEASUREMENTS ... 65

9.1. Assumptions 65

9.2. Real Admittance 65

9.3. Complex Admittance 67

10. CONCLUSIONS 69

10.1. Evaluation of Techniques 69

10.2. Results of Impedance Measurement 71

10.3. Effect of Impedance on Source Emission Measurements. . . .73

10.4. Further Possibilities 73

Appendix A. Symbols 75

B. Use of Admittance in Favor of Impedance 80

C. Derivation of Cross-Correlation Functions 81

D. Exact Ray Path Integral 85

REFERENCES

IV
LIST OF FIGURES

Page

Figure 1. Ray Diagram for Glancing Angle Geometry 3

Figure 2. Sketch of Pure- Tone Traverse .... ....... 13

Figure 3. Block Diagram for Pure-Tone Traverse . x 4

Figure 4. Block Diagram for Pulse-Echo Xg

Figure 5. Example of Cross-Correlogram . 23

Figure 6. Source and Microphone Layout - Plan View ..... 27

Figure 7. Source and Microphone Layout - Elevation ..... 28

Figure 8. Block Diagram for Broad-Band Cross-Correlation

- Data Collection .29

Figure 9. Block Diagram for Broad-Band Cross-Correlation

- Data Reduction 3 X Figure 10. Examples of Predicted Correlograms 35

Figure 11. Sketch of Direct Accelerometer Measurement .... 43

Figure 12. Block Diagram for Direct Accelerometer Measurement 42

Figure 13. Output from Direct Accelerometer Measurement . . . 43

Figure 14. Ray Geometry in Comoving Coordinates ...... .46

Figure 15. Ray Paths in the Presence of a Wind Gradient . . . 52

Figure 16, Ray Diagram for Focusing Effect ....... ...55

Figure 17. Graph of Function F(a), for Real a (Eq. 8-2c) . . . 63

v
*

.

’

'
1. INTRODUCTION

1.1. Rationale

Noise emission measurements for several kinds of equipment are frequently

conducted at large outdoor test sites® A primary example is the measurement of

noise from new medium and heavy trucks, as specified in the U.S. Environmental

Protection Agency Regulation limiting their noise emissions . [1]

In applying such a regulation, it is necessary to be assured of a high

degree of precision in the measurement procedure. Two areas of variability have

been pointed out [2] which may affect the precision of test procedures of this

type. These areas are variability between measurement sites—-including the

geometry of the test surface, unevenness in the surface, and differences in the

acoustic properties of the surface — and variations in weather conditions.

This study addresses the question of the acoustic properties of the test

site surface, with primary emphasis on the measurement of these acoustic

properties. The surfaces of concern are "hard*' test sites, specifically asphalt

or concrete. There has been very little attention in the literature to the

measurement of surface reflection properties when the normal reflection

coefficient (defined below) may be as high as 0.99. Yet even for surfaces this

hard, the reflection coefficient at glancing angles is expected to be much

less — e.g., .90 at 5° incidence —"so that the small remaining absorption ma‘y

routinely introduce differences of the order of 1-2 dB in source emission measurements performed at different test sites.

It may thus be desirable to set quantitative bounds on the acoustic

characteristics of the surface. Surface impedance measurements performed at a variety of test sites and under differing pad temperature conditions would provide some estimate of the variability of the surface impedance, and thereby

provide an estimate on the variability of source emission measurements due to

site variation. The data base thus acquired would help to specify a pad surface

impedance qualification requirement.

A clear prerequisite for both a site characterization survey and a possible

qualification requirement is the development of a practical and sufficiently

accurate procedure for measuring these surface reflection characteristics.

Numbers in square brackets identify references which are listed at the end of

this report.
1.2. Definitions and Assumptions

By analogy with optics and with underwater acoustics, it may be expected

that surface reflection will be specular (as opposed to diffuse), since the

surface roughness is small compared to the wavelengths of audible sound.

Thus, the acoustic properties of the surface may be described by a complex

reflection coefficient C , whose magnitude and phase represent the amplitude and

phase of the reflected wave relative to the incident wave. In general, C

depends both on frequency and angle of incidence. Some authors restrict £ to

plane waves. In this paper, C is defined for an omnidirectional point source above a plane by writing the sound pressure as

l ikRx ikR2 ) p - const x

j + C

r

e

, (1-1)

where R^ and R„ are the distances of the microphone from the source via the

direct and reflected ray paths, respectively (see Figure 1). C is a complex

number; its magnitude and phase will be denoted by jc j and <j> , respectively.

Note the phase convention.

A somewhat more abstract characterization of the surface is given by its

specific acoustic impedance z_„ a boundary condition defined as the (complex)

ratio of the acoustic pressure to the normal particle velocity at the surface,

expressed as a function of frequency. (For a discussion of impedance models see (e.g.) Ref. [3].) An acoustically "hard" surface corresponds to a large value of

jzj relative to pc, the specific acoustic impedance of air.

A further simplification is introduced by assuming a locally reacting

surface, that is, assuming that waves propagating within the surface itself do

not significantly affect the reflection of waves from the surface. With this

assumption, the plane wave reflection coefficient is

C

rp

sin a - g sin a + 0

( 1-2 )

where the subscript p denotes that C is strictly applicable only for plane

waves. Here a is the angle of incidlSce of the plane wave, measured from the

horizontal, and

3 = pc/z

(1-3)

is the specific acoustic admittance ratio of the surface (pc is the specific

impedance of air). (The use of 3 rather than z is discussed in Appendix B)

.

For a "hard" surface, |b! « 1. Note that, for a locally reacting surface, 3 is

2
SOURCE

4

Figure 1. Ray Diagram for Glancing Angle Geometry

(Note that is the combined length of both legs.)

3
by definition independent of the angle of incidence. The validity of this

assumption (of local reaction) may be questioned; certainly for porous

materials serious doubts have been raised [4,5].

Corrections must be applied to the above expression for C (Eq. 1-2) to

obtain C

r

, defined by Eq. (1-1), since the waves described by . (1-1) are not

in fact plane. These corrections will be evaluated in Section 8. For the

measurements of impedance performed in this study, these corrections are generally

less than 10%, and will be ignored for the time being. Thus, although C is used

(properly) in the expressions describing measured reflected amplitudes, § will

be deduced by assuming C - C . For situations in which 3 becomes comparable

to sin a, however, the spherical wavefront corrections must not be neglected.

Throughout this report, it will be assumed that C and 8 are independent of

signal amplitude; i.e., that the surface responds linearly to incident

radiation.

In this study ray diagrams such as that in Figure 1 have been used to

derive the data reduction equations. There is thus the tacit assumption that

the reflection occurs at a single point on the surface, and that the properties

of this point are being measured. In fact, as with any wave phenomenon,

reflections are not point phenomena but involve a finite extent of the surface.

The exact area of surface participating in the reflection process has not been

determinable in this study. As long as the surface itself is uniform over the

interaction area, the properties measured can be used to describe the surface.

Section 8 examines the effect of the actual wave nature of the reflection

process, provided the surface is uniform and flat.

1.3. Scope of Experiment and Analysis

1.3.1. Experiment

Five techniques were used in this study to measure the specific acoustic

impedance of a surface. These techniques have been evaluated in terms of their

utility, under field conditions, in measuring very high values of impedance.

The techniques examined are;

1) Impedance Tube

2) Pure-Tone Traverse

3) Pulse-Echo

4) Broad-Band Cross-Correlation

5) Direct Accelerometer Measurement

The principal measurement site was the asphalt test pad surface at the U.S.

Environmental Protection Agency Noise Enforcement Facility in Sandusky, Ohio.

Tests were performed at this site both before and after the asphalt had been

sealed. The direct accelerometer measurements were added to this study some time later, and these were performed in a parking lot at the National Bureau of

Standards, for evaluation of the technique.

4
In the following five sections, each of these techniques will be described,

its limitations discussed, and the measurement results presented. It should be

emphasized that it is the techniques as used in this study which are being

compared. In several instances modifications to these techniques will be

suggested, which offer the potential of improved accuracy or precision, or easier

application of these techniques to measuring the acoustic properties of asphalt

outdoors

.

1.3.2. Further Analysis

It will be seen that the broad-band technique provided the most precise

measurement of the magnitude of the specific impedance, and offers promise of

further refinement. Realization of the full potential of this measurement

technique requires consideration of effects generally assumed "small," namely

the effects of wind and temperature gradients on acoustic ray propagation, and

the effect of the spherical (nonplanar) nature of the wavefront on the sound

field at the measurement position.

As a result, these two effects have been examined in some detail in

Sections 7 and 8. Although fairly technical and theoretical in their approach,

these two sections provide the needed foundation for the analyses discussed in

earlier sections.

In Section 9 the effects of the finite specific impedance on routine

source emission measurements are briefly investigated. Errors of 1-2 dB are not unlikely, assuming a specific admittance ratio less than .01. Further

impedance measurements might indicate a lower value of the admittance ratio,

which would reduce this measurement uncertainty proportionately.

The final section summarizes the results of this study, including the

principal advantages and disadvantages of the five impedance measurement

techniques

.

5
2. IMPEDANCE TUBE MEASUREMENTS

2.1. Description of Technique

The impedance tube measurement is the conventional technique for measuring

surface impedance under normal incidence conditions [6,7]. The apparatus consists of a long cylinder with a sound source at one end. The opposite end

terminates in the sample surface to be measured. The source is excited with a pure tone, resulting in a standing wave pattern in the tube. The locations of the

minima relative to the sample surface are related to the phase change at reflect ion , and the difference in level between maximum and minxmum (the standing—wave

ratio) is determined by the attenuation at reflection. Thus

10AL/2° -1

I0AL/20 «

(2-1)

r = tt - 4-rrfD^/c > ( 2 - 2 )

where AL is the standing wave ratio in decibels, D is the distance from the

surface to the first minimum, f is the frequency or the tone, and c is the speed

of sound. The reflection coefficient has been written as

i<f> r

e (2-3)

From C^, the specific acoustic admittance ratio 8 is computed using

6 = 1-C

1+C

(2-4)

obtained from Eq. (1-2) for the case of normal incidence.

2.2 Advantages and Limitations

Clearly this technique measures the normal incidence reflection

coefficient, and it is still necessary to validate experimentally a relation

between the normal incidence data and the reflection coefficient at the glancing

angles of interest, such as is provided by the locally reacting point impedance

model

.

The use of the impedance tube to measure acoustically hard surfaces is

limited by the size of the standing-wave ratios which can be measured ac- curately. To measure a value of the specific admittance ratio jB| of .01. for

example, it is necessary to measure a standing-wave ratio of 40 dB. Yet |B| = .01 (and real) gives rise to 2 dB of attenuation when reflection occurs at a glancing angle of 5°, and thus may significantly affect the results of source emission measurements

.

6
Further, for high values of specific impedance, the phase of the deduced

specific impedance is extremely sensitive to slight errors in the measurement of

D^, the position of the minimum. Thus measurement of the phase of the specific

impedance is not reliable. Yet knowledge of this phase may be required to

deduce glancing angle reflection coefficients from the normal incidence data.

The limitations on measuring high standing wave ratios may be caused by

losses in the walls of the impedance tube, disturbance of the field in the tube

due to insertion of the microphone probe, misalignment of the tube resulting in

excitation of cross modes, and noise, both acoustical and electronic. Some of

these difficulties can be minimized; for example, narrow band filtering can effectively eliminate noise as a problem. Other limitations may be ameliorated

to varying degrees by yet greater care in the experiment, or by using more sophisticated instrumentation. Using conventional techniques, however, it was found to be difficult to measure standing wave ratios greater than about

40 d£ consistently and reliably, under field conditions.

The impedance tube is attractive in that the measurements are conducted

in an enclosed space and are unaffected by the wind.

2.3. Experiment

Impedance tube measurements were performed under the direction of Y. M.

Chang in cooperation with International Harvester, Inc., at the Sandusky test

site on August 24-25, 1976, before the asphalt had been sealed, and again on September 8-9, 1976, after sealing. These measurements are reported on in

detail in a separate document. [8]

2.4. Results

Averaged results are presented in Table 1. It is not clear whether the

data scatter represented by the standard deviations reflects measurement

imprecision or actual spatial variation of the surface specific impedance.

The experience obtained in the field confirmed the expectation that

standing wave ratios above about 40 dB could not be reliably and consistently

measured with the apparatus used. As a result, values of | 6 | much below .01

could not be reliably measured. Further, the measurement of the phase of 6 was severely limited by uncertainties in the position of the acoustic center of the

microphone probe, uncertainty in the position of the surface itself, and

imprecision in the measurement of the distance between them. The scatter in the

measurements of arg 6 is comparable to that which would be obtained from the

selection of random numbers in the range -90° to +90°.

7
Table 1

Summary of Impedance Tube Data*

Frequency (Hz)

:

125 250 500 1000 1600

Unsealed

| B | (averaged) .029 .023 .022 .029 .036

std dev .010 .006 .006 .011 .010

arg 8 (averaged) +14° + 8° +13° +27° +30°

std dev 16° 16° 25° 31° 35°

Sealed

| 6 | (averaged) .009 .008 .012 .020 .023

std dev .002 .005 .007 .008 .013

arg 8 (averaged) +21° + 8° +72° +79° +60°

std dev 45° 58° 22° 6° 43°

*Data collected and reduced under the direction of Y. M. Chang (u.s

in cooperation with International Harvester, Inc. See ref. [3],

Notes: Values are averages of measured |g| and arg 6 taken at 15

positions. Standard deviations are computed from the scatter of

the 15 individual data points.

8
3 . PURE-TONE TRAVERSE

3.1. Glancing Angle Geometry

Three of the techniques used in this study-—pure-tone traverse, pulse-echo,

and broad-band cross-correlation—measure reflection coefficients at glancing

angles directly. In this report the term ''glancing angle" refers to angles of

Incidence a less than about 20°, measured from the horizontal. This is the

range of angles of incidence typically encountered in passby test

configurations

.

Figure 1 shows a ray diagram applicable to all three glancing- incidence

techniques. A sound source at S emits a signal, and a microphone at R receives

both a direct signal and a signal reflected off the surface under test.

The following quantities are defined for the geometry of Figure 1:

H source height

h microphone height

r horizontal distance from source to microphone

2 2 1/2

R

jL f r + (H-h) ] distance travelled by direct ray (SR on Fig. 1) .2 2 1/2

R

2 “ t r + (H+h) ] distance travelled by reflected ray (SPR on Fig. 1)

<5 = R_-R. path length difference

R Z X a angle of incidence, measured from the surface.

computed:

Note that tan a = (H+h)/r. The path length difference 6 may be readily

R

<S R

- R

2 -R

x = 2Hh/r ' , (3-la)

where

r

c = (R

2

+R

l

)/2 * r[l +

(3-lb)

1

Terms smaller by another factor of 4 Hh/r h^ve been ignored in Eq. (3-lb). (For

the glancing angle geometry [a <_ 20° ] 4 Hh/r < 0.1 and these terms represent

corrections to 6 of less than 1%.

)

R

All three glancing angle techniques measure the reflection coefficient C

defined by Eq. (1-1). In all cases, the specific admittance ratio 6 is obtained

from C , using the locally reacting point impedance model Eq. (1-2), and

assuming - C (the plane wave reflection coefficient) ; thus

9
sin a (3-2) B = 1-C

r 1+C

r

Thus values of 3, obtained from reflection coefficients measured at different

angles of incidence, may be compared directly.

In fact, however, C is the quantity of physical interest, i.e., the

quantity which directly expresses the result of a source emission measurement

(see Eq. (1-1)). Thus, measurements of Cr made at the glancine angles of interest

do not depend on the validity of the point impedance model for their utility in

assessing source emission measurements made at the same angles.

3.2. Description of Traverse Technique

The pure-tone traverse technique is similar in concept to the impedance

tube, except that the reflection coefficient at glancing angles is measured

directly. [9,10]

The source is a continuous pure tone, and the microphone is moved along the

reflected ray path (PR in Fig* 1) • As for the impedance tube measurement, the

positions of the minima along the traversed path are used to compute the phase

shift at reflection, and the difference between maximum and minimum levels is a measure of the attenuation at reflection. By following the reflected ray, the

angle of incidence is kept fixed.

For the traverse, the phase shift <f> is given by

<P r = TT-27T 6 r

/X

(3-3)

where

6 is the path length difference defined in Eq, (3-1).

R

X = c/f is the wavelength of the pure tone,

f is the pure tone frequency, and

c is the speed of sound in the air.

Having first determined the phase shift $ from Eq. (3-2), the magnitude of

the reflection coefficient is determined from

10
where AL is the difference between maximum and minimum sound pressure levels.

The specific admittance ratio 6 is obtained from using Eq. (3-2).

3.3. Advantages and Limitations

Since C is measured at glancing angles, the same value of 6 results in

lower values of jcj and smaller values of AL than for the impedance tube

measurement. The absence of any enclosure, however, means that values of AL as large as could be measured in an impedance tube cannot in practice be measured

using the traverse.

Thus the major limitation of the traverse technique is again due to the

highest AL that can be measured. Under normal field conditions, with even a small amount of wind, the associated wind turbulence refracts the sound field

irregularly, causing the precise location of the minimum to move around and

producing rapid fluctuations in level at the microphone [11]. It is the authors’

experience that a wind speed as low as 8 km/hr (5 mph) would limit the measurable

AL to 20-25 dB regardless of how long one averages. This corresponds to values

of B> .015.

The traverse technique is quite accurate, however, in measuring the phase

shift <f> r

. The estimated precision of +5° results from uncertainties in the

microphone position of +3 mm. Additional uncertainties result from the

presence of wind and temperature gradients.

To the extent these gradients are known, they can be corrected for, as discussed in Section 7. Corrections are also required for the spherical

wave-front; these are derived in Section 8.

The sensitivity of the deduced phase of 3 to the measured location of the

minimum is not nearly as serious as for the impedance tube measurement, since at

glancing angles the reflection coefficient jc^j is not so close to 1.

The traverse is limited to the higher frequencies since a pressure minimum

occurs only when the path length difference is large enough: 6^>X / 2 , for = 0.

With the configuration used in this study, frequencies were limited to 1 kHz and

above

.

A major deterrent in using the traverse technique was that the traversing

apparatus used was cumbersome and difficult to set up with the required

precision. Once set up, the traverse can be used to collect data at only one angle of incidence and only one reflection point on the surface. Each change in

these parameters requires knock down and reassembly of the apparatus. Also,

noise from the traverse motor interfered with determining the location of the

minimum, let alone the measurement of the sound pressure level there.

Performing the traverse by hand, perhaps using a string to define the traverse

path, seems to be a lot easier, particularly if the precise angle of incidence

can be determined a posteriori from the measured values of h and r, without

requiring that it take on a predetermined value.

11
3.4. Experiment

The pure-tone traverse measurement was conducted at Sandusky on August 27,

1976, on unsealed asphalt. The mechanical setup is shown in Fig. 2, and a block diagram of the electronics is presented in Fig. 3.

The source transducer consisted of 2 Phillips 5060W8 speakers, one on each

end of a 75 mm long by 200 mm diameter plastic cylinder. Lab tests had shown

this source to be fairly non-directional (within +0.5 dB) within the angles

of interest, for frequencies below 2.5 kHz. The source was mounted at a height of 1.2 m (4 ft), and the traversing microphone was adjusted to be at

ground level at a point 7.6m (25 ft) from the base of the source, and to

follow the reflected ray (angle of incidence a = 9.1°) out to 15 m from the

source, with a positional accuracy of +10 mm. A fixed microphone was used to

monitor the signal level.

3.5. Results

Data were taken at three frequencies: 1000, 1600, and 2000 Hz. Raw data

and processed results are presented in Table 2. Two values are given for the

microphone height at the position of the minimum: the first is a direct measurement, and the second is inferred from the horizontal distance and the known

height vs. distance path followed by the traverse mechanism. It is not known

which measurement is the more reliable; however, the discrepancy provides an estimate of the precision which may be inferred. The phase shift angles are com- puted from the average of the two height measurements, and rounded to the nearest 5°

The final two columns are results corrected for spherical wavefronts (see Section 8)

and source directivity (2000 Hz only). These corrections affect the specific

admittance ratio by less than 15% in each case.

The measured values of AL are as large as one can expect to measure. Thus,

the deduced values of 8 may be higher than the true value, although the deduced

values appear reasonable for unsealed asphalt.

It was not possible to run the traverse measurement on the sealed asphalt,

since the advent of rain wetted the pavement at the time scheduled for this

test, and scheduling limitations precluded waiting out the weather. Phase shift

information from such a measurement would have been interesting; magnitude

information would not be expected to reflect the actual very low specific

admittance, due to the limitations in measuring AL discussed earlier.

12
SOURCE

DR! VE

MOTOR

Figure 2. Sketch of Pure -Tone Traverse

13
SOURCE MONITOR RECEIVER

Figure 3. Block Diagram for Pure-Tone Traverse

Commercial instruments are identified in order adequately to specify the

experimental procedure. In no case does such identification imply

recommendation or endorsement by the National Bureau of Standards, nor does it

imply that the equipment identified is necessarily the best available for the

purpose

.

14
Table

2.

Pure-Tone

Traverse

Data

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15
4. PULSE-ECHO MEASUREMENTS

4.1. Description of Experiment

In this technique a short pulse is emitted by the source, and the

microphone receives and displays on an oscilloscope both the directly

transmitted pulse and a reflected version. The reflected pulse is delayed and

attenuated with respect to the direct pulse; its shape may be altered as well.

The magnitude of the reflection coefficient is readily deduced from the

attenuation of the reflected pulse:

-AL/20

(4-1)

Here and A^ represent the amplitudes of the direct and reflected signal

pressures, AL is the difference in sound pressure levels, and R and R are the

path lengths of the direct and reflected rays, respectively.

The phase shift is more difficult to determine, due in part to the fact

that a short pulse cannot be a pure tone: the finite duration of the pulse

means that more than one frequency must be present. Only if there were a phase

shift at reflection which is proportional to frequency, would this be manifested

as a simple time delay in the reflected pulse in addition to that due to the nath

length geometry. Phase shifts independent of frequency are not manifested as a uniform time translation of the waveform. Rather, the different frequency

components are shifted by different times, resulting in distortion of the

waveform. Such distortion is difficult to interpret.

Similar techniques have been cited in the literature [12-17], In general,

this previous work differed from the present experiment either in that normal

incidence data was collected, or in that oblique incidence absorption data were obtained by comparing the test surface to a "hard" surface assumed to be

perfectly reflecting. In this study it is the "hard" surface itself that is to

be measured. Further, most of this previous work was performed indoors in a laboratory environment

.

4.2. Advantages and Limitations

One advantage of the pulse-echo technique is the relative ease of

relocating the source and microphone, enabling measurements to be conducted at

different angles of incidence and different locations on the test pad.

Frequency range is limited by the necessity for direct and reflected pulses to

be non-overlapping. Thus the pulse width T _ must be smaller than the path

length difference 6x : Se

^pulse

< ^2 % 2Hh/rc

(4-2)

(the last approximation is valid for glancing angles - see Figure 1).

16
Limitations in using this technique outdoors can be serious, however.

Background noise even at 20 dB below the level of the signal makes precise

reading of the amplitude using the oscilloscope very difficult. Furthermore,

attempts to filter the received signal to lower the noise level result in

broadening the pulse and causing the direct and reflected pulses to overlap.

Also, phases are altered and waveforms distorted by the filter.

Further difficulties are caused by the wind. Wind turbulence causes rapid

fluctuations in amplitude, time delay, and waveform, further complicating the

reading of data. At a minimum, the measurement must be repeated a number of

times and the results averaged; still the precision obtained in this study was limited

Signal enhancement techniques, using a large number of recorded pulses,

offer a means of substantially improving the signal to noise ratio, and hence

the precision of the attenuation measurement. Further, the greater precision so obtained in the shape of the waveform would permit comparison of the measured

shape- with shapes predicted theoretically using various assumed values of the

phase shift, and thus allow inferences to be drawn regarding the phase of the

reflection coefficient. However, this approach involves a substantial increase

in the complexity of data collection and analysis, and lies outside the scope of

the present study.

Clearly, to obtain the attentuation jc j from the amplitudes of the direct

and reflected pulses, it is necessary for tfie source to radiate the same intensity in both the direct and reflected ray directions. A known amount of source directivity may be compensated for in the data reduction, but the

accuracy of directivity corrections depends on the accuracy with which the

source orientation is known.

4.3. Experiment

Pulse-echo measurements were taken at the Sandusky test site on August 26,

1976, before the asphalt had been sealed, and again on September 8, 1976, after

sealing. Figure 1 depicts the geometry of the test; Figure 4 diagrams the

electronic setup.

The lowest frequency obtainable in a pulse of a given duration is obtained

from a single sinusoid (half a sine wave was ruled out because of loudspeaker

constraints). Also, sinusoidal shape minimizes the high frequency content of

the pulse. Thus a single sinusoid was chosen as the pulse shape. With the test

geometries used, this allowed measurement down to 1000-2000 Hz as a lower

frequency limit. Frequencies up to 4000 Hz were used. The sound source consisted of two Phillips 5060W8 speakers mounted on a cylinder, as described in

Section 3.4.

The received signal was displayed on a dual-trace oscilloscope. The signal

from the microphone was fed directly into one channel, and routed through a calibrated time delay circuit and a calibrated attenuator into the other

channel. The time delay was adjusted to retard the direct pulse so that it

would overlay the reflected pulse seen on the other channel. Then the

attenuator was adjusted until the direct pulse amplitude was reduced to

coincide, as well as possible, with that of the reflected pulse.

The time delay circuit was adjustable in 0,01 ms steps, the attenuator in

steps of 0.1 dB. (Time delays for the September 8 measurements were determined

17
SOURCE RECEIVER

Figure 4. Block Diagram for Pulse-Echo

Commercial instruments are identified in order to adequately specify the

experimental procedure. In no case does such identification imply

recommendation or endorsement by the National Bureau of Standards, nor does

Imply that the equipment identified is necessarily the best available for the

purpose

.

18
directly from the oscilloscope trace, because of a failure in the time delay

circuit .

)

4.4. Results

Results are presented in Table 3. Values of 6 at different frequencies

were not averaged, since application of the F test for equality of the means

(see e.g.. Ref. [18]) indicated significant differences in mean values of 8 as a function of frequency, at the 95% confidence level.

The ratio of signal to noise varied between 10 and 20 dB, corresponding to

amplitude ratios of 3:1 to 10:1. The signal cannot be filtered, for reasons discussed above; and the sound source was already being operated at maximum

drive level. As a result of the large noise component, it was not possible to

match the amplitudes of the direct and reflected waveforms to better than +0.5

dB (estimated), corresponding to errors in the deduced reflection coefficient of

+.05. These errors are manifested in the large scatter of the deduced values of

specific admittance ratio 6 (see Table 3). Negative values of |b|, in particular,

clearly result from inaccuracies in the measurements propagating through the

equations

.

The noise is due both to ambient acoustic noise, and to air turbulence in

the sound path. These problems could be minimized in the future by using a stronger

signal source, by limiting observations to essentially windless conditions, or by

averaging a very large number of received signal waveforms (signal enhancement).

As a further consequence of the noise, distortion in signal waveform, which

would be expected from frequency-independent phase shifts at reflection, could

not be observed. Thus no conclusions can be drawn about these phase shifts.

The time difference between direct and reflected pulses could be measured

quite accurately (estimated error of +.02 ms), and results agree with values

predicted from the geometry alone. Thus, there was no evidence for reflection

phase shifts proportional to frequency, which would be manifested as additional

time delays.

Corrections for wind and temperature gradients were not applied, since

known measurement uncertainties alone make the pulse-echo data unreliable.

Corrections for spherical wavefronts would be even smaller. Source directivity

is believed to be a serious problem, particularly at the higher frequencies.

Directivity data at 3-4 kHz were not available, but the known directivity at 2.5

kHz indicates that uncertainties in source alignment, estimated at +3°, would

make futile any attempt to correct the measurements by measuring the source directivity a posteriori .

19
Angle of

Incidence

a (deg.)

Table 3, Pulse-Echo Data

Run

Frequency

(kHz)

Measured

Attenuation

AL (dB)

Ref 1.

Coeff

.

Specific

Admittance

Ratio I 6 I

6t 6t

meas comp

(ms) (ms)

UNSEALED ASPHALT

1 17.6 4 2.5 .79 .035 2.28 2.21

3 2.0 .83 .028 2.28 2.21

2 1.8 .85 .025 2.26 2.21

1.6 1.3 .90 .016 2.26 2.21

1 0.3 1.01 -.002 2.26 2.21

2 13.5 4 4.3 .62 .055 1.18 1.12

3 3.4 .69 .043 1.18 1.12

2 1.7 .84 .020 1.18 1.12

1.6 1.0 .91 .011 1.18 1.12

1 0.8 .93 .008 1.18 1.12

3 9.1 4 4.3 .62 .037 .60 .57

3 3.2 .70 .028 .60 .57

2 0.3 .98 .002 .60 .57

Average (std dev) 4 ,042(.011)

3 ,033(.009)

2 .016 (.012)

1.6 .014 (.004)

1 ,003( .007)

SEALED ASPHALT

4 13.5 4 1.4 .87 .016 1.04 1.12

3 0.9 .92 .010 1.04 1.12

2 0.1 1.01 -.001 1.04 1.12

1.6 0.2 1.00 .000 1.04 1.12

5 13.5 4 1.8 .83 .022 1.08 1.12

3 0.9 .93 .008 1.08 1.12

2 -0.2 1.05 -.006 1.08 1.12

1.6 -0.2 1.05 -.006 1.08 1.12

6 12.6 4 1.2 .89 .013 _ 1.05

3 0.5 .97 .003 - 1.05

2 0.0 1.02 -.002 - 1.05

1.6 0.0 1.02 -.002 - 1.05

20
Rvm

7

8

9

10

11

12

Angle of Measured Ref1 . Specific

6 t St

Incidence Frequency Attenuation Coeff

.

1! 1 Admittance tneas comp

a (deg.) (kHz) AL (dB) 1 c

r

1 Ratio | 8 | (ms ) (ms)

12.6 4 0.1 1.01 -.001 - 1.05

3 0.1 1.01 -.001 — 1.05

2 0.0 1.02 -.002 - 1.05

1.6 0.0 1.02 -.002 — 1.05

9.6 4 0.6 .94 .005 .72 .80

3 1.2 .88 .011 .73 .80

2 0.6 .94 .005 .75 .80

9.6 4 0.8 .92 .007 .77 .80

3 0.4 .97 .003 .76 . SO

2 0.4 .97 .003 .75 .80

12.7 4 1.5 .86 .017 1.62 1.59

•A- +» • • 3 0.8 .93 .008 1.62 1.59

2 0.6 .95 .006 1.62 1.59

1.6 1.2 .89 .013 1.60 1.59

1 2.3 .79 .026 1.60 1.59

17 6 4 2.0 .83 .028 2.20 2.21

3 1.1 .92 .013 2.20 2.21

2 1,0 .93 .011 2.20 2.21

1.6 0.9 .94 .009 2.20 2.21

1 0.2 1.02 -.003 2.25 2.21

16 5 4 1.3 .90 .015 2.10 2.06

XU f J

3 1.1 .92 .012 2.10 2.06

2 0.6 .98 .003 2.10 2.06

1.6 0.3 1.01 -.001 2.10 2.06

1 0.1 1.04 -.006 2.10 2.06

Average (std dev) 4

3

2

1.6

1

. 014 (. 009 )

. 007 (. 005)

. 002 C . 005 )

. 002 C . 007 )

. 006 C . 018 )

21
5. Broad-Band Cross-Correlation

5.1.

Description of Technique

5.1.1.

Data Collection

This technique also uses the glancing-angle geometry depicted in Fig. 1.

The source in this case emits broad-band noise, and a second microphone is

introduced very near the source. Signals received by both the near-field and

far-field microphones are recorded simultaneously on a multichannel tape

recorder.

Subsequently, the two recorded signals are processed using a cross-cor- relation analyzer. This is a microcomputer specifically designed to compute the

cross-correlation function R (t ) of two time-varying analog signals p (t) and

p 0 (t). is defined by

T

R

12

(t) = Um i f

Pi(t)p

2

( t+t) dt

. (5_ 1}

(In practice, of course, T does not go to infinity, but it must be chosen large

enough that the error in R due to finite T is within bounds set by the

experimenter. If p

«* p , Eq. (5-1) defines the auto-correlaticn. A detailed

discussion of correlation functions, and also of cross spectral functions, may

be found (e.g.) in Bendat and Piersol[19], The output of the correlation

analyzer may be used to drive an x-y plotter to produce a correlogram, a graph

of R^ 0 versus x (Fig. 5).

Correlation techniques have been used by others to measure absorption or specific impedance (see e.g. [20,21,22]). In these earlier experiments, normal

incidence measurements were performed. In many cases, the microphone signal was correlated with the electrical signal input into the speaker. Speaker response

distorts the correlogram, and absorption must then be determined by comparison

with data from a "perfectly reflecting" surface. In the present study it is the

impedance of the "perfectly reflecting surface" itself which is to be measured.

Thus a near-field microphone was introduced to eliminate the effect of speaker

response

.

It is also possible to extract impedance data from the autocorrelation of

the far-field microphone compared to the autocorrelation of the near-field

signal. For low values of 3, this alternative procedure provides less precision

in 3 than the cross-correlation technique studied here.

5,1.2.

Correlogram Features

A typical correlogram obtained during this study is reproduced in Figure 5.

Such correlograms reveal certain information about the propagation paths of the

signal. For example, two prominent peaks may be noted, representing the direct

and reflected signals respectively. Since the far-field microphone receives the

22
NORMALIZED

CORRELATION

AMPLITUDE

1.00

42.0 43.0 44.0 45.0 46.0 47.0 48.0

LAG TIME (msec)

Figure 5. Example of Cross-Correlogram

23
same (direct) signal as the near-field microphone, but delayed by the

propagation time A, there is a strong correlation between the two received

signals when a time lag of A is introduced into the near-field signal. Thus,

R (t ) exhibits a peak at t “A. Similarly, the reflected signal introduces a second peak. Since the reflected path is longer by 6 than the direct path, the

second peak appears at a longer time delay A + 6 . Further, since the reflected

signal is attenuated, the correlation is not quite as strong and the second peak

is not as high.

5.1.3. Extraction of Reflection Coefficient

It will be assumed in the following analysis that C

^ is independent of

frequency. Removal of this restriction will be discussed later.

If the sound source were a truly random signal, i.e. 9 infinite bandwidth

white noise, then the ratio of the amplitudes of the two correlogram peaks would

equal |c |r /R», the magnitude of the reflection coefficient times a spherical

divergence racfor, at least for real values of C . (The assumption ^ = 0 is

not unreasonable - see below.) In practice, of course, the loudspeaker is

limited in frequency response at both high and low frequencies. The use of

band-limited white noise introduces oscillations in the correlogram, also seen

in Figure 5. The correlation functions for band-limited white noise, for the

experimental geometry under study, are derived in Appendix C. A further conse- quence of the oscillations introduced by the finite bandwidth of the signal is

that the oscillations resulting from one peak add co the amplitude of the other

peak. Thus the observed amplitude ratio is in fact

A

2 + l c

r

l VR

2 (5' 2) -

l + rM(«| cr n^ •

as derived in Appendix C, and not simply (C jR^/R^. Here r (6) is the

source (near-field) autocorrelation functionr at t = 5, normafized by the mean

squared value of the signal. In the absence of a reflected wave, the cross- correlation would be a replica of the source autocorrelation, shifted by the

propagation time A, and possibly attenuated in the vicinity of the peak, due to

the presence of ambient background noise. Thus it is not surprising that the

actual cross-correlation behaves like the superposition of two source autocorrelations

.

As usual, & is obtained from using Eq. (3-2).

5.2 0 Advantages and Limitations

There are a number of advantages to this broad-band cross-correlation

technique. Since several minutes of recorded data are used to perform the

correlation, the result is an average of air turbulence evolving over a period

of time, and is not sensitive to instantaneous fluctuations in the ray paths.

24
The gathering of data is relatively simple, and long samples may be taken,

using a number of different source-receiver geometries, quite inexpensively. It

is however necessary to measure the source and receiver positions accurately (+1 cm for all dimensions has been required in this study, corresponding to a precision of +.001 in 6).

Ambient noise and instrument noise are no great problem. Noise generally

does not cross-correlate with the same delay time as the signal. The principal

effect of a moderate amount of noise is to lower the correlation amplitudes by a constant factor, which is eliminated in taking the ratio of two peaks.

Additional information is contained in the correlograms . Steady wind

causes changes in the signal propagation time; thus, the location of the first

peak is a very sensitive measure of the average wind component in the direction

of propagation. On the other hand, the propagation time difference between

the two peaks is independent of uniform wind, but it is affected by wind

gradients, as discussed in section 7. By setting up the experiment to record

simultaneously signals propagating in two opposite directions, effects of horizontal wind (which are positive in one direction and negative in the other) can easily be separated from other factors such as temperature gradients uniform

over the test pad, which have the same effect regardless of the direction of

propagation.

Additional features, such as the relative amplitudes and spacing of the

smaller peaks, are affected by the phase of the reflection coefficient, and by

its frequency dependence. These effects can be studied by mathematical

modeling: deriving analytic expressions for the cross-correlation functions

using different values of the parameters (e.g,, <J> ), and plotting the resulting

correlograms for comparison with field data.

Some assumption about , the phase of the reflection coefficient, is

necessary to infer jc | from the correlogram peak heights. Eq. (5-2) above was derived assuming <p ** 0. For a very hard locally reacting surface and typical source emission measurement geometries, 4> is quite small even for a large value

of arg 3, as a result of Eq. (1-2). Thus rthe assumption <p = 0 is reasonable to use during data reduction. (Conversely, the phase of 3 cannot be accurately

determined from a, measured d> ,) r

1

A difficulty in using this technique is the need for a source which is

omnidirectional over the entire frequency range being used, at least within the

angles spanned by the direct and reflected ray paths. The difficulty of meeting

this requirement is increased when measuring in two directions at once. Although in principle a known source directivity might be accounted for, in

practice this is limited both by inaccuracy in orienting the source, and by the

variation of directivity with frequency. In addition, microphone directivity

effects must be accounted for to obtain the greatest possible precision.

Frequency dependence of 3 raises many questions. It is likely that a study

of cross-spectral density functions - the Fourier transform of cross-correlation

25
functions - would provide information on frequency dependence (see e.g., Ref

23). Computer modeling, including frequency-dependent parameters in the

model, would also provide such information. Both of these avenues lie outside

the scope of this study. It should be noted that in the absence of spectral

decomposition, it is only a frequency-averaged impedance that is measured, and

not the impedance as technically defined for a single frequency.

An additional difficulty in using this technique is the requirement for

test source emission which is "white” within its frequency range, i.e.,

radiating equal energy per unit bandwidth. The equal energy requirement was imposed to minimize the need for detailed frequency studies, to achieve the

highest resolution consistent with the usable bandwidth, and to permit analytic

solutions of the mathematical model.

An alternative of combining a broad-band cross-correlation technique with

the impedance tube apparatus, as described in Ref. [20], would eliminate wind

problems. However, the normal reflection coefficient is so much closer to 1

than the glancing angle coefficient (at least for a locally reacting surface)

that much greater precision would be required of the measurements, to achieve

comparable precision in the inferred 6. It is not known whether the necessary

precision COuld be achieved.

5.3. Experiment

5.3.1. Configuration

Broad-band recordings were made at the Sandusky test pad on September 9,

1976, after the asphalt had been sealed. (Good data could not be collected on

the previous survey - before sealing - because the original sound source did not

have sufficient power-handling capability relative to the ambient noise

conditions which were found at the site.)

The ray diagram of Fig. 1 is applicable to this experiment. Scurce/microphone geometry is indicated in Figs. 6 and 7. Two near-field and eight far-field

microphones were recorded simultaneously, providing data for a number of

source/receiver geometries, including simultaneous upwind and downwind measurements. A block diagram of the test setup is presented in Fig. 8. The sound source

for this measurement used two KEF B-110 speakers, one at each end of a plastic

cylinder 75 mm long and 200 mm in diameter. Although similar to the original

sound source which used Phillips speakers, the second source proved to be

somewhat more directional. A spectrum shaper between the noise generator and

the speakers was adjusted so that the sound radiated by the speakers was as nearly as possible "white" in the frequency band 100-2500 Hz, and rolled off

sharply outside this band. At 15.2 m (50 ft.), the overall sound pressure level

of the band-limited signal (100 Hz - 2500 Hz) was about 70 dB. Flatness in

frequency as measured using a 5 Hz bandwidth narrow-band analysis, was achieved

to within +4 dB.

26
Figure 6. Source and Microphone Layout - Plan View

Source positions (circles) and microphone positions

(squares) are numbered for identification with the

corresponding positions in Table 4.

27
NORTH

SOURCE

Figure 7. Source and Microphone Layout - Elevation

Source positions (circles) and microphone positions

(squares) are numbered for identification with the

corresponding positions in Table 4.

28
SOURCE RECEIVER

Figure 8. Block Diagram for Broad-Band Cross-Correlation - Data Collection

Commercial instruments are identified in order adequately to specify the

experimental procedure. In no case does such identification imply

recommendation or endorsement by the National Bureau of Standards, nor does it

imply that the equipment identified is necessarily the best available for the

purpose.

29
5.3.2. Locations of Source and Microphones

Nine positions of the source were recorded: three with the source 2.4 m

(8 ft) above the ground, three at 1.2 m (4 ft), and three at ground level. The

eight microphone positions include two at 2.4 m (8 ft), four at 1.2 m (4 ft),

and two at ground level (see Figs. 6 and 7). The elevated microphones were

oriented so that the normal to the diaphragm was vertical; the ground level microphones were oriented so that the direct ray from the source was at grazing

incidence. Data obtained from the ground-level locations can be useful for

validation of detailed models, including wind and temperature effects. Such detailed modeling, however, is outside the scope of this study. Measurement of

the reflection coefficient using the two main peaks in the correlogram, as described above, requires that both source and microphone be above ground level.

Thus six elevated source positions and six elevated microphone positions provided thirty-six measurements for the present analysis.

Thirty of these measurements were used to compute average values of the

specific acoustic admittance ratio 6, and standard deviations of the measured 8.

The six configurations with the smallest propagation time difference 6 were not

included in these averages. At these small values of 6, less than 0.5 ms, the source autocorrelation r (6) varies rapidly and may be large; further, accurate

source autocorrelograms were not obtained for all source positions. The

anticipated errors in B (and in peak spacing 6) introduced by these

uncertainties in r (6) were much greater than for the remaining thirty

measurements. (HaSs the other six measurements been included, the average B would not have changed, but the scatter would have been increased.)

It is implicitly assumed that the admittance of the asphalt is the same from point to point, and that the standard deviations from the 30 measurements

reflect measurement imprecision and not actual variations in the surface. It is

hoped that this assumption will be validated or rejected by future studies in

which the sources of error are more closely controlled.

5.4. Initial Data Reduction

Data reduction (correlation) was carried out using a Saicor SAI-43A 400

point correlation analyzer, in the configuration shown in Fig. 9. Initially the

correlation was carried out using 0.1 ms resolution [i.e., R(t) was computed at

increments of 0.1 ms in x] and did not require the analog delay circuit shown

in Fig. 9.

Correlation amplitudes obtained from this analysis were used to compute

values of S, using Eqs . (5-2) and (5-3). The average, value of B based on thirty

source-receiver geometries was B = .010. However, these data showed a great

deal of scatter, as reflected in the standard deviation of the thirty

measurements of B, a = .008.

Additionally, the measured values of C r were plotted as a function of

1/sin a. (These plots are not reproduced here.) Ideally, the plotted

points would lie on a straight line with a slope of 28. It is believed, however,

30
| HONEYWELL 5600C

14-CHANNEL FM

TAPE RECORDER

NEAR-FIELD I SIGNAL

j FAR-FfELD

1 SIGNAL

8&K 2606

MEASURING

AMPLIFIER

f 1 B&K 2606

MEASURING

AMPLIFIER

i

i KROHN-HSTE 3550

BAND PASS

FILTER

100-250G Hz

j KROHN-HITE 3550 | BAND PASS

FILTER

100-2500 Hz

1 , _J

EXACT 127

SIGNAL

GENERATOR

NBS-BUILT

ANALOG TIME

DELAY

10-100 ms

DUAL TRACE

OSCILLOSCOPE

SAICOR SAH3A

400 POINT CORRELATION

ANALYZER

OSCILLOSCOPE

DISPLAY

HP 7 044

A

X-Y PLOTTER

PLAYBACK & ANALYSIS

Figure 9. Block Diagram for Broad-Band Cross-Correlation

Data Reduction

Commercial instruments are identified in order adequately to specify the

experimental procedure. In no case does such identification imply

recommendation or endorsement by the National Bureau of Standards, nor does it

imply that the equipment identified is necessarily the best available for the

purpose

.

31
that the measured relationship between Cr and 1/sin a could be nonlinear, or linear but with a non-zero intercept — due perhaps to temperature gradients,

position measurement errors, or inapplicability of the point impedance model.

In this case, the points might cluster closely around a curved line, even though

the scatter of B's deduced from the straight line model was large. Such was not

the case, however: the plot of Cr vs. 1/sin a showed a great deal of scatter,

which appeared to be consistent with the scatter about the straight line model.

It was, however, noted that the peaks displayed on the correlograms were

spikey, not smooth, and it was suspected that perhaps with 0.1 ms resolution the

true peaks were being missed, resulting in errors in the measured amplitudes. By

computing a correlogram based on an analytic model, using a Tiigh-speed digital

computer, this was demonstrated to be the case. Limiting the resolution to 0.1 ms resulted in amplitudes being underestimated by varying amounts, often

several percent, leading to substantial errors in the computed values of 8.

5.5. Final Data Reduction and Results

The entire analysis was therefore rerun with a resolution of 0.02 ms. At

this resolution, errors were expected to be less than 0.3%, less than the 0.5% errors inherent in the x-y plotter and the reading of the correlograms. These

errors, in turn, when propagated through the data reduction equations, would

lead to worst-case uncertainties in the value of 8 of about .002. This was felt

to be a useful and realistic target; useful in that effects of uncertainties

in 8 on source emission measurements could then be reduced to within a few

tenths of a dB (see Section 9)

.

Results of the analysis using 0.02 ms resolution are presented in Table 4.

The average specific admittance ratio, based on thirty measurements, is

6 ** .007; the standard deviation of the thirty measurements of 8 is o = .004.

These results clearly reflect a reduction in the scatter of data. Further,

plots of C ^ vs. 1/sin a show a similar reduction in scatter; however,

substantial scatter remains.

A number of possible sources of error were considered. These can be

divided into measurement errors, errors in the model, and simplifications used

in applying the model. The first group includes errors in correlogram

resolution - as noted above these are believed to affect 8 by less than .002 - and errors in position measurement, believed to be very small (see below).

5,6, Analysis of Delay Times

To verify the accuracy of the position measurements, measured values of the

propagation time difference 5 were compared with computed values 6 = meas comp.

2Hh/r'c, where r* = r[l 4- (h -i h )/2r J. Since 5-2Hh/rc, relative errors in

position (e.g., AH/H) translate into relative errors in 6 (i.e., errors in

(6 - 6 )/6). Statistics of the relative errors were therefore meas . comp

.

examined. The average of thirty relative errors was -0.8%, with a standard

deviation of 2.9% (of the typical 6 of 1-2 ms).

The non-zero average of the thirty errors could be attributed to a cor- responding error in average sound speed c, due to an uncertainty in temperature

1

32
Table 4. Broad-Band Cross-Correlation Data

Source Mic. H h r a

6 meas

r ( 6) ss

1 r* 1 Q o Pos. Pos

.

(m) Cm) (m) (deg) (ms)

A

1 A

2 c i 1 r

1 6 p

SI M2 2.4 1.2 15.2 13.5 1.12 .72 .69 ,09 .974 .003 .002

M3 2.4 15.3 17.6 2.18 .68 .57 -.02 .885 .018 .017

M32 1.2 15.2 13.5 1.12 .70 .68 .09 ,990 .001 .002

M33 2.4 15.3 17.6 2.22 .65 .57 -.02 .925 ,012 .013

M4 1.2 21.5 9.6 .78 » 64 .55 -.14 .904 .008 .007

M5 1.2 21.5 9.6 .78 .63 ,54 -.14 .902 .009 ,007

S2 M2 2.4 1.2 16.4 12.6 1,07 .65 .63 .11 .985 .002 .000

M3 2.4 16.5 16.5 2.06 .65 .55 -.05 .896 .016 ,014

M32 1.2 16.4 12.6 1.07 .68 .68 .11 1.025 -.003 -.002

' M33 2.4 16.5 16.5 2.10 .60 .53 -.03 .927 .011 ,012

M4 1.2 17.8 11.6 1.00 .69 .63 .06 ,922 .008 .007

M5 1.2 26.2 7.9 .61 .65 ,58 .07 . 867 .009 .008

S3 M2 2.4 1.2 21.5 9.6 .78 .62 .52 -.14 .889 .010 .008

M3 2.4 21.6 12.7 1.59 ,66 .61 .06 .938 .007 .006

M32 1.2 21.5 9.6 .82 ,63 .55 -.15 .917 .007 .009

M33 2.4 21.6 12.7 1.60 .62 .58 .07 .949 .006 .007

M4 1.2 15.2 13.5 1.09 ,70 .64 .11 .920 .010 .011

M5 1.2 34.1 6.1 .51 .63 .57 .10 .890 .006 .005

S4 M2 1.2 1.2 21.5 6.5 .43 .53 .52 .02

M3 2.4 21.6 9.6 .76 .56 .51 -.10 .940 .005 .004

M32 1.2 21.5 6.5 .44 .50 ,53 .03 — M33 2.4 21.6 9.6 .80 .51 .47 -.09 .948 .004 .006

M4 1.2 15.2 9.1 .53 .67 .60 .10 .886 .010 .007

M5 1.2 34.1 5.1 .20 - - - - - -

S5 M2 1.2 1.2 16.4 8.5 .53 .69 .64 .10 .923 .006 .004

M3 2.4 16.5 12.5 1.02 .69 , 64 .03 .946 .006 .005

M32 1.2 16.4 8.5 .55 .69 ,64 .09 .925 .006 .008

M33 2.4 16.5 12.5 1.07 .67 .62 .05 .939 .007 .008

M4 1.2 17.8 7.8 .49 ,66 .61 .10 .916 .006 .004

M5 1.2 26.2 5.3 .24 .60 .48 -.11 - - -

S6 M2 1.2 1.2 15.2 9.1 .54 .70 .62 .09 .876 .010 .008

M3 2.4 15.3 13.4 1.08 .61 .54 .06 .893 .013 .012

M32 1.2 15.2 9.1 ,56 .68 .63 .08 .926 .006 .008

M3 3 2.4 15.3 13.4 1.12 .65 .60 .06 .945 .007 .008

M4 1.2 21.5 6.5 .33 .54 ,48 -.20 - - MS 1.2 21.5 6.5 .42 .59 ,51 .00 - - -

NOTE: represents inferred values of 8 after including the effects of wind.

g u does not account for wind effects.

33
of 0.5 °C. However-, c had been computed from the positions of the first

correlogram peak. Further examination showed systematic differences in

^'northerly" vs . "southerly" measurements. These differences were attributed to

wind gradients, which are discussed in detail in Section 7. Corrections to

6 due to wind gradients are also derived there. These corrections were

agg¥S&d, assuming a 1 m/s wind speed (at 1,2 m height) from the south (reasons

for this choice are discussed later).

After applying the wind corrections, the averaged errors in 6 were -0.1%;

however the standard deviation was still a comparatively large 2.7% (0.004

ms). By grouping the measurements according to source position and

(separately) according to microphone position (Table 5), any error in

measurement of H or h would show up as a systematic error in 6 in one group

compared to the other groups. Such between group variations were not

statistically significant, and it is believed that accuracy in H and h is

confirmed to within +1 cm. Effects on <5 of similar errors in r are much

smaller; effects on 6 would also be small (the importance of accuracy in r is

its use in deriving wind and temperature information from first peak positions - see below)

,

5.7. Sources of Error

Thus the measurement errors - peak height and source-receiver positions - do not account for the observed scatter in (3. However, the assumptions and

simplifications used in reducing the data may well account for these

discrepancies. These simplifications included ignoring wind and temperature

gradients, assuming zero phase shift (<f> =0), assuming C is independent of

frequency, assuming uniform source directivity, and ignoring spatial variations

of 3 and deviations of the asphalt surface from flatness. Also, the point

impedance model itself may prove inadequate.

A number of these simplifications were removed by further study, and these

will be discussed below because of their applicability to future measurements.

In fact, the source directivity was shown to be a serious problem, sufficient by

itself to account for all the observed data scatter. Differences in radiated

intensity of direct and reflected rays were several decibels for frequencies

above 2000 Hz, for some of the geometries. Even averaged over all frequencies,

differences of more than 1 decibel must be expected. Although known directivity

could have been accounted for in the model, the source orientation was not known

with sufficient accuracy to obtain any meaningful improvement. It may be noted

that the inclusion of directivity information tends to lower the deduced values

of 3, for this particular experiment.

Microphone directivity also needs to be considered in the future assuming

source directionality is reduced. It is estimated that the difference in microphone sensitivity to the direct v^. the reflected rays may be as high as 0.5 dB

at 2500 Hz, or perhaps 0.2 dB overall, for the particular microphones used,

oriented vertically as described.

The effect of assuming $ ^ = 0 was studied by generating correlograms using

the analytic expressions derived in Appendix C. Figure 10 shows three such

correlograms, for <J> = 0, -10°, and -70°. Note the changes in the "fine

structure" of the cSrrelogram, i.e., in the smaller features, resulting from a 10° change in <f> . Deviations of 10° are readily detected, and one may

conclude that <p r

= 0 + 10°. (It is noted that for the locally reacting point

impedance model, C = (sin a - B)/(sin a + 3)

,

so that given the geometry of the

experiment and ass&ning j B | < . 01

,

the predicted value of <

p

must be less than 10°

regardless of the phase of 3.) I

34
Table 5. Measured vs. Predicted Propagation Time Differences

100 (<5 - 6 ,) /6 , percent, after wind corrections

meas pred

microphone position

Standard

Source

Position

2 3 4 5 32 33 |

row average

error of

mean

,

= std dev/'/n

1 +2 0.0 0 0 -2 -1.0 -0.2 0.5

2 +4 +1.5 +5 -5 0 +0.5 +1.0 1.4

3 0 +2.0 -1 +4 0 -1.5 +0.6 0.8

-——

4 -3 -4 - - -2 — —

-3.0 0.6 — ——

5 +4 0+4 - 0 +1 +1.8 0.9

L6 -2 -2 - - -6 -2 -3.0 1.0

average +1>6 _Q 2 +0 . 8 - 0.3 - 1.6 - 0.8

of column

standard

error of 1.2 0.8 1.7 2.6 1.2 0.5

mean, = std dev/^n

Note: For higher values of 6 , the percent deviations are known to

greater precision and are so shown.

35
il

' t

\0r^^

.A S «

0r =-1O°

I

7Wp\/

1 > > >

O

O

,

..

,

„

| y

V

v

Figure 10. Examples of Predicted Correlograms

a) <j> = 0°. b) <f> = -10°. c) <f> = -70°

r r r

36
Note that the frequency dependence of C r has been totally ignored in the

above analysis. Effects of frequency dependence can be studied by further

analytic modeling, that is, by assuming a frequency dependence and generating

the corresponding correlograms ; and also by use of cross spectral density

functions, the Fourier transform of the cross-correlation functions (see

especially Refs. 20 and 23).

5.8. Corrections for VJind Gradients

Effects of wind and temperature gradients were studied in some detail. The

principal effects are derived in Section 7, using the geometrical (ray)

acoustics limit. These effects are changes in path length difference, changes

in glancing angle (thereby changing the value of 8 to be deduced from the

measured C ), and focusing effects which change the relative peak heights in the

correlogra^. Corrections for these effects were applied to the data in the hope

of reducing the scatter.

Average wind speed was derived from the position of the first correlogram

peak. This peak is located at time delay x = A, where A = (R^-R )/c'. Here

is the distance traveled by the direct ray, R^ the distance to the near-field

microphone, and c' the effective sound speed c’ = c + w cos 0 (c is the speed of

sound in still air, w is the wind speed, and 0 is the angle between the wind

direction and the direction of propagation of the sound) . Because the wind

speed varies with z, the height above the ground, a specific dependence had to

be assumed, first to reduce all data to a standard height of 1.2 m, and then for

1/A use in determining the wind gradient. A one-fourth power law, w a z , was assumed (see Section 7.8). The wind direction was taken to be from the south,

based on observation at the time of the experiment and on EPA wind data from the

date in question. This assumption was consistent with the first peak data.

With 0 defined, w was deduced from the positions of the first peaks to be 1.4

m/s; the standard deviation of these measurements, however, was 1 m/s. The

large scatter in wind data is attributed to actual variability of the wind, poor

knowledge of the wind direction, incorrect assumptions about the z-dependence

,

insufficient precision in the measurement of microphone positions, and

incomplete calibration of the analog time delay unit in the data reduction

circuitry.

Each wind measurement is itself an average over several minutes of wind

history, since the datum is inferred from a correlogram computed from several

minutes of recorded sound signal. Since the six far-field microphones provided

data simultaneously, the complete set of 30 wind measurements were grouped, each

group consisting of six simultaneous measurements. Scatter within these groups

was not significantly lower than the scatter between groups, as determined by an F-test (Ref. [18]). In other words, the variability between simultaneous

measurements (same source position, different microphones) was comparable to the

variability between successive measurements (different source positions) . Thus

the actual variability of the wind cannot account for the measurement

uncertainty. Positional errors of 2-3 cm in microphone placement could account

for these uncertainties, as could substantial departures from a dependence

on height. (Analysis of the noise data taken at ground level might well resolve

some of these questions; this additional investigation is beyond the scope of

the present study.)

37
In fact a wind speed of 0,9 m/s was actually used in applying the

corrections. This was based on the observation that a gross overestimate of

wind corrections would result in "corrected" data scattered as much as the data

without any corrections; however a large underestimate should still show

improvement (reduction) in the data scatter. Results of applying these wind

gradient corrections are also presented in Table 4, The notation gW

is used in

Table 4 to represent wind-corrected measurements. Although individual measurements of 6 were affected, the averaged specific admittance ratio is still $ « .007, with a standard deviation of .004. No improvement resulted in the plot

(not shown) of C r vs. 1/sin a either.

5.9. Corrections for Temperature Gradients

The ray acoustics corrections for a simple temperature gradient were also

applied. Average air temperature was deduced from the measured sound speed; pad

temperatures were known from on-site measurements. Differences between air and

pad temperatures ranged from 2°C to 8°C during the testing period, the pad being

always warmer. In order to underestimate rather than ^y^restimate, a change in

temperature of 1.5°C was postulated, together with a z

' dependence on height.

The effect of even this small a temperature gradient on the wind-corrected data

would be to lower the averaged specific admittance ratio to 8 * .006, leaving

the standard deviation unchanged at .004. However, the propagation time

difference 6 would then change by amounts of 0.05 ms or so, easily large enough

to be ruled out by the measured values of <5. Thus the measured propagation time

differences indicate that there was no significant temperature gradient during

this test. In other words, although a measured difference existed between pad

temperature and air temperature, the gradients implied by these differences and

extending from ground to microphone height were ruled out, based on ray acoustics

analysis. However, it is recognized that large gradients may exist in a small

boundary layer at the pad surface. In this case the ray acoustics analysis would

require more detailed examination.

Analysis of the noise data taken at ground level, and autocorrelations of

the far-field data, might provide yet additional information, especially regarding

wind gradients. Cross-correlation of filtered signals might eliminate much of the

directivity effects.

38
6. DIRECT ACCELEROMETER MEASUREMENT

6.1, Description of Technique

This technique measures the response of the asphalt itself to an applied

sound field, A pure tone source is directed at a point on the surface, the

sound pressure at the surface is picked up by a microphone, and the normal

velocity of the surface is sensed by an accelerometer rigidly mounted directly

on the surface. The specific acoustic admittance, which is by definition the

normal velocity of the air at the surface divided by the sound pressure, is

computed directly. Both magnitude and phase can be readily measured, although

calibration of the phase response of the microphone and accelerometer (as well

as the amplitude sensitivity) is required.

However, the accelerometer at best measures only the motion of the surface

material. Motion of air molecules permitted by surface roughness or porosity

cannot be detected. Thus this measured ratio of asphalt velocity to acoustic

pressure must be considered as a lower bound of the specific admittance (upper

bound of the specific impedance) of the surface as an acoustic element.

Such a lower bound is useful, however, in that it indicates a limit to the

sensitivity which may be required of other impedance measurement techniques.

Equivalently, the lower bound obtained from this measurement, together with the

sensitivity-limited values of B obtained by other techniques, may be considered

to bracket the true specific acoustic admittance ratio of the surface,

6,2, Advantages and Limitations

Since the measurements are conducted using steady pure tones, both microphone and accelerometer output can be filtered with (matched) narrow-band

filters. Thus the signal-to-noise ratio can be made so high as to be of no further concern. Moreover, since only the sound field in the immediate vicinity

of the accelerometer Is of interest, wind turbulence, source directivity,

non-planar wavefronts, and other perturbations of the propagation path are all

irrelevant.

Several potential difficulties with this procedure were resolved during the

experiment. The effect of the accelerometer cable capacitance on accelerometer

sensitivity may be substantial, but can be accounted for by proper calibration.

The accelerometer mass may affect the surface response, and the physical size of

the accelerometer may shield the asphalt from the incident signal sufficiently to

affect the measurement. Direct acoustic excitation of the accelerometer may also

occur. By repeating the experiment with two accelerometers of quite different

size and mass, and obtaining comparable results (within 10-20%), dominating effects

of size and mass were felt to be ruled out. Further, any such shielding or mass loading would generally be expected to lower the measured value of B; thus the

interpretation of this measurement as a lower bound to the. true value of B is unchanged. Direct excitation of the accelerometer was ruled out by measuring much

higher impedances on another surface. Clearly, mounting of the accelerometers

must be done with care, to effect a rigid bond with the surface.

39
6.3. Experiment

The direct accelerometer measurement was performed on the asphalt in a parking area at the National Bureau of Standards, Gaithersburg, MD. (This

experiment was added to the study well after the other techniques had been

tested at Sandusky.) The experimental setup is depicted in Fig. 11, and the

equipment configuration is presented in Fig. 12. A siren horn was used as the

sound source because of its high efficiency. (Uneven frequency response and

harmonic distortion are irrelevant, since both microphone and accelerometer

signals are filtered. Also, the frequency is swept slowly, so that the variation in sound pressure, resulting from changes in speaker efficiency with frequency, occurs slowly compared to amplifier, filter, and detector response

times.) With the siren, sound pressure levels of 100-110 dB at the surface

were easily achieved. These produced acceleration levels greater than 60 dB

re 10~ B g which, when filtered with the 5-Hz bandwidth tracking filter, were

at least 30 dB above ambient vibration levels and instrument noise at all

frequencies from 300 Hz to 2000 Hz.

A frequency scan, with automatic plotting of the specific impedance ratio, was performed under several conditions. Two different accelerometers were used:

a Bruel and Kjaer #4332, 14 mm in diameter with a mass of 30 g, and a Bolt

Beranek and Newman #501, 7 mm in diameter with a mass of 2 g. Source height was also varied from 15 cm to 1 m. The larger accelerometer was screwed onto a threaded mounting plate that had been epoxied onto the asphalt; the smaller

accelerometer was bonded to the smooth surface made by applying a few drops of

epoxy resin on the top of the asphalt.

6,4. Results

The output from any particular run shows rapid fluctuations with frequency,

anywhere from + 10% to factors of 2 in the magnitude of the specific impedance,

and fluctuations of about +10° in the phase (Figure 13). The details of these

fluctuations differ from one configuration to another (accelerometer, mounting,

source height). However, visual inspection of these fluctuating curves allows

one to estimate a smoothed response, i.e., a curve following roughly the mean of

the excursions. These smoothed data are consistent, from one configuration to

another, to within about 20%, over a frequency range from 500 Hz to 2000 Hz.

These smoothed data indicate magnitudes of the specific acoustic admittance

ratio 6 of about .00035 from 500 Hz to 1000 Hz, and gradually falling to about

.00020 by 2000 Hz 0 The phase of 6 as measured (raw data) decreases gradually

from about 0° at 500 Hz to perhaps -40° by 2000 Hz.

The phase lag of the microphone cannot be neglected. Manufacturer’s

specifications indicate a lag in response rising from 0° at 500 Hz, to about 10°

at 1000 Hz and 20° at 2000 Hz. The accelerometer introduces no such phase shift

in this frequency range; this has been confirmed by calibration relative to a known reference accelerometer. As a result, the phase in 3, after correction

for the microphone, is -60° at 2000 Hz rather than the -40° indicated by the raw

data.

40
Figure 11. Sketch of Direct Accelerometer Measurement
SOURCE RECEIVER

Figure 12. Block Diagram for Direct Accelerometer Measurement

Commercial instruments are identified in order to adequately specify the

experimental procedure. In no case does such identification imply

recommendation or endorsement by the National Bureau of Standards, nor does it

imply that the equipment identified is necessarily the best available for the

purpose.
PHASE

MAGNITUDE

\(]\

(ARG 3(

)

.00012

00019

00030

00047

00074

FREQUENCY (Hz)

Figure 13 Output from Direct Accelerometer Measurement
Cause of the rapid fluctuations with frequency is not known. One possible

cause is resonance effects within the asphalt itself (e.g., reflection from the

bottom of the asphalt layer) . Variation in the details of the fluctuations may

well be due to changes in asphalt temperature over the period of testing. Other

possible causes have been discussed, including various artifacts of the instrumentation system.

6.5. Validation

As a check on the sensitivity of this technique, the measurement was also

performed on the 50 cm thick painted concrete floor of the NBS reverberation

room. The smoothed data from these runs indicate specific admittance ratios as low as .00003, a factor of ten below those measured for asphalt. Moreover,

these values are consistent with the value of pc for the concrete, i.e., they

represent the specific admittance due to longitudinal waves within the concrete.

Variations of the smoothed data, from one configuration to another, are greater

at these low measured values of B. The rapid fluctuations with frequency are also considerably greater for the concrete. If these fluctuations in fact arise

from resonances within the slab itself, their increased size may result from

there being less damping in concrete than in asphalt.

44
7. EFFECTS OF WIND AND TEMPERATURE GRADIENTS

7.1. Description of Effects

The effects of both wind and temperature on sound propagation are manifested by changes in the speed of propagation of the sound. Thus a uniform

wind will increase the effective sound speed (shorten the propagation time) of a signal traveling with the wind, and decrease the speed of a signal traveling

against the wind. Similarly, a higher temperature increases the local speed of

sound in all directions 0

In practice, wind and temperature are rarely uniform. Generally they vary

with height above the ground. Wind, in particular, is constrained (by the

viscosity of the air) to be zero just at the surface, and increases with height.

As a result of these gradients in the wind speed and temperature, sound is

refracted as it propagates. Ray paths are no longer straight lines, and

propagation times are affected as well.

There are three principal effects of these gradients on measurements using

the glancing-angle geometry (see e.g.. Fig. 1). These are changes in the propagation time difference between direct and reflected rays; changes In the angle

of incidence of the reflected ray [which therefore changes the reflection coefficient actually observed - Eq. (1-2)]; and focusing or defocusing of the re- flected ray (i.e., differences in the refraction of "neighboring" rays resulting

in increased or decreased intensity at the microphone) . Two important "noneffects" ought to be noted: there is no significant focusing of the direct ray,

and there is no change in the propagation time difference in the presence of

uniform wind (zero gradient)

.

7.2. "Non-effect" of Uniform Wind

Although this last conclusion can be drawn from suitable limiting cases of

the wind gradient corrections, a separate derivation is of interest. Care must

be exercised because the propagation time difference is already a small quantity

relative to the total propagation time; hence neglect of quantities small

relative to the total time may not be justified. Thus, for example, having

first shown that the time difference in still air is "to first order" 5 t - 2Hh/rc, one might suggest that in the presence of a wind component w parallel to

the direction of propagation, the difference might become 6x =» 2Hh/rc } , where

now c

? = c + w is the effective speed of sound. This is incorrect, since the

reflected ray (at least) is not traveling parallel to the ground, but at a small

angle a with respect to the horizontal. Thus the speed of this ray is only c 4- w cos a. Although the difference is slight, it applies to the entire time of

propagation, and its effect on the propagation time difference cannot be

neglected.

An elegant computation of this time difference can be performed in a coordinate system comoving with the wind (Figure 14). In this frame of

reference the sound wave travels in still air, and Its behavior is fully

understood. However, the source and the microphone are moving with speed -w.

45
Figure 14. Ray Geometry in Comoving Coordinates

46
Thus in the time taken by the direct ray to travel to the microphone, the

microphone has moved by an amount -wT^. The distance traveled by the direct

signal, in the comoving frame, is

c (7-la)

= r - wT

i

when both source and microphone are at the same height; here r is the horizontal

separation of the source and microphone. In general, for source height H and

microphone height h, the distance traveled in the comoving frame is

R

1

C = V(r_wTi)^ + (H-h) 2

.

In either case, the transit time of the direct signal is

(7-lb)

T = O

c

z (7-lc)

X 1 ~ R

i /c •

Noting that wT^/r is of the same order as w/c, these equations can be solved to

give

T

i _ Vr2 + (H-hT w r 2 ? - “f + 0(w"/c 2

) (7-ld)

Similarly,

r

2 ° = V(r-wT

2

) 2 + (H+h ) 2

T

2 = R

2

/c

(7-2)

T = V^

2

+(H+h) 2 w r 22

2 c - - - + o(w /c ).

Thus

6t" = T,-Tl = j^+ (H+h)

2 _ Vrf+ (H-h)

2

+ q(w2

/c2 ( 7-3)

z l c c

’

identical to the still air case to first order in w/c.

2

0 Throughout this section terms which are second-order in wind speed (i.e. s w /c ) will be neglected. Source emission measurements are generally not

conducted in winds above 20 km/hr, corresponding to w/c of about .017.

Neglected terms will be at worst 2% of the included terms. In the present

study, w/c was consistently below .005.

47
7.3. Fermat’s Principle

Derivations of the effects of wind and temperature gradients will be based

on Fermat’s principle, that the path followed by a ray between two points is the one taking the shortest time (technically, an extremum) [ 24] . Fermat’s principle

is succinct and allows efficient derivation of results. The final path

equations are, of course, identical to those obtained by other authors (e.g.,

Morse and Xngard[25], Kriebel[26]) using brute force techniques. In addition,

Fermat’s principle leads directly to a useful "time-of-f light" lemma, that ray

path curvature may be neglected to first order in computing propagation times.

In mathematical terms, Fermat’s principle states

6T = 0 , (7-4)

where T «= Jds/c is the propagation time, and 6 signifies an infinitesimal

variation •'of the ray path, keeping its end points fixed. The path itself may be

parameterized by z(x), where z is the height above the ground and x is the

(horizontal) distance along the ground.

Wind and temperature are assumed to vary only with height, so that the

sound speed £ is a function of z only. For a wind component w(z) parallel to

the plane of propagation, c = c q

+ w(z) cos a(z), where c^ is the sound speed at

the reflecting surface, z = 0. Writing z’ = dz/dx, one gets

2 -1/2 2 2 1/2 2 1/2

cos a(z)=[l+z’ ] ; also ds = [dx + dz ] = dx[l + z’ ) For a temperature profile 0(z),

c(z) = c .V

9(z)

0( 0)" c o

1 0(.Q

2 °o 0(o)

(7-5)

2 2 (neglecting terms in [o(0)-0(z)] / [0 (0 ) ] ) so that the effects of temperature

gradients can be equally well described by using an equivalent wind profile

w

T

(z) = - lr 9(°) ~ 9 (z) . (7-6)

2 o 0(0)

T

Then c(z) = c + w(z) cos a + w (z). Only the true wind term will be examined

here; the derivation of temperature effects clearly follows the same logic. It

should be noted, however, that the ray acoustics limit which has been used may

not be valid near the surface, since temperature in particular (but also wind

speed) may change markedly within a wavelength.

7.4, Time-of-Flight Lemma

This lemma states that, in computing the propagation time of a ray to first

order in w/c, it is sufficient to integrate along the straight line (zero wind)

48
path, rather than the curved path actually followed in the presence of wind

gradients

.

This lemma is based on the following reasoning. For small values of w/e,

the change in ray path due to the wind is also, small, i.e., first order in w/c.

Since, however, the new ray path is an extremum of the time of flight T (by

Fermat's principle), small deviations from the new path - for example the old

zero-wind path - give rise to the same time of flight, to first order in w/c.

It is important to recognize that the integrand still includes the wind terms.

In other words, the wind affects the propagation time integral in two ways: it

changes the integrand itself, and it changes the path of integration. This

time-of-flight lemma allows one to neglect the change in the path of

integration, when computing first-order effects.

Formally, let the actual ray path be written as

z(x) = z q

(x) + Az

1

(x) , (7-7)

where z Q

i s the path in the absence of wind, and X = w/c<<l. (If w<0, the inequalities are reversed; the proof is otherwise identical.) Then the time of

flight integral may be represented as a function of the path of integration as T(A). Using Taylor’s theorem [27],

T(A) = T(0) + AT’ (0) + j T

2 T”00

(7-8)

for some A such that 0£A<_A, Similarly,

*

(7-9)

T' (A) = T’ (0) + f t"(A) , where also 0<_A<A.

But Fermat's principle implies that T'(A) = 0 for A = w/c. Thus Eq, (7-9) becomes

T ’(°) = -AT" (A).

(7-10)

Inserting in Eq. (7-8),

T(w/c) = T(0

)

- (w/c)tT"(?) + |T2 T"(A)

.

Since both A and A are less than w/c, these last two terms are of second order

in w/c, and can be neglected provided T" is suitably bounded. Thus T(w/c) = T(0) to first order in w/c.

49
7.5. Effect on Propagation Time Difference

Using this lemma, the time difference between the direct and reflected

signals can be computed by integrating along the zero-wind straight line paths

z(x) = z + z ' x, where z and z ’ are constants. For the glancing-angle

O ^0 oo 2 geometry, z’ << 1 and power series expansions in z ' as well as in w/c can be

® ® 4 ? applied. As mentioned earlier, however, terms up to z ’ and z

' w/c must be

retained in computing path length differences. Thus

x

,

T =

/• _ds_

. / dx V

1 + z

' 2 = 1_ f

dx V

1 + Z 7 c(z) 7 C

Q

+ w(z) cosa cc 7 +

w (z)

7^

Vl+z' 2 (7-12)

/

*L

dx (1 + 4z'

2 - Tfz'^ - w ( z ) / c ) / o o

-t/

5.

2 dx d+p' 2 - f

*

,4

> - ^jt/

2

dz

.

o o z, o

x

Writing T° for the zero-wind time, and defining

Iw

(z) - j

w(z) dz", o

(7-13)

one obtains

v T

o

1

w<

z 2>-Iw<

Z

l

) T - T —

c z o

(7-14)

For a source height H and receiver height h, and horizontal distance r, the

direct ray propagation time becomes

T

W

.. = T° - ~- [I (H)-I (h)]

dir dir 2 H-h w w

c

o (7-15a)

if H t h; or

T

W

= T° — w ( h) dir dir c c o o (7-15b)

if H = h, as obtained directly from the integral dx.

The reflected ray is computed as two straight line segments. Combining the

propagation time for the two segments, the total reflected ray propagation time

becomes

50

/
( 7 -16 )

T

W

= t° 1 r r re£l refl ‘

2 55 tynj+^Ch)].

o

The wind-corrected path delay difference is thus

2r

6t + 2 cO L

hi (H) -HI (h)

w w

2 2

H - h

(Hfh) ( 7 - 17 )

w _ w w

CT - T -1 , . refl dir

x ° , r 6x + —

;

w(h) -

1/n

Iw

(h)

(H=h)

For the frequently postulated power law w(z) « z , these expressions

become

w

6t =

. o . n r 2Hh

6t +

n+1 c u

2 2 o H -h

o , 1 r w(h)

6T

w(H) w(h)

c (H^h)

(H=h)

( 7 -18 )

For a uniform wind (zero gradient), n-x» and 6 t W

= <5t° as proved earlier.

Physically, the time delay difference arises because the reflected ray

travels closer to the ground, hence through regions of lower wind speed, than

the direct ray. It is not carried along as fast, and arrives later than it

would were the wind uniform with height. For rays traveling against the wind

(w < 0), the reflected ray is not slowed as much as the direct ray, and arrives

sooner.

7,6. Effect on Angle of Incidence

To obtain corrections for the angle of incidence and intensity of the

reflected ray, it becomes necessary to solve for the curved ray paths in the

4

presence of wind gradients (see Figure 15). In this case, terms in z' and

z* w/c can be neglected, as these will modify the already small corrections by

at worst 10% for the glancing-angle geometries with a<20°, and more typically by

only a few percent.

Fermat’s principle again provides a direct derivation. Using the

expression derived above:

51
w

SOURCE

Figure 15, Ray Paths in the Presence of a Wind Gradient

V

52
(7-19)

w(z)/c

Q

)

Variation of the integral (6T *= 0) gives rise to

(It

- lb aT

7-) (

1 +

I

Z

’ 2 - w^)/c

0) - 0 (7-20)

for the ray path equation (see any standard text on classical mechanics; e.g.,

Goldstein[10] ) . This can be integrated to obtain ~ = /C- 2w(z) /

c

dx 1 o (7-21)

where C. is a constant determined by the end points of the ray path. Thus,

choosing x = 0 at the reflection point (for the reflected ray) s and using the

physically necessary condition w(0) = 0, one sets C = z ' , and

"I o

dz

dx 2w(z)/c

o

This can be rewritten in integral form as

(7-22)

(7-23)

dz

j2. - 2w(z)/c

o o

l/n

For the case of w(z)«z * Eq, 7-23 can be integrated analytically (Appendix D)

.

The resulting expressions, for n>2, are complicated.

2 Instead, for situations in which z

* >>2w(z)/c , the integrand may be

o

2 ^ expanded as a power series in 2w(z)/c

q

z o

’

. For the truck measurement geometry

(as used in this study) , this approach is valid for wind speeds up to a t least 2 m/s (5mph). Then

o o

(7-24)

53
1 (7-25)

z (x) ft; z *x - o

c z o o T I (z( x>)

% £- w

Since Iwo/c is already of first order in w/c, its argument z(x) need onlyJ be

carried to "zeroth” order, i.e., z(x) = z 'x, o

z ' has been defined as the slope of the actual curved ray path at the

o w w

point of reflection. Equivalently, ' = tan , where a is the angle of

incidence (measured from the horizontal) in the presence of a wind profile.

In the absence of wind, the angle of incidence a,° is given by

tan a^

0 = (H 4- h)/r. Further simplification of Eq. 7-25 may thus be obtained.

since z

' = tan a + 0(w/c), and

o 1 o o o 3

tan a -a - (a^ ) / 3 = (H + h)/r + terms smaller by

(H+h) (ct^° must expressed in radians for the power series

expansions in this section.) To first order, then

z (x) ft; z’x -

o

I _(z(x))

2 w

(7-26)

In order to obtain the wind-corrected angle of incidence (arctan z 0

'), it

is necessary to know the point of incidence, which also may shift in the presence

of wind (see Fig. 15). Letting x^ represent the horizontal distance between the source and the point of incidence, and x

2 the corresponding distance between the

point of incidence and the microphone, Eq. (7-26) becomes

H = z ’x, -

o 1 c (a.°)

o 1

2 w

I .(H)

(7-27)

J o x

2

-2 Vh)

Since x. 4- x.

1 2 r and - (H + h)/r, these equations reduce to

2 , = H+h l o r

„

I (H)+I (h)

r w w

c o H+h H+h

(7-28)

or

£an - tan L + Wr\2 Iw

(H)+I

w

(h)

\ c 0 \

H+h

/ H+h (

(7-29)

54
For the case wCz^z ^ T\ this becomes

W o 5 n 1 i r tan a,

i * tan a . <1 + —- —I—

i

I n+1 c Q

lH+h

^ Hw(H)+hw(h) I (7-30)

H+h

Note in particular that, given that n>2, the precise value of n is not terribly

important in determining an approximate correction. Since

1 - C

3

s* '£ sin a. and sin a - tan a,, the multiplier in braces in Ec„(7~30)

1 + C

1 1

IT can be applied directly to the 3 obtained from impedance measurements at

glancing angles, to account for the change in angle of incidence due to wind

gradients.

7.7 Focusing Effect

An additional correction is required, however, due to focusing of the

reflected rays in the vertical plane,. Figure 16 presents a diagram. Two

"neighboring" rays are emitted, one (solid line) with slope z the other

(dashed line) with slope z

f + <5z *, from the source at height H. The first ray

impinges on the surface at

e a point away (horizontal distance), and at an angle of incidence a,. This ray is reflected, and arrives at the microphone at

height h, a (horizontal) distance r from the source, or equivalently r - x, from

the reflection point. The second ray impinges on the surface at a somewhat

different point, x + 6x

J away, and at an angle of incidence + Sa^. This ray

reaches the height of the microphone at a point r + 6r from the source. At the

microphone, however, its height is z(r), different from h by an amount 6z r

. The

distance between the two rays is Sz r cos ar , where ar is the angle of the ray

path with the horizontal. The energy contained in the emitted ray bundle 6z e

! is

also contained in the rays spanned by <5z r cos ar

. The intensity is thus proportional to (6z r cos a

r

) -1, and the pressure amplitude is proportional to the square

root of this. Thus, defining as the ratio of the amplitude in the presence of

wind to the amplitude in still air,

H = w

provided the same emission angle is used.

The derivation is straightforward but tedious,

described by

(7-31)

The first ray path is

z

x (x) =

I (a.x)

z 'x - 2 w i o c a

.

o i

(7-32)

35
w

Figure 16. Ray Diagram for Focusing Effect

56
where x » 0 is at the reflection point. On the incident side , the source is

r at x x., z “ H: the emission angle is z

1 - . On the reflected

1* e dx'x-=x.

JL dz f side, the microphone is at x * r - x , z = h; a =arctan —| . The

x=r-x. second ray path is described by

z

9 (x) = (z *+6z ')x -

l o o

c (a.+6a.)

o x x

9 I ( (a ,+6a , )x) Z W X X

(7-33)

Again x = 0 is taken at the reflection point. As a result, the source

dz | is at x = x, + 6x, , z = H; z® + <5z

9 = — . The microphone is at

1 1* e e dx x=x.; +6x„ r

x = r -(x + <$x ) , at which point the ray is at height z(x) = h + 6z^. It is

only necessary to evaluate 6z_ as a function of 6z ’. Terms which are second-order (or higher) In tfie infinitesimal quantities are always discarded

Thus the condition H = z (x ) = z (x + <$x,.) leads to XI Z X X

z ®6x1 + (x -a. x, +

o 1 1x1 c a

.

O Xr VH» 40 i

(using also z

* = tan a. - a. + a.~*/3). The definitions

o ill

dz.j dz

?

' z

’ = -—j and z * + 6z ® = -j—

I

e dx x=x., e e dx S x=x+6x. . , . ' 1 1 lead to

(7-34)

^ 1 0\ + w(H) /c

o

0t i

Z

j

6a

i

The condition h = z. (r - x.) and the definition h + 6z = z.(r - x.

^ 11 r 2 1

to

6z

r = " z o'

6x

l + (r"x !)

'

JL O

21 (h)

+ —-—J-Sa - ^ c -.

3 1 ca . 2 r Ox OX

(7-35)

6x„ ) lead

1

(7-36)

Combining,

& = r 6z r e

, i, , 2 <VH> +I

w< h» w(H)+w(h)

' "3 _

2

' c a . r o i c a . O X

(7-37)

2 where terms of order a w/c have been neglected. The factor

2 9 cos a-^ ~ 1 - a_^ /2 [see Eq.(7-3D] leads to corrections smaller by a factor of a

than the remaining terms, and will be neglected for this study. Note in

passing, however, that although still-air expressions for a ( may be used in the

terms which are already of order w/c, wind-corrected values x for a. would have to

be used in the a_, /2 term, in the expansion of cos a„.

57
It may be noted that there is also some focusing in the horizontal plane.

Rays are curved horizontally as well as vertically (i.e., their direction of

propagation in the xy-plane changes) as they pass through layers of differing

wind speed. Neighboring rays emitted at the same vertical angle z’ but at

slightly different azimuthal angles (in the xy-plane) are bent by aiffering

amounts in azimuth, arriving with greater or lesser Intensity at t^e microphone.

This effect is smaller than the vertical focusing by a factor of a . Thus, it

is comparable to other terms which have been neglected in this study, and it

will be neglected as well.

The correction factor for the reflected signal amplitude in the presence of

a wind profile is thus approximately

w

M = — ~ 1 w o

P

I <H)+I (h)

r \ w w

H+hJ cQ

(H+h)

+ ±

2 \H+h j

w(H)+w(h)

c (7-38)

For the case w(z) “z

1/n

M

w

« 1 + x2 (H+h) Ha«) n+1

w(h)

c

(7-39)

The factor Mr

, representing a change in the amplitude of the reflected

signal, is manifested as an apparent change in the reflection coefficient.

Thus, in the glancing angle measurement of reflection coefficient discussed in

earlier chapters, the quantity measured was in fact rather than just C^.

This effect of wind can be compensated for by dividing the apparent C by M to

obtain the actual C . r A similar derivation would show that focusing of the direct ray bundle is

negligible. If H f h, the direct ray path equation is similar to Eq. (7-25), but

with a constant added on the right hand side. For the special case of H = h

(a = 0), the constant C, in Eq. (7-21) becomes C, = 2w(z )/c , where z is

i i p o p

the highest (or lowest) value of z occurring in the ray trajectory. From

symmetry arguments, it is clear that z occurs at the midpoint. Using also

P

z = H + 0(w/c), one can derive

P

zP

2 r w* (H) /

c

o (7-40)

(7-41)

z(x) - H + - x(r-x) w’(H)/c

for the direct ray path when H = h. In either case, the derivation of the

focusing effect would proceed similarly to the above.

58
7.8. Wind Profile

In applying the corrections for wind gradients to the measurements

performed in this study (see e.g. , Section 5), a wind profile had to be

postulated. A power law profile w(z) was assumed and the value n = 4 was

used. The selection of n = 4 was based on some sparse data found in the

literature (Thornthwaite , [29] ) . Non-turbulent flow theory predicts n = 7 [26];

however, as noted earlier, the wind correction factors are not very sensitive to

the choice of n, for n > 2.

Using n = 4, the corrections become:

w

'« T ° +

2Hh w(H) w(h)

5 c 2 2 r n = < o H -h

—l 0

j 0

1

6 t ° + I w (h)

b c c o O

(H^h)

(H=h)

w _ o L . 4 r Hw(H)4-hw(h) tan a. = tan a. {1 + — o —— i l ] 5 /it . u\ 3 c

( (H*rh) o

(7-42)

(7-43)

M = w (H+h)

HM + (H-i, «OHl

Co

5 co j

(7-44)

The Wind speed was to oe specified at a height of 1.2 m, and corrected to other

v4i_^es of H or h using the assumed z power law. Since only three pairs of value for (H, h) were used in the broad-band experiment, the corrections were

further simplified as follows

:

6t =6t +kA^-

1 c c o o (7-45)

„ w o 1 tan a.

X = tan a . { 1 + K

X J 2 w

2 r

c (7-46)

M = 1 + K r w

2 — 3 Co (7-47)

59
where w is the wind speed parallel to the propagation plane, at a height of 1.2

m; and the constants take on the following values:

(meters) 1.2,1.

2

1.2, 2.

4

2. 4, 2.

4

(s' 1

) .20 .20 .24

(nT

2

) .135 .068 .040

(m"

2

) .034 .015 .010

The results of applying these corrections to the measurements obtained in this

study have been discussed in earlier sections.

60
8. SPHERICAL WAVEFRONT CORRECTIONS

8.1. Correction Expressions

For a flat locally reacting surface with a given specific acoustic

admittance ratio 8, the reflection coefficient for plane waves is given by

C

rp

sin a - 6 sin a + 8

(8-1)

For the case of sound emission from a localized source s the incident waves are not plane; at best they are approximately spherical, and the reflection co- efficient given by C (above) is no longer applicable. Conversely, the

specific admittance ritio 8 inferred from the measured reflection coefficient

by using Eq. 8-1 will be in error.

The problem of computing the sound field due to a point source above a plane has been addressed by a number of authors [30-34] . In deriving analytic

expressions for the sound field in the limiting case of a high impedance

boundary, these authors all make the assumption (among others) that

k(H + h) 2/r<<l. This condition requires that the observer be so far away that

the path length difference (R^ - R^) becomes a great deal smaller than one-sixth

of a wavelength. For the truck measurement geometry and typical truck noise

spectra, this condition is not met.

Thomasson[34] has derived an exact solution containing a single integral.

Using the notation of the present paper, Thomasson*s expression for the velocity

potential ip can be manipulated into the following form:

ikR. ikR2

e e

\b = + C r

4ttRx

r 4 ttR 2 (8-2a)

C

r ‘ C

rp + Hr F < a > (8-2b)

where

(8-2c)

y - sin a

C = (y-8)/(y+8) (plane wave coefficient)

rp

2(1+ By) a =

2 • kR

2 (6+y)

61
In this form F(a) corresponds to the function F defined by Ingard, and a is

inversely proportional to his numerical distance [30] . These expressions apply

for j 3

| <1 and kR

2 >>l.

For small ja, F(a) - ia/2 and the plane wave approximation is very good.

For very large a, F(a) ^ 1 - /jT/2a 4- i/rr/2a asymptotically; this is equivalent

to the expressions derived by Wenzel [31] and others. For the truck noise

measurement geometry, a_ ranges roughly from .01 to 10. The linear (small a)

approximation suffices for the lower portion of this range; neither limit is

very good in the upper portion.

F(a) can be written exactly as

(8-3)

where w here is a function (related to the error function) which is tabulated in

Abramowitz and Stegun, p. 325 [35]. F(a) for real values of a. is presented as a graph in Fig. 17.

Note that the reflection coefficient Cr given in Eq. (8-2b) does not become

zero at some small angle of incidence, as it would in the plane-wave case.

8.2. Application to Impedance Measurement

In this study 8 has been computed from measured values of Cr using the

plane-wave expression [Eq. (1-2)]. Thus, corrections due to the spherical

wavefront are required. Using F(a) << 1, Eq. (8-2b) can be solved for 6 to give

6 ^ 8

p

(1 + 2F(a)/(l+C

r

)) (8-4)

23 3 d (1 + F (a)

)

, if c a 1 . P r Here 8 is the value obtained using the plane-wave approximation. Since

P

8«y« 1 , little error is introduced by using 8 (or even any nominal value of 8)

in computing a. ^ For the pure-tone traverse measurement this correction has been included in

the tabulated results (Table 2), Values of a are .37, .27, and .22 at 1000 Hz,

1600 Hz, and 2000 Hz respectively, and the resulting corrections are less than

15%.

The broad-band measurement is not so easy to correct, since a_ and therefore

P(a) depend strongly on frequency, whereas the data are frequency averaged.

Approximate values of F(a) averaged over frequency were estimated for several

pertinent values of a’ = a/f ranging from 75 to 450:

62
Figure 17. Graph of Function F(a), for Real a (Eq. 8-2c)

63
averaged F(a)

75 .01 + .041

150 .03 + .081

300 .06 + .121

450 .10 + .151

Even the worst case (a* = 450) resulted in corrections to the deduced value

of 8 of only 10%. This is well below several other sources of error at the

present time, and has not been included in the data reduction of Section 5.

64
9. EFFECT OF SURFACE IMPEDANCE ON SOURCE EMISSION MEASUREMENTS

9.1. Assumptions

The asphalt impedance measurements conducted for this study yielded

specific admittance ratio values 6 of less than .01 for the sealed asphalt, and

.02 - .06 for the unsealed, at the EPA Sandusky test site. It is of interest,

then, to address the question of how serious is the uncertainty of an enforcement measurement, as a result of the uncertainty in surface impedance,

given that the test site surface is a sealed asphalt surface. Further, to what

extent could the measurement uncertainty be reduced, were the value of surface

impedance at a given test site known more precisely.

Several assumptions will be made for this analysis. First, the surface is

assumed to be locally reacting, so that the reflection coefficient is given by

C

r & ±

~ 6)/(Y

i + 3) ( 9- 1 )

where

y ^ - sin a_, , and a_^ is the angle of incidence (measured from the horizontal) of the reflected ray. Second, the truck passby measurement geometry

will be assumed, so that y^«l (and also for the direct ray: y

= sin a^«l).

This assumption will continue to hold for other products testeS in a similar

configuration. However, a will be assumed large enough - and g small enough - that

i

26/ Y. «i

( 9- 2 )

so that

C * 1 - 26/ Y. r i

( 9- 3 )

For specific admittance ratios of .01 and truck geometries (y,>0,l) this

assumption is valid. For sources closer to the ground - truck tires,

motorcycles, Xawnmowers - this assumption is borderline for g = .01, and becomes

invalid if g is any larger. (Additional effects of wind and temperature

gradients and of spherical wavefront corrections will not be considered in this

Section.

)

9.2. Real Admittance

For a point source, the measured signal results from the superposition of

the direct and reflected signals. The reflected signal is attenuated and

shifted in phase relative to the direct signal, both because of differences in

the path length, and because of the effect of the surface as described by the

reflection coefficient.

65
The mean-squared pressure for a pure tone is thus

2

A 2 A 2 (9-4)

p A

d + A

r + 2A

d

A

r cos * »

where A, is the amplitude of the direct signal, A of the reflected signal, and

<t> is the total phase difference between them. In

r general,

(9-5)

(the attenuation on reflection multiplied by a spherical divergence factor), and

4r = d>r + <j> 0

r r (9-6)

where <j> = 2ir(R ~ R-j^/^ -*- s t^ie P^ase change due to the path length difference,

and d> ?s that due to reflection,

rFor real values of 8, using the approximation 2B/y<<l, <j> « 0 and

S:

ss d-2(s/Y

r + |yd

1 2 ‘ Fr ) A,

so that

p

2 (B) 2(l+cosij> o

) (l-2B/y

r

+~-Yd

2 -^Y r

2

)

(9-7)

(9-8)

When cos <{> = -1, there is cancellation between direct and reflected rays, and

° 2 terms in (2$/y) become important; these terms will be neglected in general.

(The cancellation is incomplete in the presence of even a small amount of

turbulence - see, e.g.. Ref. 11.) The error, in decibels, resulting from 3 not being zero (B = 0 is the idealized measurement) is then roughly

e

6 = 10 log « - 9 B/y r (9-9)

This is independent of <j> (except when cos <J> = -1), so that the only

dependence on frequency ?s through 8. To the degree that 8 is independent of

frequency, the error in measuring a broad-band source is also given by Eq. 9-9.

For 8 = .01 and y

= .1 (the smallest angle generally encountered in truck

passby testing) e « -

r 0.9 dB, i.e., the measured level is about 1 dB less than

that which would be achieved over an infinitely hard reflecting surface. (For

the unsealed surface with 8 = .02 - .06 the approximation

28/y«l is no longer valid. It is conjectured from physical grounds that the

largest frequency-independent error possible is 3 dB.)

66
A value of .01 is believed to be an upper bound of £ for the sealed as- .It surface of the Sandusky site. A better measurement, using the experience t;a : - ec in this study, could indicate actual surface values of £ considerably small-r.

Uncertainties in measured values of noise emission due to uncertainties lr. t re- value of £ would be reduced proportionately. Similarly, demonstration that the

values of £ at several sites were very close together would limit the noise

emission measurement variability attributable to differences in site surface

However, it is not known what range of £'s might be found in comparing existing

test sites, or even in comparing different points on the same test pad. It

not even known if £<_.01 is typical for sealed asphalt in general.

If indeed £ is real, measurements performed at a particular site could lr principle be corrected for the effect of known impedance, using Eq. (9-9 above-.

In practice, however, sources (e.g., trucks) are not point sources, and ofter

radiate noise from several areas, each having its own value of y . Thus the

correction is not known unless the relative contributions of the differer.

sub-sources are known.

Assuming some realistic source distribution for truck noise emission

(excluding tires), one conjectures that corrections of -0.9 dB (for 3 = .01

from source components at lower heights are averaged with corrections :f :er. -0 . 3 dB for higher components (particularly the exhaust). Thus even comparer.,

source emission measurements at sites having £ = . 01 with measurements at sices

having £ = 0, systematic corrections would be well under 1 decibel. Or.e mav further conjecture that site to site variations in £ are in fact smaller than

this. It seems more promising to validate these conjectures on site variabil i tv

rather than to attempt to develop meaningful site corrections.

For lower and smaller sources, however, such as truck tires, nioc> r . .. : lawnmowers, Eq. (9-9)may be useful in correcting measurements from different

sites to some "standard” value of £.

9.3. Complex Admittance

The problem of £ having an imaginary part is more complicated. Eq .(9-6

must then be written as

P

2

(£) « A

d

2

^2(l+cos(<()o

+4> r )) (1-2 Re £/y

r + -

- 1 o > (9-10)

where the phase shift now includes a term <J> due to reflection, at; veil as chat

due to the path difference <f> . Again, the £rror is e

; = p (£) /p*’ (0)

If <f> is less than about tt/2 radians, this error can be expressed rough I', o

e

£ - “ /Re_i + ImJ. tan M

V Yr Y r (9-11)

67
Even this restricted case depends on frequency, since <f> Q = 2 tt(R

? -R

1

) f/c, so

that the error in a broad-band measurement depends on the source spectrum.

Moreover, the restriction <j> <tt/ 2 limits the validity of this expression to frequencies below 200-500 Hz, depending on the precise source-receiver geometry.

One result of a phase shift on reflection is to alter the frequencies at

which cancellation between direct and reflected signals occurs. If the source

has a very strong pure tone, the measured level could be altered by many decibels as a result of the phase shift on reflection. For realistic spectra,

errors of a few (2-3) decibels are quite possible. Quantitative correction of

such errors is not practical, since they would be highly sensitive to tone

frequency and source-receiver geometry, and the slightest wind or temperature

effects would alter the correction. Such pure-tone errors can be minimized by

appropriately averaging measurements taken at several different microphone distances and/or heights.

At the other extreme of a broad-band source with a flat spectrum out to

high frequencies, all frequencies are averaged, and the resulting error

becomes e =-9 Re B/y , as the case with no phase shift (B real). For

broad-bana sources in'general, it may be expected that errors at some frequencies will be partially offset by errors of the opposite sign at other

frequencies, when reporting overall A-weighted levels.

By bounding j B | and arg B, and assuming some typical source spectra,

realistic error bounds for use in field measurements may be obtained. It may be

anticipated that these errors would lie in the range of 1-2 dB for j 8 j = ,01,

Additional measurements of B, at several sites, and information on actual source

spectra, would be desirable before such an analysis is undertaken.

68
10. CONCLUSIONS

10.1. Evaluation of Techniques

Five techniques were used to measure the specific acoustic impedance of an asphalt surface. Two of these - impedance tube and direct accelerometer

measurement - measure normal impedance and require the assumption of a model

(such as the locally reacting point impedance model) to deduce reflection

coefficients at the glancing angles of incidence relevant to source emission

measurements. The other three - pure- tone traverse, pulse-echo, and broadband

cross-correlation - measure reflection coefficients at glancing angles directly.

It is after all these glancing angle reflection coefficients that must be known

in order to interpret source emission measurements. The quantity 8, as defined

in Eq. (3-2), may be a convenient way of expressing the reflection coefficient.

Only its interpretation as a specific admittance ratio, and its use to relate

reflection coefficients at different angles of incidence, depend on the

assumption of a locally reacting point impedance model.

Advantages and disadvantages of each method as used are summarized in Table

6. It is clear that further refinements are possible in all of the techniques.

Some of these refinements have been discussed in earlier chapters.

The commercially available impedance tube proved to be awkward to use,

limited in how high a value of impedance could be measured, and very inaccurate

in measuring the phase of the impedance for very hard surfaces.

The accelerometer technique on the other hand was very sensitive. However,

it measures only the mechanical part of the impedance, and ignores the effects

of porosity and surface roughness on both magnitude and phase of the acoustic

impedance. Its usefulness lies in providing an upper bound to the actual

specific impedance, while the less sensitive methods provide lower bounds.

The pure-tone traverse provided a relatively precise measurement of phase

shift on reflection, but was limited in the values of impedance that it could

measure. The apparatus used was awkward to set up; thus only one point and one angle of incidence were measured. A hand-held meter traversing a path defined

by a string would be simpler to set up, and would still provide useful phase

shift information.

The pulse-echo experiment provided no useful impedance data. Ambient noise

limited the accuracy with which signal amplitudes could be compared, although

the addition of signal enhancement techniques could be expected to alleviate

this problem. Phase shift information was entirely buried in noise; even under

better circumstances, waveform distortion would be difficult to interpret in

terms of phase shift. Speaker directivity, dependent on frequency, complicated

this measurement even further.

The broad-band cross-correlation method proved to be the most promising

technique for immediate application. It required a relatively simple setup,

permitting collection of a lot of data, including data at different angles of

69
Table

6.

Advantages

and

Disadvantages

of

Impedance

Measurement

Techniques

as

Used

in

This

Study

60 60 3 3 X X 3 CJ CJ o 3 3 3 3 c co C0 3 cO co co H CO r—1

X)

x 60 cu 60

CD Cu • 60

cu

x a; «—i

CO 3 <U

B 60 00 o CO 0)

X CJ

X 3 X 3 o 3 3 3 X X X> X o 3 X 3 3 a) 3 3 3 (U

co 3 CO X) cO 60

O X)

COX CO

x

X

a CO

X

X CO o 3 o o <u

<u t>0 3 a; CO 4J

X 3 1 .H 3 X

CO

X >4

B CJ 3 a 3 3 3 x 3 3 X 3 CU (3 X 3

CO CO 3 co <3 0> ph o (U CU <U 3 O 5 co X X! X 3. X 3 3 X cu 00 3 o a> 3 O O 3 CO cO Q o

CO

x S <3 Cu

a 3 O

X pH o a 3

4-1

u 3 B Cu CU 3 cu e X 3 0) CO X CO CO CU X CO CO

cO 3 O

x 3 (1) 3 3 a CO cu cu cu > o 3 3 3 i—) 3 X O 3 3 r-4 X X 4-i 3 CJ « -H 60 3 3 X X 60

cd •3

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O

x 3 3 CO •3

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CJ 3 3 3 O 3 O

O o

X) 3 3 X 3 »3 CO Cu 3 X 3 3 3 3 3 3 3 CD CU 3 O 3 3 3 O 3 X rH 3 a 3 CO 3 3 3 3 3 CO 3 X 3 X x X PH 3

rH X> 3 CU iH 3 O PH 3 X> 3 a X 3 X 3 X O

CU 3 3 3 3 3 •3 3 3 3 3 3 3 • 3 O O O N 3 CJ 3 x X X O O X X cr 3 3 3 60 3 X) 3 3 3 3 3 iH 3 3 cr

CU

X 3 3 CJ 3 O 3 3 X 3

CO

x 3 X 3 3 3 3 3 X <3

X 3

CO & 3 > > a 3 CJ i <3 o [5 3 X X X 3 X 3 3 3 3 o 3 Ph x 3 3 3 XI Cu 3 3 O 3 x a X i X X 3 1 3 C0 CO 3 X 3 3 3 3 3 3 3 X X rH s 3 X) O 3 3 3 O 3 S X CO 3 o X 3 3 3 3 3 G 3 C0- 3 3

CJ 3 3 3 a 3 X S 3 x X 3 3 O CD 3 x :d a *r"! 3

•3

X 3 3 X) Ph 3 XI 3 cXX 3 X •3 a O •3 3 3 3 O 3 cr 3 cr £ X X •3 3 3 3 O 3 3 O 3

X) rH 3 3 > P> CU 3 3 3 3 a3

CO 3

CO X

O

Xc Xo3

a.

CO3O 3 X 3 X

60 i—l o 60

X 3 CJ 3 3 3 3 3 3 <3 3 o 3 C O 3 X i—1 > Cu O X 3 3 3 3 X 3 3 3 X X 3

60 3 3 3 X 3 3 60 3 3 3 3 O 3 3 3 i—1 O 60 X 3 3 3 o > X X X iH X X 3 3 3 X X X 3 3 3

CO CD

X 3 t X

O

X

B 3 X

X 3 3 o

rH

a 3 •3 > X 3 3 3 3 X 3 60 3 a 3 o 3 3 3 3 PH 3 3 Cu 3 X 3 o rH 3 X 3 > 3 a 3 ?H 3 4-1 3' X 3 3 X 3 o 3 3 XI O 3 PH S X 3 3 3 X CO <4-1 O 3 X 3 X 3 o <3 cC rH 3 3 > 3 X 3 X <3 3 3 X 3 3 > 3 X

CO 3 3 X iH

O

X

X

OX 3 X

X 3 o 3 a

O

X o 3 3 Cl iH 3 <3 3 3 3 3 3 B 3 X 1 3 CJ 3 3 3

60

O CO S X 3 3 3 Cu 4J 3 3 3 3 3 3

co 3 XX o O

x 3 3 e X

O 3 3 X o O o 3 X 3 3 3 3 X 3 3 3 3 60 3 3 X 3 3 CO 3 3 3 3 3 3 3 60 X 3 3 1 X 3 3 3 X 3 3 CH X 3 3 3 B fr 3 X 3 3 r—i St 3 3 g 3 PH 3 C3 X CH CJ X 3 3 PH PH B 3 > X > iH 3 3 g 3 3 B O X 3 3 Cu 3 3 3 3 C X 3 O X X 3 o o O 3 3 g 3 3 O 3 3 3 > 3 a < s 3 cu O 3 > 3 3 CU CU 3 3 CJ 3 Cu 3 XOX3s 333 3 3 3 3 O a 3 X o 3 3 3 3 > 3 X 3 3 3 X 3 X 3 3 3 H 3 3 H 3 o a O X 3 3 CJ 3 3 X 3 O O < 3 C a 3 O 3 3 O w PQ 1 3 3 3 H i ! 3 X CJ 3 1 3 X 3 3 3 3 3 3 3 O

Cu 3 X 3 X

O 3 a X 3 3 3 O X Q X X

70
incidence and different points on the test site surface. It averages over air

turbulence, and can provide its own information on wind, temperature, and their

gradients. In the present experiment the results were limited in precision by

source directivity. It is believed that directivity errors can quite easily be

reduced to the point where values of the specific admittance ratio (reciprocal

of specific impedance ratio) as lov? as 3 = .002 can be measured with some confidence, provided that corrections are applied for wind and temperature

gradients. These corrections have been derived as part of the present study

(Chapter 7)

.

The principal drawback to the broad-band measurement is the difficulty of

extracting information on frequency dependence. Further work is required to

demonstrate the feasibility of extracting such Information from the broad-band

recordings. This avenue seems promising, and the required analysis ought to be

undertaken.

10.2 Results of Impedance Measurement

Table 7 presents a comparison of the specific admittance ratios as measured

by the various techniques, for a particular frequency (1000 Hz). (Clearly the

broad-band result is available only as a value averaged over the frequency band

100-2500 Hz.) The observed discrepancies are readily explained by the

limitations of each technique, as discussed above. Thus, for the unsealed

asphalt, the phase measurement with the impedance tuoe is not reliable because

of the sensitivity to small imprecision in the measurement of the position of

the minimum. Conversely, the measurement of | 3 • using the pure tone traverse

may well be limited by the ability to measure low values of |3| with this

tehnique. The accelerometer, of course, measures only the mechanical protlon of

the admittance, and sets a lower bound to |$j. The phase also refers only to

the mechanical portion of the specific admittance.

The most reliable values of the specific admittance ratio of the unsealed

asphalt are felt to be those obtained from the impedance tube data, j 3 j = .03 +

.01, (Data using the broad-band technique were not taken for the unsealed

asphalt due to unexpectedly high ambient noise conditions.) The more reliable

phase measurements are believed to be those obtained from the traverse, ranging

from -5° to -70°. No confidence interval has been established for this

measurement, although known imprecision in position measurement limits the

accuracy of these results to ±5°.

For the sealed asphalt, the broadband results are believed to be the most

reliable. These average to M .007 ± .004. The value .004 is the standard

deviation of the 30 measurements about the mean. Systematic effects of source directivity and temperature gradients (which might shift the mean value of 3)

have not been included. These uncertainties in systematic effects can be re- duced in future measurements by greater control of the source, and by careful

measurement of temperature gradients. The phase of Cr , deduced from comparison

of measured cross-correlograms with theoretically predicted correlograms , is

believed to be 0° ± 10°, averaged over frequency. Traverse data for the sealed

asphalt was not available for comparison due to adverse weather at the time of

the scheduled test.

71
Table 7. MEASURED SPECIFIC ADMITTANCE RATIO AT 1000 HZ

UNSEALED ASPHALT SEALED ASPHALT

|e| arg B |e| arg 3

IMPEDANCE TUBE .03 +25° .02 +79°

ACCELEROMETER — — .0003 -30°

PURE-TONE TRAVERSE .06 -70° — —

PULSE-ECHO .00 XX .01 XX

BROAD-BANlf CROSS-CORRELATION XX .007 XX

* (BROAD BAND VALUES OF &)

NOTEr XX indicates quantities which cannot be obtained using the methods of

this study. — indicates that data were not obtained during this study.

72
10.3. Effect of Impedance on Source Emission Measurements

Analysis of the effects of surface impedance on noise source emission measurements (Section 9) suggests that uncertainties of 1-2 dB including systematic

errors as high as 1 dB can be expected in typical measurements conducted on a surface with a specific admittance ratio 8 of .01. The broadband measurement

technique is certainly capable of measuring values of 8 as low as .01; and, by

controlling source directivity, reliable determinations of 8 (obtained by

averaging a number of individual measurements) as low as .002 to .005 are to be

expected. (Clearly, the "reliability" of such determinations depends also on the

validity of the model.)

Correction of source emission measurements for surface impedance effects

appears impractical. Frequency-dependent effects are far too sensitive to

instantaneous air turbulence conditions. The systematic (frequency-independent)

effects (e

g - - 9 8/sin a ) require knowledge of the source location to

determine the angle of incidence; for complex sources such as trucks, which

radiate from several areas, location of a dominant source is not well defined.

Meaningful systematic corrections based on large distributed sources would be

expected to be substantially less than 1 decibel. For small sources close to

the ground, such as truck tires, motorcycles, or lawnmowers , the corrections

would be both larger and more reliable.

For truck passby measurements, the importance in measuring the surface

impedance lies in being able to bound the uncertainty in source emission

measurements resulting from uncertainty in the acoustical characteristics of the

test site.

Additional corrections for spherical wavefronts such as have been described

in Section 8 become important only for values of the specific admittance ratio

8>.01 or for angles of incidence a<5°; both of these are considered unlikely for

source emission measurements on concrete or sealed asphalt, and for which the

microphone is at 1.2 m.

10.4. Further Possibilities

Further work on the broadband cross-correlation technique seems desirable.

By controlling source directivity, a precision of ±.002 on 8 (standard deviation

of individual measurements of 8) appears achievable. Such a careful measurement

of 8 at the Sandusky test site may well indicate that a value of 8 lower than

.01 is appropriate. The knowledge that 8 is much less than .01 would allow the

estimate of uncertainties in source emission measurements due to uncertainties

in site impedance to be reduced from the 1-2 dB suggested in the present study.

Extraction of frequency dependence from the broad-band data appears a practical possibility, based on a preliminary theoretical examination of the

problem. The analysis required to develop this technique should be performed.

Frequency dependence of impedance, together with actual source spectra, can be combined to provide a more rigorous estimate of source emission measurement

errors due to finite impedance effects. Such analysis is also desirable.

73
Surface impedance measurements ought to be performed at a variety of test

sites and under differing pad temperature conditions to gauge the variability of

the surface impedance. Measurements should also be performed at several points

on the same test pad, to evaluate the variability of the surface impedance from

point to point at a given site. The data base thus acquired would help to

specify a pad qualification requirement. It could also show whether careful

specification of pad surface construction may guarantee the required acoustical

performance without the need for a rigorous qualification test.

74
Appendix A - Symbols

Note that the phase convention used in this paper defines the time dependence

of a wave by e . Some authors use a plus sign, with resulting sign inversions

in <(> r and in arg 8. Note that A* denotes the complex conjugate of A.

a dimensionless inverse distance parameter [Eq.(8-2)]

a* a/f

A1’ A

A

d>

A

A(w)

2

r

B

c

co

c

*

C

r

C

rp

ds

amplitude of first and second correlogram peaks

pressure amplitude of direct and reflected waves

Fourier transform of p(t)

/2tt (bandwidth)

speed of sound in air

speed of sound in air at ground level

c + w cos 6 (speed of sound in wind)

oconstant of integration [Eq.(7-2l)]

reflection coefficient of test surface (at some angle of

incidence), for spherical wavefront [defined by Eq. (1-1)3

magnitude of

reflection coefficient of test surface, for plane waves

element of path length

D

1

distance from test surface to first minimum (impedance tube)

e

e error in measured source emission level as a result of test

pad surface having a specific admittance ratio of 8 rather

than 0

f frequency

f (to.+cu, ) /4tt (center frequency)

o 2 1

F(a) Ingard’s integral [Eq. (8- 2c) 1

h microphone height

75
H source height

i /IT

I (z) w

wind integral [Eq.(7-13)]

k 2tt/X (wave number) WK

3

coefficients used in wind corrections, Eqs . (7-45, 46, 47)

M

w wind correction factor for focusing effect: ratio of pressure

amplitude in the presence of wind to the amplitude in

still air [Eq.(7-31)]

n positive integer

P (acoustic) pressure

P

1 (t) ,P 2 (t) pressure time histories of two sound signals w

P acoustic pressure in the presence of wind

o

P acoustic pressure in the absence of wind

r horizontal distance between source and microphone

r' "adjusted" r [ Eq . ( 3—1) ]

r (t) ss normalized "source" (near-field) autocorrelation function,

R (t)/R (0) ss ss R

1

slant distance from, source to microphone:

R

x = [r + (H-h)

Z

] i/Z

R

2

slant distance via. ref lected ray, from source to microphone:

r

2 = [r~ + (H+h rr'

R

s c c *1 ’ R

2

distance from source to near-field microphone

R. ,R„ as measured in coordinate system comoving with the

wind (Section 7.2).

R

12

<t) cross-correlation function between p^(t) and p

2 (t).

R (t) ss "source" (near-field) autocorrelation function

sign (u) sign function: = 1 if w is positive, -1 if negative,

0 if zero.

76
t time (as a parameter)

T Propagation time (esp. Section 7); also used as length of

available time history [Eq. (5-1); Appendix C)

^pulse

Pulse width [ Eq . ( 4— 2 )

]

T

1 ,T

2 T

W

propagation time of direct and reflected rays, respectively

propagation time in the presence of wind (also T

W

, . , T

V

r r ..) ° dir refl

T° propagation time in the absence of wind (also T

0

^

, T° )

w wind speed; also specifically wind speed in the direction of

propagation

w(z) same, expressed explicitly as a function of height z T w equivalent wind profile, i.e., that wind profile which causes

the same effects as a given temperature profile [Eq.(7-6)]

w- function in Eq.(8-3), as defined in Ref. 35 - related to error function

X horizontal coordinate in plane of propagation

X

1 horizontal distance from source to reflection point

X

2

' horizontal distance from reflection point to microphone:

X

2 = r"X

l

z specific acoustic impedance

z vertical coordinate (distance above surface)

z (x) propagation path, expressed as height as a function of

distance

z' dz/dx

z o z(0)

z

’ o

2 * (0)

z p

1

value of z(x) at highest (or lowest) point on direct ray

path

z

' e dz/dx at source

9

77
a angle of incidence, measured from the horizontal. In

general, angle of the ray path with the horizontal

a (z) same, with explicit dependence on height z (in the presence

of gradients)

OL iw

a

i 0 a .X

angle of incidence, measured from the horizontal a_^= a(0)

a_^ in the presence of wind

a_^ in the absence of wind

a

rw o a ,a r r

a(z) at the microphone

in the presence, absence of wind, respectively

6 specific acoustic admittance ratio pc/z

e w

6 inferred from C and including wind effects in the

computation

6° same, not including wind effects

8

P

8 inferred from C r using the plane-wave approximation

C = C [Eq. (8-4)]

r rp

n

Y sin a V \

y for direct and reflected waves, respectively

6 propagation time difference between direct and reflected

rays, in cross-correlation configuration; same as 6 t

6 comp

value of 6 computed from source/microphone geometry, with or without wind effects.

6 meas value of 6 obtained from cross~correlogram

6 r path length difference between direct and reflected ray:

S R

- VR

1

6T (used in Fermat's principle. Section 7) - change in

propagation time T resulting from an infinitesimal change in

the propagation path, keeping the end points fixed

6X

1

infinitesimal change in x as a result of following a "neighboring*’ ray rather than the original ray. Similarly

6z 6z , 6z

W

, 6z °, 6z 6a.

e r r r o i

78
6t propagation time difference between direct and reflected

rays: <5 t = (R^-R-^/c in the absence of wind

, w _ o

6 t ,6t 6t in the presence or absence of wind, respectively

A difference in propagation time between direct rays to

4 near-field and far-field microphones, in cross-correlation

configuration; A = (R -R )/c in the absence of wind

1 s AL difference in levels between maximum and minimum (impedance

tube or traverse)

0(z) temperature profile (function of height z)

- e angle between wind direction and horizontal component of

direction of propagation of the sound signal

X wavelength

X parameter used in proof of time-of-f light lemma [Eq.(7-7)j

X is set equal to w/c to complete the proof

a, = X, X specific values of the parameter x , arising from application

of Taylor's theorem

P density of air

pc specific acoustic impedance of air

a standard deviation of a set of measurements; for n

1 7

measurements x., a =%'X^

X

i

v

1 ’ "-1

where x - 2^x^/n is the mean

*r

arg C^_: phase of reflection coefficient

phase difference between direct and reflected ray, at

microphone

*o

that part of <f> attributable to geometry:

<f> = 2 tf(R -R ) /X

o Z 1

'p

b velocity potential [Eq.(8-2a)]; p

= 34>/3t

to angular frequency: w = 2irf

w r

w

2

lower and upper limits of w for band-limited signal

79
Appendix B

Results have been expressed in terms of the specific acoustic admittance

ratio 8 (8 = pc/z) rather than its inverse t, (used in earlier reports*), for

several reasons. First, the plane-wave reflection coefficient can be written

as Cr = 1 - 28/ot + small corrections of order (8/a) 2

, for the small values of

specific admittance ratio 8 and the glancing angles a of interest. Thus, errors in the measured quantity Cr propagate linearly in computing 8, and

therefore it appears more reasonable to average the values of 8 obtained from

repeated measurement of C r rather than averaging C. Similarly, data scatter

as measured by the standard deviation of 8’s is considered to be more meaningful, again because of the linear relationship with the measured quantity Cr

. Also, since values of |

C

r [ greater than 1 arise from inaccuracies in measurement, these determinations of Cr must not be discarded when computing the mean

and measures of scatter of the data. By computing 8, rather than £ , one avoids

introducing a discontinuity at C r

= 1 (8=0), thereby allowing inclusion of

the measurements for which |cr j>l in the determination of the mean and standard

deviation of 8. Finally, theoretical analyses such as "finite impedance"

(spherical wavefront) corrections seem more naturally expressed in terms of the

small number 8

.

*Letter reports to the U.S. Environmental Protection Agency dated November 1,

1976, and December 14 , 1976 .

80
Appendix C. Derivation of Cross - Correlation Functions

The following conventions regarding Fourier representations will be

observed. To avoid confusion, the convention of discarding the imaginary part

of complex quantities will not be presumed; it must be stated explicitly.

Thus, Ae

lujt does not represent the sound pressure of a pure tone: sound

pressure must be real, whereas e“

laJt is complex. A(e~ la)t + e

lu)t

) = 2A cos wt

is real and represents a pure tone.

For arbitrary sound pressure p(t),

/

COA(oj) e~

1U)t: dm

—00

and hence

A(oj) - p(t)e

+1UJC

dt . —oo

(c-1)

(C-2)

Since p(t) is real, A(-co) = A*(w), where the asterisk denotes complex

conjugate. By changing w to -w in Eq. (C-1) and applying A(-w) = A*(m),

P(t) J

A* (w)e+la,t

(C-3)

as well. The identity

00 dt = 2tt6 (w) (C-4)

is required. With two pressure signals p^ and p 0 , their Fourier transforms

will be distinguished by A^ and A9 , respectively

t

The cross-correlation function R^C1 ) is then

T/2

r

12

(t) = t

^

dt p l^p 2^ t+T ^ » -T/2

where the limit T -> 00 will be taken freely. Writing

P

2 (t)

OO J

A^*(m)e+iu)t dm

—00

(C-6a)

OO = J

A2^ e

t -ia) e

T P

2 dm' (t+t)

—oo (C-6b)
and substituting gives

R

1 2

( T )

oo oo oo

dlu 'A

1

*(u)A

2

( M ')e'

i( “'"t" )t e" ia,

' T (C-7)

Interchanging the order of integration and performing the integration over t gives

oo oo R

12 ( t > =

%/"dcJ J

da) ' (to) A

2

(

uj

T ) 6 (to' -to)

-iw ? T (C-8)

—OO —oo

“1031

using Eq.(C-4). The to' integration gives

OO

R

1 2

( t ) = J" doj (oj) A

2

(to) e —OO

For the case of band-limited white noise from a point source,

(C-9)

A (to)

A +itoR/c

4ttR e

0 (otherwise)

(C-10)

where to. and to 9 are the lower and upper frequency limits of the noise, R is

the distance from the source, and A is a measure of source energy per unit to (A is real). As usual, frequency (Hz) is given by f - co/

2

tt

.

At the

near-field microphone, R = R . At the far-field microphone, the reflected

wave must be included: s

A

2 (w)

__A

4 Tr R

1

itoR^/c

e + C

r

itoR

2 /c

e (C-ll)

A

4 ttR_1

itoR^/c

e +

itoR

2 /c+i4> (to) e 5

where R„ and R

2 are the distance from the source and its image, respectively.

The assumption that C is independent of frequency is now applied. However,

the requirement that X(-to) = A* (to) — necessary because p(t) must be real —

forces C (-to) = C *(10 ), i.e., <}> (-to) «= -<p ( 00 ). Thus, even for C independent

of frequency, <p (w) = sign (to) x const. ^From a physical standpoint one would

consider only positive frequencies, to > 0. The spectral decomposition looks

like A(to) + A(-to) , and the reflected signal looks like |C | cos (to r

2 /c + 4> - tot),

where now <f> is independent of frequency.)

82
Substituting Eqs . (C-10) and (C-ll) into (C-9) gives

on _ ,

x 2 tt R

12

(t) "

T

f

2

a

2 _iajR

s /c

I da) —-a

- e

, 2

to R

s

ia)R /c

e +

|C

| iuR /c+i<j> \ Lli e

2 r

+ 2_tt T r

.2 -ia)R /c /. iwR, /c A s si x

2 —U)_ 4 IT R

2 s

(C-12)

iC | ia)R / c-i4> \ -ia>x

LJLL e

2 r fe

2A 1 sin ttB(A-t)

4ttR^T Jr A-t

COS 2TTf^(A-T) 4-

c 1-jl sj-. n cos f2Trf (A+6-t)+<J>

A+6-t

where now B is the bandwidth (to- - u).)/2ir; f is the center frequency

(oj^ + a) ^ ) /4tt ; A is the propagation time difference between the near-field and

far-field microphone, of the direct signal: A = (R., - R )/c; and 6 is the time

difference, (R

2 - R1 )/c, between direct and reflected signals to the far

microphone.

In computing the source autocorrelation, both p and p- are given by Eq

.

(C-10). Then

x

R (t) = ss 2A sin ttBt

4ttR

2

T

s

cos 2 tt £ t. o (C-13)

Thus, for the case 4> = 0, Eq. (C-12) can be rewritten as

R

12

(t) -

j

R

ss

(4-^> + l C

ri r

2

R

ss (A+S_t)

|

•

The ratio of correlogram peaks is thus

(C-14)

4 = R

l2 (A+6 ) _

r ss

(<S) + l C

r l R

l /R2

l l ’ R

12

(A) ' 1 + r ss

(,5) iql Rl/^

(C-15)

83
where r (x) = R (x)/R (0) is the normalized source autocorrelation function, s s s s s s and the identity R

gs

(-x) = R (x), which follows directly from Eq . (C-5) if p

p 9 , has been used.

S 'S bS ^

In generating correlograms , both by using the cross-correlation analyzer

and by modeling on a high-speed computer, the normalized cross-correlation function

has been used. The normalization is performed with respect to the

root-mean-squared signal amplitudes

:

r

i2 (t) = R

12 ( t >

’/K1,1<°) R

22 <0)

(C-16)

where p^ (rms) = /R^, (0) follows from Eq. (C-5). Thus the actual sound pressure

level is factored out of the normalized correlograms. In the presence of noise,

the rms amplitudes increase, but R.^( T )> for x near A, is unchanged. Thus the

amplitude of r^^) decreases, but its shape (and relative peak heights) remains

the same

.

84
Appendix D. Exact Ray Path integral

From Eq.(7-23), the ray path in the presence of wind is described by

x

dz

2w(z)

/

c

q (D-l)

For the case w(z) this can be integrated analytically to give

x

(-l)

r (n-1)

!

(2r+l)r! (n-l-r)

!

(D-2)

(setting y

= z

1 / 11 and using Integral 134 in Ref. 36).
Numbered List of References

1. Environmental Protection Agency Noise Emission Standards for New Medium

and Heavy Duty Trucks, U.S. Federal Register , 41, Section 205.54, p. 15544

(April 13, 1976).

2. Holmer, C.I., and Kilmer, R.D., A Proposed Experiment to Evaluate

Environmental Dependence of the Measurement of Noise from Medium and

Heavy Duty Trucks, NBS Report NBSIR 76-1138 (National Bureau of

Standards, September 1976).

3. Morse, P.M., and Ingard, U., Tehoretical Acoustics, pp. 259-270 (McGraw-Hill,

Inc., New York, 1968).

4. Shaw, E.A.G., The Acoustic Waveguide. II. Some Specific Normal Acoustic

Impedance Measurements of Typical Porous Surfaces with Respect to

Normally and Obliquely Incident Waves, J. Acoust. Soc . Am. 2_5 (2), 231-235

(March 1953) .

5. Attenborough, K. , The Prediction of Oblique-Incidence Behavious of

Fibrous Absorbents, J. Sound Vib. 14_ (2), 183-191 (1971).

6. Beranek, L.L., Acoustic Measurements, pp. 309-325 (John Wiley & Sons, Inc.,

New York, 1949)

.

7. Standard Method of Test for Impedance and Absorption of Acoustical Materials

by the Tube Method, ASTM C384-58 (R1972) , in 1977 Annual Book of ASTM

Standards, Part 18 (American Society for Testing and Materials, Philadelphia,

1977).

8. Chang, Y.M., The Acoustic Absorption Coefficients and Impedances of the

Test Pads of the U.S. Environmental Protection Agency Noise Enforcement

Facility, Sandusky, Ohio (Draft), EPA report 336/9-77-001 (U.S. Environmental

Protection Agency Noise Enforcement Facility, Sandusky, Ohio, October 1976).

9. Embleton, T.F.W., Piercy, J.E., and Olson, N. , Outdoor Sound Propagation

Over Ground of Finite Impedance, J. Acoust. Soc. Am. 59_ (2), 267-277 (Feb.

1976) .

10. Jaeger, E.W., Oblique-Angle Determination of the Reflection Coefficient R

of Flat Sound Absorbers, J. Acoust. Soc. Am. j^4 (1), 280(A) (July 1973).

11. Daigle, G.A., Piercy, J.E., and Embleton, T.F.W., Degraded Interference

between Directional Ground-Reflected Sound Waves Due to Atmospheric

Turbulence, J. Acoust. Soc. Am. &L, Suppl. 1, p. S14(A) (Spring, 1977).

86
12. Gatley, W.S., and Cohen, R. , Methods for Evaluating the Performance of

Small Acoustic Filters, J. Acous. Soc. Am. 46_, 6-16 (1969).

13. Spandock, F. , Experimentelle Untersuchung der akustischen Eigenschaf ten

von Baustoffen durch die Kurztonmethode . Ann. Physik, 5, 328-338 (1934).

14. Walker, B., and Delasso, L.P. , Integrated Pulse Technique for the

Measurement of Acoustical Absorption Techniques. Seventh International

j Congress on Acoustics, Budapest, 1971.

15. Rogers, C.L., and Watson, R.B. , Determination of Sound Absorption

Coefficients Using a Pulse Technique, J. Acous. Soc. Am. 32 (12),

1555-1558 (Dec. 1960).

16. Powell, J.G., and Van Houten, J.J., A Tone-Burst Technique of

Sound-Absorption Measurement. J. Acous. Soc. Am. 4ji

, 1299 (1970).

17. Tsau, M.C.C., and Musa, R.S., Transient and Steady State Sound Absorption

Coefficients of Fiberglass and Polyurethane Foam. Inter-Noise 72

Proceedings pp . 445-450 . (Wash., D.C., Oct. 1972).

18. Lapin, L.L., Statistics for Modern Business Decisions, p. 353-365

(Harcourt Brace Jovanovich, Inc., New York 1973).

19. Bendat , J.S., and Piersol, A.G., Random Data: Analysis and Measurement

Procedures (John Wiley & Sons, New York, 1971)

20. Seybert, A.F., and Ross, D.F., Experimental Determination of Acoustic

Properties Using a Two-Microphone Random- Excitation Technique. J.

Acous. Soc. Am. 61 (5), 1362-1370 (May 1977).

* 21. Hollin, K.A., and Jones, M.H., The Method of Sound Absorption Coefficient

in Situ by a Correlation Technique. Acustica 37(2), 103-110 (March

1977),

22. Fowweather, F. , Hollin, K.A., and Jones, M.H. , The Measurement of

Absorption Coefficient Using a Correlation Technique. Presented at

British Acoustical Society Meeting, Sept. 1971.

23. Schomer, P.D., Measurement of Sound Transmission Loss by Combining

Correlation and Fourier Techniques. J. Acous. Soc. Am. 51 (4),

1127-1141 (Apr. 1972).

24. Goldstein, H. , Classical Mechanics, pp. 231, 312 (Addiscn-Wesley

Publishing Co., Reading, Mass., 1959).

k

r

87
NBS-1 14A (REV. 7-73)

U S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLIC AH ION OR REPORT NO.

NBSIR 78-1541

2. Gov’t Accession

No.

4. TITLE AND SUBTITLE

Techniques for the Measurement of Acoustic Impedance of

Aspha 1

3. Recipient’s Accession No.

5. Publication Date October 1978

6. Performing Organization (otic

7. AUTHOR(S)

Peter A. Mansbach, Curtis I. Holmer

8 . Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

ask/Work Unit No.

11. Contract/Grant No. EPA-IAG-D6-0755

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Environmental Protection Agency

Office of Noise Abatement and Control

Crysta 1 Mall ft2 Washington, D. C. 20460

13. Type of Report & Period

Covered

Final Aug. 76-Mar. 77

14. Sponsoring Agency ( ode

15. SUPPLEMENTARY NOTES

16. ABSTRAC T (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Five techniques were used in an attempt to measure the very high acoustic impedance

of an asphalt surface. These techniques are: Impedance Tube, Pure-Tone Traverse,

Pulse-Echo, Broad-Band Cross-Correlation, and Direct Accelerometer Measurement. These

techniques are described and evaluated in some detail, and the results of the measurements are presented. Of the five techniques, the broad-band cross-correlation

proved to be the most effective, and also is capable of even further improvement. The

value of the specific acoustic admittance of the sealed asphalt surface obtained with

this technique is .007.

The effects of wind and temperature gradients on ray propagation are derived theoretic

ally, as well as spherical wavefront corrections to plane-wave reflection. These

refinements are necessary to realize the full potential of the broad-band measurement

techn i que

Effects of the finite test surface impedance on source emission measurements are discussed. Measurement uncertainties of the order of 1-2 dB due to surface impedance

are considered likely.

17. KEY WORDS (six to twelve entries; alphabetical order, capitalize only the first letter of the first key word unless a proper name seoarated bv semicolons) Acoustics: acous t i c . i mpedance of aspbalt; correlation, used to

meS^ufe ifcousp 1 Fermat's pcinciple Tn acoustics: impedance, acoustic, , measurement or; impedance, acoustic, or aspnaTt; noise measurement, truck passby, sound

propagation, in wind gradients; sound propagation over impedance surface; wind gradients

effect on sound propagation.

' 21. NO. OE PAGES

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