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NASA CR- 13 2 4 5 1

The Pennsylvania State University
The Graduate School

ABSORPTION CHARACTERISTICS OF

GLASS FIBER MATERIALS AT NORMAL

AND OBLIQUE INCIDENCE

A Thesis in

Engineering Acoustics

Barry R. Wyerman

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

June 1974

NGL 39-009-121

1^1

The Pennsylvania State University
The Graduate School

Absorption Characteristics of Glass Fiber Materials at
Normal and Oblique Incidence

A Thesis in

Engineering Acoustics

Barry R. Wyerman

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science
June 1974

Date of Signature: Signatories:

Gerhard Reethof

Alcoa Professor of Mechanical Engineering

Thesis Advisor

Jiri Tichy

Chairman of ' the Committee on Engineering

Acoustics

11
ACKNOWLEDGEMENT

The author wishes to express his gratitude to Dr. Gerhard
Reethof, Alcoa Professor of Mechanical Engineering, for his guidance
and advice during this research. Sincere appreciation is also ex-
tended to Oliver McDaniel, Research Assistant, and Dr. Jiri Tichy,
Chairman of the Committee on. Engineering Acoustics, for their assist-
ance in formulating the theoretical aspects of the study.

This research was funded by Grant No. GK-32584 from the National
Science Foundation's Engineering Division and by the National Aero-
nautics and Space Administration's Langley Research Center, whose
financial support is gratefully acknowledged.

1X1

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ii

LIST OF FIGURES v

LIST OF TABLES . . ix

NOMENCLATURE x

ABSTRACT xii

I. INTRODUCTION 1

1.1 Sound Absorbing Materials 1

1.2 Methods for Measuring the Sound Absorption of
Materials. . . ...... 2

1.3 Statement of the Problem ........ 5

1.4 Purpose of Research. ........ 5

II. MATERIALS AND THEIR PROPERTIES 7

III, THEORETICAL BACKGROUND 13

3.1 Introductory Theory 13

3.2 Beranek's Theory for Porous Materials 17

3.3 Ford's Theory for Porous Materials ......... 23

3.4 Pyett's Theory for Non-isotropic Materials ..... 28
IV. PROCEDURE AND TECHNIQUES 36

4.1 Standing Wave Tube 36

4.2 Flow Resistance 40

4.3 Surface Pressure Method 43

V. DISCUSSION OF RESULTS 55

5.1 Standing Wave Tube 55

5.2 Flow Resistance 59

5.3 Surface Pressure Method 62

Page
5.4 Theoretical Results 92

VI. CONCLUSIONS AND RECOMMENDATIONS 113

BIBLIOGRAPHY 118

APPENDIX A - NEWTON-RAPHSON ITERATION METHOD 120

APPENDIX B - INTERFERENCE PATTERN CALCULATION 123

LIST OF FIGURES

Figure Page

L Plane Wave Propagating Normal to the Surface of

Material ..„,... 16

2 Plane Wave Propagating at Oblique Incidence to the

Surface of Material. 16

3 Incremental Volume of Material 19

4 Plane Waves at Normal Incidence to a Material with

Rigid Wall Backing , . . 24

5 Plane Wave at Oblique Incidence to a Material with

Rigid Wall Backing 24

6 Plane Wave at Oblique Incidence to a Material with

Rigid Wall Backing 32

7 Wave Propagating at Oblique Incidence to Material. ... 32

8 Standing Wave Tube . 37

9 Flow Resistance Apparatus. 41

10 Surface Pressure Method Measurements .......... 46

11 Phase Relations Between Measurements with the Surface
Pressure Method. ......... 46

12 Schematic for Surface Pressure Method Tests. 49

13 Sound Source for Surface Pressure Method Tests 51

14 Mounting Arrangement for Reflecting Panel 53

15 Panel with Hard Wall Surface 54

16 Panel with Surface of Sound Absorbing Material 54

17 Specific Normal Impedance Measured by the Standing

Wave Tube for O.C. 703 Fiberglas -1.0" 56

18 Absorption Coefficient Measured by the Standing Wave

Tube for O.C. 703 Fiberglas -1.0" 56

19 Specific Normal Impedance Measured by the Standing

Wave Tube for O.C. 704 Fiberglas -1.0" 57

Figure Page

20 Absorption Coefficient Measured by the Standing

Wave Tube for O.C. 704 Fiberglas -1.0" . 57

21 Specific Normal Impedance Measured by the Standing

Wave Tube for O.C. 705 Fiberglas -1.0" 58

22 Absorption Coefficient Measured by the Standing

Wave Tube for O.C. 705 Fiberglas -1.0" 58

23 Specific Flow Resistance Vs. Flow Rate for O.C.

700 Series Fiberglas Samples 60

24 Pressure as a Function of Incident Angle at the

Surface of O.C. 705 Fiberglas -1.0" 63

25 Measured Surface Pressure "P 2 in dB Below Hard Wall
Pressure "f, for O.C. 705 Fiberglas -1.0" 65

26 Measured Phase Difference Between the Hard Wall Pressure

and Surface Pressure for O.C. 705 Fiberglas -1.0". .. . 66

27 Hard Wall Pressure as a Function of Incident Angle for
Three-Foot Square Panel 67

28 Hard Wall Pressure as a Function of Incident Angle for

Two Different Panels ..... 71

29 Hard Wall Pressure as a Function of Incident Angle with
Edges Treated ...... 72

30 Hard Wall Pressure as a Function of Incident Angle with
Corners Covered 74

31 Hard Wall Pressure as a Function of Incident Angle for a
Circular Panel ,. ., 75

32 Hard Wall Pressure as a Function of Incident Angle for

Six -Foot Square Panel 77

33 Phase Variation for Temperature Changes 79

34 Pressure Ratio in Decibels as a Function of Incident
Angle Between the Hard Wall Pressure T, and the Free

Field Pressure "^ . . . , 83

35 Phase Difference as a Function of Incident Angle Between

the Hard Wall Pressure and the Free Field Pressure ... 83

36 Absorption Coefficient at Normal Incidence Measured by
the Standing Wave Tube and by the Surface Pressure Method

for O.C, 705 Fiberglas -1.0" 85

Vll

Figure Page

37 Specific Normal Impedance Measured by the Standing
Wave Tube and by the Surface Pressure Method for

O.C. 705 Piberglas -1.0" 86

38 Impedance Vs. Incident Angle for O.C, 705 Fiberglas

-1.0", 2000 Hz 88

39 Impedance Vs. Incident Angle for O.C. 705 Fiberglas

-1.0", 3000 Hz 88

40 Impedance Vs. Incident Angle for O.C. 705 Fiberglas

-1.0", 4000 Hz , 89

41 Impedance Vs. Incident Angle for O.C. 705 Fiberglas

-1.0", 5000 Hz 89

42 Absorption Coefficient Vs. Incident Angle for O.C.

705 Fiberglas -1.0", 2000 Hz 90

43 Absorption Coefficient Vs. Incident Angle for O.C.

705 Fiberglas -1.0", 3000 Hz 90

44 Absorption Coefficient Vs. Incident Angle for O.C.

705 Fiberglas -1.0", 4000 Hz .... 91

45 Absorption Coefficient Vs. Incident Angle for O.C.

705 Fiberglas -1.0", 5000 Hz 91

46 Impedance Measured with Standing Wave Tube and Calculated
from Beranek's Theory for O.C, 705 Fiberglas -1.0" ... 94

47 Absorption Coefficient Measured with Standing Wave Tube
and Calculated from Beranek's Theory for O.C. 705
Fiberglas -1.0". ....... . 95

48 Specific Normal Impedance Calculated from Beranek's

Theory for T$ s = 26,3, £l = . 94 and A = .99 97

49 Specific Normal Impedance Calculated from Beranek's

Theory for~R 5 = 78.0, II = .94 and U = .99 98

50 Specific Normal Impedance Calculated from Beranek's

Theory f or Jfl = .961 and "R s = 20, 60, 100, 140 99

51 Absorption Coefficient Calculated from Beranek's Theory
forfl= -961 and "R^ = 20, 60, 100, 140 100

52 Impedance Calculated from Ford's Theory for O.C. 705
Fiberglas -1.0", "R a = 26.3, Cl= .961 103

VX11

Figure Pa 9 e

53 Absorption Coefficient Calculated from Ford's Theory

for O.C. 705 Fiberglas -1.0", "R 4 = 26.3, D-= .961 .... 104

54 Impedance Calculated from Ford's Theory for O.C. 705
Fiberglas -1.0", "R s = 78.0, Q = .961 105

55 Absorption Coefficient Calculated from Ford's Theory

for O.C. 705 Fiberglas -1.0", *R, = 78.0, £1= .961 .... 106

56 Attenuation Constant and Phase Constant for O.C. 703
Fiberglas 109

57 Attenuation Constant and Phase Constant for O.C. 704
Fiberglas , 110

58 Attenuation Constant and Phase Constant for O.C. 705
Fiberglas ........ ..... » 111

59 Geometry for Surface Pressure Method Tests. .*..... 125

LIST OP TABLES

Table Page

1 Performance Characteristics of Owens Corning 700

Series Fiberglas . 11

2 Statistical Absorption Coefficient for Owens Corning
700 Series Fiberglas Materials Mounted with a Rigid

Backing 12

3 Frequency Limits for Standing Wave Tube Measurements ... 40

4 Flow Rate Versus dB Level, re. 0.002 Microbars, for

Flow Resistance Measurements .......... 44

5 Flow Resistance Data for Owens Corning 700 Series

Fiberglas , 61

6 Cut Off Angles for Hard Wall Pressure Measurements .... 64

7 Pressure Maxima and Minima at 1 KHz, 3 -Foot Square

Surface 70

8 Pressure Maxima and Minima at 3 KHz, 3 -Foot Square

Surface ■ 70

9 Error in Phase Measurements Due to Temperature Variations. 81
10 Beranek's Flow Resistance Data 102

NOMENCLATURE

o. = length of a side of the square sample

c = phase velocity of sound

o = thickness of material

t = frequency

F * temperature in degrees Fahrenheit

j = (-0*

t< = compressibility of medium

K = modulus of elasticity of medium

m = integer number

fl = integer number

b = pressure

IP = acoustic reference pressure

Q = flow rate

IT = reflection coefficient

rS* = flow resistance

r\. = flow resistance per unit thickness

S = area

SVJft = standing wave ratio

"t = time

U = particle. velocity

V = particle velocity

X = coordinate axis

y = coordinate axis

2 = coordinate axis

~l_ - specific normal acoustic impedance

NOMENCLATURE CONTINUED

o^ = absorption coefficient

B = phase constant

T = ratio of specific heats at constant pressure and at
constant volume

% = change in density

J = propagation constant

9 = angle designation

A = wavelength

7) = angle designation

5 = normalized specific acoustic impedance

f* = density

0* = attenuation constant

$ = velocity potential

•f = phase angle

XI = porosity

XIX

ABSTRACT

A free field method for measuring the specific normal impedance
and absorption coefficient of a material at oblique incidence has
been investigated. The surface pressure method, developed by Ingard
and Bolt, compares the pressure and phase of an incident wave at a
point on the surface of an absorbent material to a similar measure-
ment at the surface of a perfectly reflecting boundary. A contin-
uous recording of these quantities as a function of angle of inci-
dence yields the impedance and absorption coefficient for oblique
angles. Measurements were taken with a six foot square sample
mounted in an anechoic chamber. Due to the finite size of the
sample, the measurements are limited both with respect to frequency
and angle of incidence because of diffraction effects. Without
having analyzed the diffraction problem, the limitations of this
method are determined from experimental results. The low frequency
limit for measurements is inversely proportional to the sample size
which must be large enough relative to the wavelength so that it
behaves as an infinite surface. The upper frequency limit is de-
termined by the accuracy in measuring the phase angle, upon which
the results depend strongly. Further limitations including diffrac-
tion effects, sample geometry, and temperature problems are also con-
sidered with recommendations included for improvement of the surface
pressure method.

The absorption characteristics of several fibrous materials of
the Owens Corning 700 Fiberglas Series were measured to determine the
variation in impedance as a function of incident angle of the sound

Xlll

wave. The results indicate that the fibrous absorbents behave as
extended reacting materials. The poor agreement between measurement
and theory for sound absorption based on the parameters of flow re-
sistance and porosity indicates that this theory does not adequately
predict the acoustic behavior of fibrous materials. A much better
agreement with measured results is obtained for values calculated
from the bulk acoustic parameters of the material.

CHAPTER I

INTRODUCTION

1.1 Sound Absorbing Materials

Several types of sound absorbing materials are currently avail-
able for noise control applications. These materials are generally
of a porous nature , constructed either from plastic foams or from
organic or glass fibers held together with a binder, and are avail-
able in the form of flexible blankets or semi-rigid and rigid sheets.
Materials of woven and sintered metals or perforated sheet metals are
also used for special noise control applications under adverse environ-
mental conditions.

There are four factors to be considered in choosing a sound ab-
sorbing material for a particular noise control problem - acoustic
performance, environment, appearance, and price. Since the primary
purpose of these materials is to control the sound reflection from
a surface and thus reduce the overall noise level , the most important
factor is the acoustic behavior of the material. Secondly, the envi-
ronment in which the material will be used must be considered so that
it will not interfere with its acoustic performance. As a minor fac-
tor, the appearance of the material becomes important in certain ar-
chitectural applications. Finally, the cost of the material must be
considered so that a material which meets the desired acoustic stand-
ards is an economically feasible solution to the problem.

The sound absorbing properties of a material are most often de-
scribed by two parameters - the absorption coefficient and the specif-
ic normal impedance. When a sound wave impinges on the surface of an

absorbing material, part of it is reflected and part of it is absorbed
and dissipated. The amount of sound energy dissipated is called the
absorption coefficient, which ranges in value from zero for a perfect-
ly reflecting surface to 1.0 for a totally absorbing surface. If the
material is placed in a diffuse field where sound waves are incident
at all angles, the random incidence or statistical absorption coef-
ficient is used to describe the amount of sound energy absorbed by
the material. The specific normal impedance is the ratio of the acous-
tic pressure to the normal particle velocity at the surface of the
material. These two properties are a function of the surface charac-
teristics , internal structure and thickness of a material , the mount-
ing conditions, the frequency, the sound intensity, and the angle of
incidence for the sound wave.

1.2 Methods for Measuring the Sound Absorption of Materials

The absorption characteristics of a material are commonly measured
using two standard techniques.

1. Standing Wave Tube Method

If a sample of material is placed at one end of a rigid-
walled tube and a sound source at the other, an incident wave
will be reflected from the surface of the material and, for pre-
ferred modes of propagation, will generate a standing wave be-
tween the source and the material. The properties of the
standing wave can be measured to yield the absorption coefficient
and impedance of the sample for a plane wave at normal incidence
to its surface (1).

2. Reverberation Room Method

The reverberation time - the time for the sound pressure to

decay to a value one thousandth of its original value - is mea-
sured for a room with acoustically highly reflecting surfaces.
When a large sample of material is placed in this room, the
statistical absorption coefficient of the sample can he deter-
mined from the change in reverberation time (2).

Several relationships and graphs (3) are available for calculating
the statistical absorption coefficient from data obtained with the
standing wave tube method.

The majority of work with sound absorbing materials has been
restricted to absorption at normal incidence. This is because ex-
perimental procedures become much more difficult when considering
sound absorption at oblique incidence. A few of the more common
measurement techniques for determining the acoustic behavior of a
material at oblique incidence are listed.

1. Interference Pattern Method

A large sample is mounted in an anechoic chamber in the
presence of an obliquely incident sound wave. An interference
pattern similar to the pattern generated in a standing wave tube
is investigated to determine the absorption characteristics of
the material at oblique incidence (4).

2. Pulse Method

If a sound source located a distance from a material at an
angle to the normal to the surface emits a short sound pulse,
a microphone can be positioned to measure two pulses - the direct
pulse from the source and the reflected pulse from the sample
surface. A consideration of the geometrical configuration and
the measured intensities reveals the absorption coefficient of

the material for a particular angle of incidence. A large sample
and free field conditions are required for this method (5).

3. Standing Waves in a Rectangular Room

The natural modes of a rectangular room determine the angles
of incidence for plane waves reflected at the walls. By covering
certain walls of the room with a sound absorbing material, the
absorption coefficient can be determined for specific angles of
incidence and frequencies (6).

4. Acoustic Waveguide

Similar to an electromagnetic waveguide, an acoustic wave-
guide is a rigid walled duct which limits wave propagation
within it to its principal and transverse modes. If a sample
of material is placed at one end of the tube and a sound source
at the other, the oblique incidence behavior of a material can
be measured by employing the transverse modes of the duct
(7, 8, and 9).

5. Surface Pressure Method

A sample is mounted in an anechoic chamber in the presence
of obliquely incident sound. The pressure and phase of the
incident wave at the surface of the absorbing material are
compared with similar measurements at the surface of a perfectly
reflecting boundary. From this data, the absorption character-
istics can be determined as a function of incident angle (10).
In all of these measurement techniques, it is important to first
determine the limitations inherent with each method before the absorp-
tion characteristics of a material can be measured. The surface
pressure method will be the subject of further study and is discussed

in detail in Section 4.3.

1.3 Statement of the Problem

Of all the sound absorbing materials commercially available,
glass fiber absorbents are one of the most economical products in
terms of noise reduction per cost of material. These materials are
fairly inexpensive and possess high absorption characteristics over
a very broad frequency range. Although the absorption characteristics
of these materials at normal incidence are fairly well known, their
oblique incidence acoustic behavior has not been completely investi-
gated. This is because measurements at oblique incidence are much
more difficult to perform than measurements at normal incidence.
Therefore, a suitable oblique incidence measurement technique should
be investigated for determining the acoustic properties of these
materials at oblique angles of incidence.

It would also be helpful to be able to predict the acoustic
absorption of a fibrous material from a knowledge of its physical
properties and parameters. Although several theories for sound
absorption by a porous material have been developed , these theories
cannot be universally applied to all materials because the mathematic
models characterize some absorbents better than others. Therefore,
the limitations of these theories with respect to fibrous materials
must be determined.

1.4 Purpose of the Research

The purpose of this study is to determine the validity of the
surface pressure method as a technique for measuring the specific
normal impedance and absorption coefficient of a material at oblique

incidence. It is hoped to determine under what conditions and over
what frequencies this technique provides reasonable measurements of
the acoustic characteristics of a material. The behavior of fibrous
absorbents of the Owens Corning 700 Fiberglas Series is investigated
to determine the variation in specific normal impedance as a function
of incident angle. Furthermore, measured values of absorption are
compared with calculated values to determine if theories for porous
absorbents adequately predict the acoustic behavior of glass fiber
materials.

CHAPTER II
MATERIALS AND THEIR PROPERTIES

Because of the low cost and the high acoustic absorption of glass
fiber materials , the acoustic properties of several samples of this
type material were investigated. These samples are marketed by Owens
Corning as the Fiberglas 700 Series of Industrial Insulation and are
used for insulating duct work and equipment operating at high tempera-
tures. The Piberglas 700 Series products are constructed of inorganic
glass fibers held together with a binder and pre-formed into semirigid
and rigid rectangular boards of varying densities. These materials
are available in 24" x 48" boards and in thicknesses of 1" to 4" with
1/2" increments. Although these products have been designed as insu-
lation materials , they also possess highly desirable sound absorbing
properties. However, the acoustical properties of each material are
quite variable, both within manufacturing tolerance specifications
and from one position to another in the same board or in different
boards. The acoustical properties of greatest importance will be
described.

The porosity of a sample is defined as the ratio of the volume
of voids within the sample to the total volume of the sample. The
porosity of a fibrous material can be calculated if the densities of
the material and the glass fibers which comprise it are known. For
a fibrous material, the weight and volume of the binder which cements
together the densely packed fibers must also be included. For a
material with negligible binder by weight, the porosity fl is

_PL s — - - \ - -T7- (2.1)

(2.2)

where

V = volume

rrt = mass

fi = density
The subscripts Q , m , and £ refer to the voids within the material
(therefore air), the material, and the fibers respectively. Since
the density of the material is much greater than the density of air ,
we will assume that the mass of the material is approximately equal
to the mass of the fibers.

m ~ m (2.3)

tin \

This is a reasonable assumption as can be seen by considering the
error for the extreme cases of a porosity of . 90 and . 99 for the
materials. The relationship between the mass of the material and
the mass of the fibers is

ir.

M + ^ (2.4,

* n v *

For a porosity of . 90 we have

m ?

* - \ 4- o.oo4 ~ I

3
where the density of air was taken as 1.18 kg/m and the density of

3 3

the fibers as 2.5 x 10 kg/m . Similarly, for a porosity of .99

^ = \ + o.o47 2 \

Therefore, the expression for porosity (11) can be written as

XI - I - £2- (2.5)

where

/ia = density of material

3 3
A = density of glass fibers (2.5 x 10 kg/m )

The specific flow resistance of a layer of material is defined

as the pressure drop across the specimen divided by the particle

velocity of air through and perpendicular to the two faces of the

layer. Thus ,

where

2
&^> = pressure drop across the sample (dynes/cm )

U = particle velocity (cm/sec)

The units ofl^are dyne-sec/cm or CGS rayls. For bulk materials,

the flow resistivity or specific flow resistance per unit thickness

of material is commonly used. Thus ,

^« d

where d is the thickness of the material. In all future work, the
term "flow resistance" as applied to a sound absorbing material will
mean the specific flow resistance per unit thickness. The flow re-
sistance is essentially constant for values of U from to some small
value and increases rapidly with increasing values of U above this
linear range.

Table 1 lists the performance characteristics of the Owens Corning
700 Fiberglas Series including flow resistance values (12, 13). Due to
the manufacturing tolerances in both density and fiber diameter for
these materials, a corresponding range of flow resistance values would
be expected. The discontinuity in the flow resistance versus density
curve for the 704 and 705 samples is due to a coarser fiber for these
two products. The statistical absorption coefficients determined by
the reverberation chamber method for these materials mounted with a
rigid backing are listed in Table 2 for data furnished by Owens
Corning ( 14 ) .

TABLE 1

Performance Characteristics of Owens Corning 700 Series Fiberglas

Specific Flow Resistance - (cgs rayls/inch thickness)

Type

Density
(lb/ft 3 )

1.58

Porosity
.990

Average Density
Fiber Diameter

and

Range

Within Manufacturing
Specifications

701

19-35

702

2.25

.986

27-56

703

3.00

.981

42-87

704

4.20

.973

35-57

705

6.00

.961

60-99

TABLE 2

Statistical Absorption Coefficient for Owens Corning 700
Series Fiberglas Materials Mounted with a Rigid Backing

Statistical Absorption Coefficient

Type

Thickness

250 Hz

500 Hz

1000 Hz

2000 Hz

4000 Hz

701

' 1"

.20

.57

.88

.86

.79

.58

.92

.93

.86

.79

702

.19

.50

.85

.76

.54

.91

.97

.87

.77

703

.22

.62

.95

.90

.82

.59

.93

.98

.87

.78

704

.18

.51

.89

.88

.80

.47

.90

.97

.86

.78

705

.19

.57

.93

.90

.83

2 >T

.55

.91

.97

.87

.78

J-3

CHAPTER III

THEORETICAL BACKGROUND

3.1 Introductory Theory

We begin the theoretical analysis of sound waves and acoustic
absorption by considering the simplest case of a plane wave at nor-
mal incidence to the surface of an absorbing material as in Figure 1.
The wave equation for the pressure p

-&- = .!_&. (3 1}

3x 2 C* **■

has a solution of the form

C3.2)

f = r { e + w e

The first term represents an incident wave propagating in the positive
X direction b[ and the second term represents a reflected wave fc> r so

that

(3.3)
(3.4)

(3.5)

The particle velocity for a plane wave in terms of the pressure is

v. -i- ^

Since the velocity is also a solution to the wave equation, we have

v = Jj- e - -^ e (3.6)

where the incident particle velocity is V; and the reflected particle

velocity is V .

\( wt-kx>
Vi= V,e J (3.7)

/, = V, e i

(3.8)

The minus sign in Equation 3.6 occurs because the velocities are vector
quantities traveling in opposite directions. The acoustic properties
of a material are defined by the specific normal impedance Z which is
the ratio of the pressure to the normal particle velocity at the sur-
face. Unless otherwise noted, the term "impedance" as used in this

study will refer to the specific normal impedance defined above and

will have the dimensions of Nt sec/m or MKS rayls. If the pressure

and velocity are out of phase , the impedance will be complex , having
a real and imaginary component.

Z - "R + jX (3.9)

"R is called the resistance and X is called the reactance. Thus, we
have

z = -

*=0 (3.10)

(3.11)

Rearranging Equation 3.11 yields the reflection coefficient T

r . IL . Z -f C (3.12.

The absorption coefficient is a measure of the energy absorbed by the
material and is defined by

X " \ - \H (3.13)

In terms of the impedance of the material and Equations 3.12 and 3,13,
the absorption coefficient is

06 " ^ 71 T (3 - 14)

(R+fcy-i. x*

The impedance and absorption coefficient for plane waves at normal
incidence to a material are not constant but are a function of several
factors including frequency, material properties and thickness, and
mounting conditions.

For some materials, the impedance and absorption coefficients are
also a function of the incident angle of sound (15). A material is
termed locally reacting if the impedance is independent of the angle
of incidence, while extended reaction occurs for materials whose
impedance varies with angle of incidence. For a plane wave incident
at an angle © to the normal to the surface in Figure 2 , the impedance

13 Z -„ ^ P

Z s - (3.15)

-U. cos e — 5-t- cos e

From the diagram, V; and V* are the normal components of the incident

and reflected particle velocities so that

Therefore, the impedance is

-p.
Vi= ^COSe (3.16)

Vrr--7t < ** @ (3.17)

Z% ^ _ T. + P.

and the reflection coefficient \- is

COS0 - — 1- (3.18)

* ""

< s ' .' .

*V -: '-

— ►■

*. _

; v - j ■ *

v - - % *

Figure 1 Plane Wave Propagating Normal to the Surface of Material

Figure 2 Plane Wave Propagating at Oblique Incidence
to the Surface of Material

r = 5L , Z,C0S8-/>C (3.19)

T\ Z s cos 9 + ^ o

If the material is locally reacting, then for all angles of incidence,

we have

Z 9 " Z (3.20)

If the impedance for any angle 9 is known, the absorption coefficient
can also be determined. From Equation 3.13, the absorption coefficient
for oblique angles is

oC = . 1- \r\ Z (3.21)

where the plane wave reflection coefficient h is defined by Equation
3.19.

It would be highly desirable to be able to predict the acoustic
behavior of material from knowledge of its physical properties. Sev-
eral theories for sound absorption based on the acoustic properties of
a porous material have been developed. The theories of both Beranek
and Ford have used the parameters of flow resistance and porosity to
predict normal and oblique incidence behavior of a material. Using
normal impedance measurements , Pyett has determined the bulk acoustic
parameters of a material which are used in predicting the acoustic be-
havior at oblique incidence. Each of these theories will be presented
in the following sections.

3.2 Beranek 's Theory for Porous Materials

Beranek (16) has developed an expression for the specific normal
impedance of a porous material in' terms of three constants - the flow
resistance, porosity, and the density of the enclosed air. The

continuity equation and force equation are derived and then combined
to give the wave equation for propagation within the material. We
begin by considering the incremental volume of material in Figure 3.
This volume Stoc contains a volume of solid matter S &X, and a volume
of air 5in.,+ 5,tx, such that the porosity is

H--I- ^ , ' (3.22>

where S = S.+ S .
1 z

The continuity equation for the air passing through the material is

^ ( © uA - _ r\ ~h

■k^-n-%

(3.23)
which becomes

p<^_ v ci ■$£— =o (3 - 24>

Beranek assumes that the cycles of condensation and rarefaction of the
enclosed air in the material occur isothermally, which he states is
valid for many acoustic materials and especially for frequencies below
2000 Hz. Therefore, for an isothermal process at atmospheric pressure

-£- = - (3.25)

■ft, v

and

^ - ^- (3.26)

If &f> is the acoustic pressure b, the continuity equation becomes

3— + — — -*rf- ^O 3.27)

which for steady state conditions is

^-4^ !- -O (3.28)

The net force applied to the incremental volume of Figure 3 is

-4 AX, JU &X 4 »-|

Figure 3 Incremental Volume of Material

^S - (, p + ^ Ml) S = - -|£- 5 A% (3.29)

This force will be opposed by the sum of the mass times the accelera-
tion of the air and solid particles in the incremental volume and by
a force dependent on friction - therefore the flow resistance. If u
is the average velocity of particle motion through the face S, then
the following continuity equation holds

Su = 5,U, + S^ (3.30)

where U t is the velocity of particle motion through S ( and U^ is the
velocity of the solid matter of area 3 . The flow resistance of the
material introduces a force that will oppose the air flow through
the material. This force due to friction r. is then

F^ = T^fcxSu (3.31)

where "R is the specific flow resistance per unit thickness. Since U 2
will be zero for non-moving solid, we have

Fr^n = ^^\ 5 ^, (3.32)

and

-r, "Kb.*

X, = — 2 — (3.33)

The forces on the areas 5 H and S 4 are

tS^/>S t M z Q + ^5,^,(^-0 (3.34)

T S x= ?* S Z **^ -^* s . **»K- O

(3.35)

where t> x is the density of the solid matter. Therefore, the force
equation becomes

_^E_5li* = pSfcX, ^AL + />S,AX ^ + P Z S Z ^, IU3. (3.36)

Using Equations 3.30, 3.34, and 3.35, u, and U^can be eliminated and
for steady state conditions we have

+ S a &** / A _

(f-')

s,w \/» /(t*VM + K^X w PAl

If f x »f and ~~ -^-X. ^ P*/lO >> ^ » the ec 3 uation yields

(3.37)

(3.38)

where

-R.-XsO* -f^

(3.39)

(3.40)

P^P

\, ^(^

S.-1

(3.41)

It can be shown that the approximations T^, = TVj and p = P are valid
for the materials we are interested in. Combining the force equation
and continuity equation, we obtain a wave equation

3x ?

/° c

(3.42)

i^i

with

''7

(3,43)

This is of the form of the wave equation in free air

XL, -5

(3.44)

with k now being a complex quantity K to account for losses as the
wave propagates in the material

~ (3.45)

c 4 c,\ /=,<* )

and

c».-

V Aw j

The solution to the wave equation can be written in the form

p = 2 J\ + e

cosk(^x + <v^

(3.46)

(3.47)

where

vy,-

M-**

(3.48)

^ + and N_ are the amplitudes of the forward and backward traveling
waves respectively. The normal component of velocity is obtained
from the force Equation 3.39, and the ratio of ^ to u, at the surface
of the material gives the impedance

£.

i*-i

n !

;<ytV\

if('-^Vr

(3.49)

The value of V is obtained by boundary conditions for the material
as determined by its mounting procedure. For the rigid wall backing

of Figure 4, u = at V = 0, so that Z = 00 at X = 0. Therefore, the
impedance with a rigid wall backing is

1*.

z 5

* 1

H >t('-^y^J

(3.50)

For a plane wave incident at an arbitrary angle 6 as in Figure 5 , we
assume that Snell's law holds

c - c.

(3.51)

and derive the impedance for any angle of incidence. By applying the
boundary conditions for a rigid wall backing, the impedance is

/°.t,

(-fey

n> cos ©,

cotVi

j^-cose.a

(3.52)

and from Equation 3.51

cos e 4 = [ \ — ^ 5\m e )

(3.53)

3.3 Ford's Theory for Porous Materials

Ford, Landau, and West (17) derive an expression for the reflec-
tion coefficient and impedance of a hard porous absorbent in terms of
the porosity and flow resistance of the material. An air wave incident
at the surface of the absorbent propagates through and within the pores
of the rigid material. It is assumed that the pores are interconnected
in a random manner and are of variable diameter and also that the
porosity is constant over an area which is small compared to a wave-
length.

We begin by examining the force equation and continuity equation
for the porous material. Introducing a coefficient of viscous friction
^.s (or flow resistance per unit length) , the force equation for an

Rigid Wall

X---J

y = o

Figure 4 Plane Waves at Normal Incidence to a
Material with Rigid Wall Backing

'Rigid Backing

x*-4

x=o

Figure 5 Plane Wave at Oblique Incidence to a
Material with Rigid Wall Backing

incremental volume of material is

where 6 is the velocity potential in the material.

Therefore

where

<jraA$ t = -!><jt-ad ^ (3>55)

1) - 7™— (3.56)

The equation of continuity for propagation within the material is

dlV V i "- " -J- ~^T (3.57)

From Morse and Ingard (18), the following expressions are developed

% - f> K\> (3.58)

C Z - -±r- (3.59)

where K i- s the compressibility of the medium, p is the density, % is
the change in density, and C is the speed of sound in the medium.
From Equation 3.57, we have

_l_<Ji V v^ - -|^ (3.60)

Using Ford's notation, this becomes

fc'Jix \i i= - ^ (3.61)

where K is the modulus of elasticity of the air in the pores. '. Sub-
stituting Equations 3.54 and 3.55 in Equation 3.61 results in the
wave equation

Now, we let

■ ¥- (-r

C i =l- Li ^|= \ 4^~ ) (3.63)

The speed of sound for isothermal (?"= 1.0) and adiabatic (T= 1.4)
conditions is C and C respectively so that

* I

C, - C3.64)

T />K T

2- I

G = — ■ — C3.65)

where K T and K s are the isothermal and adiabatic compressibility of
the medium. Since K T =T K s , the speed of sound for either condition
or for a value of f between these two extremes is

C = _i^c s - ^ C s (3.66)

where the speed of sound for adiabatic conditions C s is the same as
the speed of sound in air. The speed of sound in the material becomes

C = ^ r (3.67)

where

A (3.68)

*-m

The field in air is described by a velocity potential <fc such that
the pressure and velocity are given by

f V »- <3 - 69)

M= - qxod. <$ (3,70)

Similarly, the field in the absorbent is characterized by a potential
^> so that the resulting pressure and velocity are given by

where D is the coefficient derived from the force equation. Consider-
ing the absorbent with a rigid backing in Figure 6 , the fields can be
described by the following potentials.

> iKvt- a) cose -jH'-y)"'^^

(3.73)

^z-d^sG^ -^ x (2~3^cose ± \-\^^Qt

t> t = B (,& t e )

e (3.74)

We now apply the boundary conditions and introduce Snell's law

^- = ^^ (3.75)

ks'me = K t sm9^. (3.76)

Continuity of pressure from Equations 3.69 and 3.71 requires that at
Z. = q

$ = $t (3.77)

^ / (3.78)

where T = ~T~
n

Continuity of normal flow from Equations 3.70 and 3.72 requires that
at z = ti

(3.79)

'^tcso(\-r) = DA jHtCos © t (l-

jV t idcose^

(3-80)

From Equations 3.78 and 3.80, we solve for r to obtain

yOC^COS© CPU {j M COS e-t") + Dfl /^O CQ5 e ^

Using the potential fa, the impedance at the surface for a plane wave
incident at an angle 6 is

_L_ =

V Z = d

(3.81)

z = a

-'j\c cose ( I - rN

(3.82)

From equation 3.81, we obtain

/>C^ cot k ( j fct d COS e t )
^ = _

For normal incidence, 0=0, this becomes

£.

c A c<jth( j^<0

DXI

(3.83)

(3.84)

3.4 Pyett's Theory for Non-isotropic Porous Materials

Pyett (7) has derived an expression for the specific normal im-
pedance in terms of two experimentally determined propagation para-
meters of a homogeneous porous material. A treatment of wave

propagation in an isotropic medium is first presented and then general-
ized to the case of propagation in a non-isotropic medium using tensor
notation.

The force equation for propagation of an acoustic wave in an
isotropic medium such as air is

- qy-aci \> - />

(3.85)

where f> and M are the acoustic pressure and velocity respectively.
The equation of continuity is

ii v U = __L_ |£ (3.86)

— dW U = - ^- (3.87)

K >t

^'diM V\ = - -£- (3.88)

where K is the compressibility of the medium. Now, talcing the diver-
gence of Equation 3.85 and the derivative with respect to time of
Equation 3.88 and combining, we obtain the wave equation

v T -^ -W (3 - 89 '

Assuming a time dependence e this becomes

■*b = **fe (3.90)

where a.

(3.91)
The characteristic impedance Z of the medium will be defined as

Z = (f>K')* (3.92)

From the force equation ;

- arad f = Ztn

we obtain the velocity

(3.93)

u = __ olradi Y (3.94)

(3.95)

The quantity $ has a real and imaginary component

where o' is the attenuation constant and 6 is the phase constant which
corresponds to Y, in air. For a plane wave propagating in the positive
X direction, the pressure and velocity are

u= JL_ (3.97)

For an anisotropic medium, P and K' may depend on the direction of

U so it is necessary to use tensor notation for Equations 3.85 and 3.88.
These relations in tensor form become

-!r(/Vi>'-i£;

If the P and K tensors are both symmetric and have the same principal
axes, then, referred to these axes, both matrices are diagonal. The
wave equation becomes

£l = £l Hh. = yN o.ioo)

with

* = i- gy

and

The velocity is now given by

U- = — |^- (3.103)

The specific normal impedance of a thickness i of a homogeneous
porous material can now be calculated for a plane wave incident at
angle © to the normal. The layer is assumed to be backed by a rigid
wall having an infinite impedance as shown in Figure 7 a If the plane
of the incident acoustic ray makes an angle fi with the y axis , then
the sum of the sound pressures of the incident and reflected waves is

where the time factor £ will be omitted. The transmitted pressure,
including the component reflected from the rigid backing is

Y = \f\* +Se )& (3.105)

r = JKcosT/sme (3.106)

5= jKsinT/sifte (3.107)

Substituting the pressure in Equation 3.100, we obtain

i„, Oy Jz

where

£=4

*-y

Figure 6 Plane Wave at Oblique Incidence to a
Material with Rigid Wall Backing

*-d

x=o

Figure 7 Wave Propagating at Oblique Incidence to Material

The velocity in the x direction is

1Z>

u , - -3-r

'X^x

^U^-Be^V*

•S7-

(3.109)

We now apply the boundary conditions at the surface and backing of

the material. Continuity of pressure across the surface X = requires

A + S= /\ a + 3 Q (3 ' 110)

Because of the rigid backing, U x = at X=^ and

Therefore, the specific normal impedance at X= is

Z(d^ - ^* Z * coth(ad^

(3.112)

The value of a from Equation 3.108 is

— I <t

i.kV^e^.^lj

(3.113)

The

expression for Q can be simplified if either fl = or ^ = "^

Y J Z

*.:..*- xt

vj? . , k sin Q

(3.114)

We define an arbitrary angle (^ by

k5'mo= ^sin<t> (3.H5)

The angle <b has no meaning beyond the above definition except for the
case of zero attenuation 0^= 0, when it is identical with the angle of
refraction given by Snell's law

jKs'ia© - 3y S '^<)v (3.116)

When cr; and Oy are small compared with fly and 6^, Equation 3.114
becomes

<\ =

CW(|)

LQS X $> + ^n 4> -"-

^?y

+ j ^kCos^ (3.117)

Furthermore, when ^^/fty = ^i/ft* Equation 3.117 is simplified to

1 COS(j)

j^COS^

(3.118)

The two propagation parameters for the material are a , the propagation
constant, and 2 X , the characteristic impedance. For normal incidence
Q = and Q = ^ x , and Equation 3.112 becomes

ZU.O^l = 2 X CotlA^c!^ (3.119)

The two propagation parameters can be determined from normal impedance
measurements for samples of different thickness. If the thickness of
the samples are in the ratio of 1 to 2 , and the samples are backed by
a rigid wall, the impedance will be

ZU,O^Z x CoU(!,d V )=K+ jX (3.120)

Z^oW^coUfe^W+j*

(3.121)

so that

Z(d,o) _ \ -t ooshU3x^
Rearranging Equation 3.122, we have

(3.122)

(3.123)

where

V =

xl"R^)--R'lX-X')

(3.124)

^-•R'^-V it-*')*-

Using a standing wave tube for normal impedance measurements , the
measured values of K, X , "R , and X from Equations 3.120 and 3.121
are inserted in Equations 3.124 and 3.125. The values of £:J X <A in
Equation 3.123 may be determined from nomograms for the hyperbolic
cosine of a complex argument (19) or, as in this case, determined by
an iteration technique (20) for complex numbers using the IBM 370
computer. This iteration technique is discussed in Appendix A. From
Equation 3.95, the value of $ is

S = 0"+ \% (3.126)

Once 5 has been determined, Z^ may be calculated from Equation 3.119.
Using these values , the impedance can be calculated as a function of
incident angle from Equations 3.112 and 3.118, The validity of these
theoretical approaches will be discussed in Section 5.4 in connection
with the presentation of the experimental results.

CHAPTER IV

PROCEDURE AND TECHNIQUES

4.1 Standing Wave Tube

Measurements of the normal incidence behavior of absorbing
materials can be made using a standing wave tube, also known as an
impedance tube or constant length acoustic interferometer. A Bruel
and Kjaer Type 4002 Standing Wave Apparatus, which meets the specifi-
cations of ASTM Standard C3S4-58 CD, was used for measuring the
impedance and absorption coefficients at normal incidence. The
apparatus consists of a rigid walled tube with a sound source at one
end and the sample of absorbing material to be tested at the other
end as shown in Figure 8. The sound field in the tube is generated
by the loudspeaker and pressure levels are measured with a moveable
probe microphone. The formation of a reflected wave at the absorbing
material generates a pattern of standing waves. The sound pressure
at a distance X from the sample is the sum of the incident wave p.
and the reflected wave p p (2l).

f> = Ae J + Be J (4.D

Trie wavenumber t< is complex, to account for attenuation in the tube.
However, if we assume that losses in the tube as the sound wave
propagates are negligible , then the wavenumber K has only a real part
k so that K. = Vs = U}/c. If we can determine the phase and magnitude
of the reflected pressure amplitude's! relative to^j the specific
normal impedance can be determined. If the reflected pressure ampli-
tude is complex , it may be written as

microphone
probe

loudspeaker

material
rigid
backing

Figure 8 Standing Wave Tube

^ ( =Be J ^lr\Ae^

(4.2)

so that

where K, = k* + ~j and "T^ - wi: + --|- .

The acoustic pressure is the real part of Equation 4.3

f - R/\ * Bff cos 2 K, + (f\-tf s'm* K ( l cos (T+ € )

and the amplitude of the standing wave pattern is

T= [(M^W(U + |) + (fK-^sinV+4-)

(4.3)

(4.4)

(4.5)

From Equation 4.5 pressure maximum and minimum will be located at
antinodes and nodes respectively such that

L h = A--B- h{\-\r\)

(4.7)

For the conditions of a minimum at a point x » Equation 4.7 indicates
that all nodes will be located at positions such that svn(^x + -^-J
will be a maximum. Therefore,

*"". + t- z

(4.8)

and

(4.9)

The wavelength \ can be measured directly by taking the distance
between successive minimums at X„ and ^ . The standing wave ratio
5WR is the ratio of maximum and minimum pressures

tmm \ - \A

Thus,

r - — - - (4.11)

The specific normal impedance at the surface of the material is then

(4.12)

Finally, the absorption coefficient can also be determined and is

06= \ - \r\* (4 * 13)

A moveable probe microphone is inserted in the tube so that V n and SVJR
can be measured.

ASTM Standard C384-58 lists specifications for low and high
frequency limits of measurements based on impedance tube dimensions.
The lower limiting frequency T, is determined by the length L of the
tube in feet and is given by

s; jj>oo_ (4 . 14 )

Similarly, the upper limiting frequency Ty for measurements is given
by

*.*™°- (4.15)

where b is the diameter of the tube in inches. Within this frequency
range higher order modes of propagation are restricted, and we have
only plane wave propagation in the tube. Because of these limitations,
two tubes of different sizes were used to take measurements over the
frequency range of interest. These limits , together with the limits
for measurements specified by Bruel and Kjaer (22) for each size tube,

are listed in Table 3. The standing wave tube provides a quick and
inexpensive means of determining the relative absorption properties
of many materials in a short period of time. However, the absorption
of most materials is higher at oblique incidence than at normal inci-
dence. Since propagation within the tube is limited to plane waves,
the impedance tube measurement gives an absorption coefficient which
is usually the minimum performance expected for a material.

TABLE 3

Frequency Limits for Standing Wave Tube Measurements

Tube

Size Length

Diameter

Frequency Limits
Bruel and Kjaer ASTM Standards
Low High Low High

Large 40" 1 1/8" C 3cm) 90 Hz 1800 Hz
Small 13 3/4" 3 7/8" (10cm) 800 Hz 6500 Hz

99 Hz 2030 Hz
288 Hz 6780 Hz

4.2 Flow Resistance

The apparatus used for flow resistance measurements, shown in
Figure 9, follows specifications outlined in ASTM Standard C522-69 (23).
Since the fiberglass samples are not completely rigid, caution must be
used in inserting the material in the sample holder because compressing
the material would yield a high flow resistance. With the flow control

Pressure
Regulator

Flow
Pressure Control

Gauge Valve Jet Nozzle

Figure 9 Flow Resistance Apparatus

valve closed and the pressure regulator adjusted, the inlet supply-
valve is opened to fill the reserve tank. As the flow control valve
is opened, flow through the jet nozzle causes a vacuum at its center
section which draws air through the sample and the rotameters and then
out the jet. With the rotameter valves fully opened, the flow control
valve is opened until the maximum flow rate of 1700 cc/min is achieved.
The flow can now be regulated from to 1700 cc/min using the micro-
valve alone. Since the pressure drop across the specimen for these
flow rates is on the order of thousandths of an inch of water, it is

measured using a micromanometer. The pressure drop in units of dynes/

2 .
cm xs

4f»= ^CjK = 24^0 h (4.16)

where n is the pressure drop in inches of water. The particle veloc-
ity in cm/sec can he determined from the cross sectional area A of
the sample and the flow rate Q in cc/min

60 A

(4.17)

The specific flow resistance per unit thickness d in cgs rayls/inch
is then

IV- — , = — E (4.18)

2
For an area of 8. 73 cm , this becomes

"R 5 = 1.3 X 10 6 ~ (4.19)

where

n = pressure drop in inches of water

Q = flow rate in cc/min
Since the flow resistance increases rapidly with U , it is important
that measurements of flow resistance be performed within a range of
values for u corresponding to particle velocities encountered in sound
pressure levels appropriate to noise control problems. The sound
pressure level in decibels, re. 0.002 microbars, which corresponds
to a certain particle valocity can be ■ determined from the following
equation

\>_

s?l - zo Ho4r = ^° \«%^r (4 - 20)

and is listed in Table 4 for the range of flow rates used in testing.

4.3 Surface Pressure Method

Measurements of the absorption characteristics of a material
at oblique incidence were taken using a free field measuring technique
first presented by Ingard and Bolt (10). This method, known as the
surface pressure method , compares the pressure and phase of a plane
wave measured as a function of incident angle at a point on the sur-
face of an absorbent material to a similar measurement at the same
point in space at the surface of a completely reflecting panel. A
sufficiently large sample is assumed so that the theory of reflection
from an infinite plane boundary can be used in the analysis. The
two measurements are illustrated in Figure 10. In future use, the

TABLE 4

Flow Rate Versus dB Level, re. 0.002 Microbars,
for Flow Resistance Measurements

Flow Rate (cc/min)

dB Level

71.8

77.8

83.8

89.8

100

91.8

150

95.3

200

97.8

400

103.8

600

107.4

800

109.9

1000

111.8

1200

113.4

1400

114.7

1600

115.9

2000.

117.8

pressures s> and t>^ will be referred to as the hard wall pressure
and the absorbing surface pressure respectively. Assuming the pressure
of the incident wave is t>- , the following relationship can be written

for b t and b

^rfi + h _ - Z\>^^^ ' (4.22)

h=^ + N=^e J * (4.23)

The relationship between these expressions can be visualised by the
vector diagram in Figure 11, where Mf = ty t - 4^, . From Equation
4.22 and 4.23

f; - — (4.24)

W^fVh = fr'^t < 4 -25)

and from the vector diagram

^^T a (cos^ 2 + jsin^-0 (4.26)

^T.CwsY.+jsin^

(4.27)

Now, combining Equations 4.26 and 4,27

.a,

IM*'^+ ^T - ?,?*««¥ (4.28)

Dividing Equation 4.28 by \^\\ we obtain

,*- f T-v*

fcr_„ ?*: ?,

= ^ ^t -4r-cos^-H-i- (4>29)

M* ^ T, +

Hard Wall Pressure y>.

tf :

1 ©^

*^'

-■•\ l

\ *\ ->

Absorbing Surface Pressure b

Figure 10 Surface Pressure Method Measurements

|V *V\

Figure 11 Phase Relations Between Measurements
with the Surface Pressure Method

The absorption coefficient as a function of incident angle is then

oC.' \ ' ^

°V * ^ («** " f^ (4 ' 30)

For an incoming wave at angle Q , the incident and reflected pressures
are related from Equation 3.19 as follows

— - — (4.31)

where Z e is the specific normal impedance of the material at the
angle e • Letting ? = -^ and W = -^- = -Is. 6^ we have

w , ^2ie_ (4>32)

S cos © + 1
Therefore, for measurements using the hard wall pressure b as a
reference, the normalized impedance is given by

S--* 4 C4.33)

\ - w cos e

If the reference pressure is measured for free field conditions instead
of at the surface of the perfectly reflecting boundary, we then have
for the free field pressure V> 3

fa" \=i T -^ (4.34)

The expressions for absorption and impedance, using a free field
pressure as a reference, are now

OC e = ■^^cos'P " ^\ (4.35)

| = * L_ (4.36)

2 - \W COS

Therefore, a reference pressure for measurements can be taken either
for free field conditions or at the surface of a perfectly reflecting
boundary.

The experimental arrangement for the surface pressure method is
shown schematically in Figure 12. The material to be tested is mounted
on a panel and placed in an anechoic chamber. The measurements at the
material surface and at the reflecting surface (or free field position)
must be made at the same point in space so that no additional phase
shift between the two is introduced. The surface of the material
and the surface of the reflecting panel must then occupy the same
plane in space. A probe microphone located either at the surface of
the material or at the surface of the reflecting boundary measures
the pressure. The phase difference between the electrical driving
voltage to the loudspeaker and the acoustic pressure at the reflecting
surface is ^V,' • The corresponding phase difference for the measure-
ment at the surface of the material is V, « Thus,

V , 4^'_ if/ = Y 2 - <f, (4.37)

As the sample rotates in the presence of an approximately plane wave,
the pressure and phase are recorded continuously as a function of
incident angle.

A few differences exist between the surface pressure method as
performed by Ingard and Bolt and as performed in this study. The
"hard wall" used by Ingard and Bolt was an eight-foot square panel
rotated about a vertical axis at a speed of approximately one half

microphone
amplifier

temperature
recorder

oscilloscope

dynamic
analyzer

speaker

X? —

^ microphone
©

temperature
probe

anechoic chamber

x-y
recorder

sweep
oscillator

slave
analyser

phase
meter

frequency
counter

amplifier

turntable
control

x-y

recorder

Figure 12 Schematic for Surface Pressure Method Tests

revolution per minute. A large horn speaker was used as a sound
source and the pressure at the surface of the material and panel
was measured using a probe tube connected to a 640-AA condenser
microphone .

The initial attempt to investigate this method was made using
a three-foot square panel as the reflecting surface. To satisfy
the conditions of a hard walled rigid surface, the panel was con-
structed of a three quarter inch plywood board with a one-eighth
inch thick aluminum sheet bonded to its surface. The panel was
mounted on a turntable in the anechoic chamber and rotated about its
vertical axis at a speed of approximately 1/3 revolution per minute.
The pressure and phase were recorded continuously as a function of
incident angle with a Bruel and Kjaer Type 4136 quarter inch con-
densor microphone mounted at the center of the board. Preliminary
tests concluded that the three-foot panel was too small for the assump-
tion of an infinite reflecting surface to be valid for the frequency
limits of interest in this study. Limitations regarding sample size
will be discussed in Section 5.3, As an alternative, a larger six-
foot square panel of similar construction was used as the "infinite"
reflecting surface. Due to the size and weight of this board, it
was held stationary while the sound source was mounted at the end
of a boom and rotated about the vertical axis of the board at a
fixed distance of 8'4". The sound source, a CTS 4 1/2" diameter mid-
range speaker enclosed in a 4" x 4 1/2" x 7 1/2" wooden box, was
suspended from the boom as shown in Figure 13 at the same vertical
height as the Bruel and Kjaer Type 4136 quarter inch microphone
mounted at the center of the panel.

Figure 13 Sound Source for Surface Pressure Method Tests

The distance between the source and microphone must be kept con-
stant for testing samples of varying thicknesses so that no additional
phase shift is introduced in the phase angle measurement. The piston-
type mounting arrangement shown in Figure 14 allows the panel to be
moved horizontally so that the surface of the absorbing material and
the reflecting boundary can be placed at the same plane in space for
each measurement. The positioning of the surface was facilitated by
using a plumb bob suspended from a fixed point above the reflecting
panel. The apparatus is shown in Figures 15 and 16 for a hard wall
pressure and surface pressure measurement respectively. Since
measurements of the pressure and phase for several different samples
were recorded and then compared to a reference measurement, it was
necessary to monitor temperature variations in the chamber. The
effect of temperature changes on phase measurements will be shown
in Section 5.3.

Figure 14 Mounting Arrangement for Reflecting Panel

Figure 15 Panel with Hard Wall Surface

Figure 16 Panel with Surface of Sound Absorbing Material

CHAPTER V
DISCUSSION OF RESULTS

5.1 Standing Wave Tube

The results of absorption measurements at normal incidence using
the standing wave tube described in Section 4.1 are shown in Figures
17 to 22 for one-inch thick samples of Owens Corning 703, 704 , and 705
Fiberglas. These three samples were taken from the materials used in
the surface pressure method tests. According to limits set by both
Bruel and Kjaer, and the ASTM Standards, measurements with our appara-
tus should be possible for frequencies up to 6000 Hz. However,
successive pressure minimums did not repeat at half wavelength inter-
vals for measurements at 6000 Hz. This effect would tend to discredit
absorption measurements at the high frequency limit of the standing
wave tube. There is some question as to whether these limits are
valid for measurements with both locally reacting and extended reacting
materials. For an extended reacting material, the behavior at a point
on the surface of the material is affected by the behavior, at an
adjacent point. In this case, then, there is a possibility modes
would be generated that would interfere with plane wave propagation
within the tube.

The agreement between measurements for an overlapping frequency
range using the large tube and the small tube is quite good, indicating
that the absorption of the fibrous materials is independent of sample
size. For each material, the resistive component of the impedance is
positive, being essentially constant over the frequency range from
500 Hz to 3000 Hz, while the reactive component of the impedance has

f>C

-6

L Tl

■

- r

1 1 1

.1 .2 .5 1.0 2 5 10
Frequency (KHz)

Figure 17 Specific Normal Impedance Measured by the

Standing Wave Tube for O.C. 703 Fiberglas -1.0"

of.

1.00

.8
.6
.4
.2

8 @ ° °

.2 .5 1.0 2 5 10

Frequency (KHz)

Figure 18 Absorption Coefficient Measured by the

Standing Wave Tube for O.C. 703 Fiberglas -1.0"

e 00
© u

1 1

-2 -

-4 .

-6
-8

.1 .2 .5 1.0 2 5 10

Frequency (KHz)

Figure 19 Specific Normal Impedance Measured by the Standing
Wave Tube for 0,C, 704 Fiberglas -l o 0"

«u

0®

@
%

1 i

•1 .2 .5 1.0 2 5 10
Frequenc y ( KHz )

Figure 20 Absorption Coefficient Measured by the Standing
Wave Tube for O.C. 704 Fiberglas -1.0"

- r

g g © ©CD ° ®

©^ © g
©°

■

l i i

■ i

.5 1.0 2
Frequency (KHz)

Figure 21 Specific Normal Impedance Measured by the Standing
Wave Tube for O.C. 705 Fiberglas -1.0"

.l.

i .. .

i i

.5 1.0 2
Frequency ( KHz )

Figure 22 Absorption Coefficient Measured by the Standing
Wave Tube for O.C. 705 Fiberglas 1.0"

a large negative value at low frequencies which increases as the
frequency is increased. Although the reactance is approximately the
same for each of the 1" thick samples of Owens Corning 703, 704, and
705 Fiberglas, the resistance for each sample is different. This can
be explained by the difference in flow resistance for each material.
In Section 5.4, it will be shown that an increase in the flow resist-
ance of a material will raise the value of the real component of the
impedance but will not affect the reactance. This result is consistent
with the flow resistance measured for each sample and described in
Section 5.2.

5.2 Flow Resistance

The specific flow resistance per unit thickness of the material

was measured using the apparatus discussed in Section 4.2. Each

sample tested was one inch thick and had an area of 8.73 cm . The

samples were taken from the materials used for the surface pressure
method tests and were removed from a position adjacent to the sample
used for standing wave tube measurements. In this manner, a smaller
variation in acoustic properties between the two samples would be
expected. The flow resistance for two samples is shown as a function
of the flow rate in Figure 23 , and is in general constant for the
linear velocity range. The increase in flow resistance for low flow
rates is due to the error in measuring pressure drops of only a few
thousandths of an inch of water rather than to the properties of the
material. The value of flow resistance per unit thickness for each
of the fibrous materials was determined by taking the average of
several measurements at the maximum flow rate of 1600 cc/min. These
values are listed in Table 5 , together with the range of values

ft (cgs rayls/inch)

□

D Q

O.C.
Q O.C.

705
703

□

— L

_ ,1.

200 400 800 1200 1600 G* (cc/min)

Figure 23 Specific Flow Resistance Vs. Flow Rate
for O.C. 700 Series Fiberglas Samples

specified by Owens Corning for manufacturing tolerances. The measured
values were roughly one third of the flow resistance values for nominal
density and fiber diameter and were below the lower limit of the range
of values for manufacturing specifications. These low values prompted
further testing to determine the validity of these measurements. To
guard against air leaking around the sides of the sample, vaseline
was used as a seal between the material and the sample holder with no
appreciable change in flow resistance measured. These abnormally low
values cannot be explained unless the samples tested all came from
high tolerance production runs. It will be shown later that neither
the measured value or nominal value of flow resistance is high enough
to calculate impedance values from theory that are comparable to
standing wave tube measurements. This indicates that the flow resist-
ance for a fibrous absorbent does not provide a complete means of
specifying its acoustic properties. This limitation will be discussed
in connection with the theoretical results in Section 5.4.

TABLE 5
Flow Resistance Data for Owens Corning 700 Series Fiberglas
Specific Flow Resistance (cgs rayls/inch)

Average

Dens

;ity and

Range

Within Manufacturing

Type

Fiber

Diameter

Specifications

Measured

701

19-35

702

27-56

12.83

703

42-87

20.77

704

35-57

15.55

705

60-99

26.30

5.3 Surface Pressure Method

Before considering the impedance and absorption coefficient
measured by the surface pressure method, we will investigate the
pressure and phase measurements. It is obvious that measurements with'
this method are limited both in frequency and angle of incidence due
to diffraction effects from the finite size of the sample and the re-
flecting surface. Not having analyzed the problem theoretically,
these limitations will be determined from experimental results.

The surface pressure to measured as a function of incident angle
at the center of a six-foot square sample of one-inch thick Owens
Corning 705 Fiberglass is shown in Figure 24 for several frequencies.
As the incident angle increases from normal incidence, the pressure
decreases slowly until a cut-off angle is reached where the pressure
drops rapidly. Furthermore, as the angle of incidence approaches 90
degrees or grazing incidence, b^ approaches zero. The pressure t>
at the surface of an absorbing material with a specific normal imped-'
ance % is given by Equation 4.32 in terms of the pressure to, at the
reflecting surface. Thus,

K _ *5™S« (4.32)

fr I (jose + 1

For a finite impedance, ^> z will approach zero as approaches 90
degrees, and for a surface with an infinite impedance, t> a approaches
b . However, since we can never have an infinite impedance, even
for the perfectly reflecting surface, a similar pressure drop will be
observed for the hard wall pressure as approaches 90 degrees. The
angle at which this pressure drop occurs for the hard wall measurement
will limit oblique incidence measurements. The experimentally

-80 -60 -40

-20 20
Incident Angle

Figure 24 Pressure as a Function of Incident Angle at the Surface of 0.C 705 Fiberglas -loO"

determined cut off angles for measurements with the six-foot square
reflecting surface are taken at the point where this pressure drop
begins to occur and are listed in Table 6.

TABLE 6

Cut Off Angles For Hard Wall Pressure Measurements

Frequency

Cut Off Angle

1000

68°

2000

75°

3000

78°

4000

80°

5000

82°

6000

82°

The pressure ratio *f\ /\ in decibels between the surface pressure
for a sample of Owens Corning 705 Fiberglas and the hard wall pressure
is shown in Figure 25. The corresponding phase measurements are
shown in Figure 26. As seen by the curves, the surface pressure
relative to the hard wall pressure approaches zero as Q approaches
90 degrees.

The hard wall pressure measured as a function of incident angle
at the center of a three-foot square perfectly reflecting panel is
shown in Figure 27 for several different frequencies. As the incident
angle is increased , the pressure alternately passes through a series
of maximum and minimum values. As would be expected, the phase

(dB)

14 -

8 •■-

4*?

01800 Hz
B 2000 Hz
V2500 Hz
A 3000 Hz
05000 Hz

B
©

w
s

8 ©

20 40 60
Incident Angle

Figure 25 Measured Surface Pressure *P a in dB Below Hard
Wall Pressure T, for O.C. 705 Fiberglas -1.0

(Degrees)

10 -

-10 a

-20

-30

A
O

1800 Hz
H 2000 Hz

V 2500 Hz
A 3000 Hz
© 5000 Hz

v v

V V

20 40 60
Incident Angle

Figure 26 Measured Phase Difference Between the Hard Wall

Pressure and Surface Pressure for O.C. 705 Eiberglas
-1.0"

-80 -60 -40

-20 20
Incident Angle

Figure 27 Hard Wall Pressure as a Function of Incident Angle for Three-Foot Square Panel

-J

component exhibits a similar pattern with a peak-to-peak; phase varia-
tion of 20 to 30 degrees. The behavior of this pattern indicates
some sort of diffraction effects due to the finite size of the re-
flecting panel. As the frequency is increased , the magnitude of this
pressure fluctuation decreases. Therefore, at high frequencies where
the wavelength is much smaller than the dimensions of the surface,
the reflecting panel better approximates an infinite surface and
diffraction effects are less prominent. Recent work by Hughes (24)
indicates that an incident wave diffracted by the sharp discontinuity
at the edge of a finite size panel produces a significant secondary
source at this edge. The wave from this secondary source travels
along the face of the panel and is measured together with the incident
wave by the microphone at the center. For certain angles of incidence,
these pressures will combine so that the total pressure will have
maximum and minimum values. From Appendix B, these maximums and
minimums will be located at angles © such that

1. © = 90°

2. sir\© = x\ — ft = 0, 1, 2...

where

C{ = the horizontal dimension of the panel

\ = the wavelength
The first condition is satisfied at grazing incidence where the pres-
sure will approach zero as 8 approaches 90 degrees. This result
was previously verified by the hard wall pressure measurements and
by Equation 4.32. The second condition locates the maximum and
minimum pressures as a function of incident angle. For a three-foot
square panel, the angles at which the measured and predicted pressure

variations occur are listed in Tables 7 and 8. Diffraction patterns
from perfectly reflecting square baffles in an anechoic tank (24)
reveal pressure fluctuations similar to those measured with the re-
flecting panel in air for equal ratios of the length of a side of the
panel to the wavelength. Thus , we would expect similar results for
tests at different frequencies and with different board sizes if the
ratio between the length of a side of the panel <\ and the wavelength
\ were the same. Therefore, comparing equal values of Ka , where k
is the wavenumber, similar pressure patterns for measurements with
different size reflecting surfaces would be obtained for frequencies
related by

\<Q-- kV (5.1)

$=-^' (5.2)

where i- and a are the frequency and horizontal dimension of the panel
respectively for each measurement. These results are confirmed in
Figure 28 for the hard wall pressure measurements at the surface of a
two-foot square reflecting panel.

To reduce or eliminate the fluctuations in pressure and phase
due to diffraction, several modifications were investigated using the
three-foot square panel. Since the secondary pressure waves originate
at the edges , it would seem that treating the vertical edges of the
panel with sound absorbing material would eliminate the effect of
diffraction. When a three-inch thickness of Owens Corning 705
Fiberglas was placed along each vertical edge of the panel , the
results of Figure 29 indicate that this treatment has no appreciable

TABLE 7

Pressure Maxima and Minima at 1 KHz, 3-Foot Square Surface

X Q Q

V\ ^_a CALC MEAS

1 .376 22.1 22

2 .752 48.6 49

TABLE 8

Pressure Maxima and Minima at 2 KH2, 3 -Foot Square Surface

CALC

MEAS

.188

10.8

.376

22.1

.564

34.3

.752

48.6

.940

q = 2\ 1500 Hz

°*= 3', 1000 Hz

q = 2 ' , 3000 Hz
q = 3 * , 2000 Hz

-80 -60 -40

-20 20
Incident Angle

Figure 28 Hard Wall Pressure as a Function of Incident Angle for Two Different Panels

-80 -60 -40

-20 20
Incident Angle

Figure 29 Hard Wall Pressure as a Function of Incident Angle with Edges Treated

effect on the hard wall pressure. This is because the discontinuity
at the edge of the rigid panel is still present despite the fact that
the material is highly absorbing. To remove this discontinuity, the
edge must be completely covered by the material. With the edge of the
panel covered, the incident wave is attenuated as it travels through
the material to be diffracted at the edge. Furthermore, the diffracted
wave is also attenuated as it travels outward through the material
and toward the microphone at the center of the panel. The resulting
pressure is shown in Figure 30. Modifications to the edges, such as
rounding the corners, would have no effect on edge diffraction for
the frequencies we are interested in. This would only become effective
when the wavelength is the same size or smaller than the diameter of
the rounded corner, The importance of surface geometry was investi-
gated by measuring the pressure as a function of incident angle at
the center of a perfectly reflecting three-foot diameter circular
boarda The results, shown in Figure 31, indicate that the geometry
of the circular panel strongly reinforces the diffraction effects.
In fact, this result would be expected since each secondary source
at the circumference of the panel is the same distance from the
microphone at the center. Since the pressure fluctuations are more
pronounced for this geometry, a square or rectangular panel would be
preferred for the reflecting boundary.

As seen by the curves in Figure 27, the magnitude of the fluctua-
tion in pressure as a function of incident angle decreases as the
frequency is increased. At high frequencies where the dimensions of
the panel are large compared to the wavelength, the effect of
diffraction is less pronounced, and the panel is a better approximation

-80 -60 -40

-20 20
Incident Angle

40 60

Figure 30 Hard Wall Pressure as a Function, of Incident Angle with Corners Covered

1000 Hz

-80 -60 -40

-20 20
Incident Angle

Figure 31 Hard Wall Pressure as a Function of Incident Angle for a Circular Panel

<_n

to an infinite boundary. Therefore, the largest surface possible
should be chosen for measurements with the surface pressure method in
order that the assumption of an infinite boundary be valid at the
lowest frequency of interest. Furthermore, the diffracted wave from
the edge will be attenuated by the additional distance it must travel
to the microphone at the center of a larger panel. It can be seen
then that the use of a larger surface would reduce diffraction effects
and also result in a lower limit for measurements. For these reasons,
a six-foot square panel was used instead of the three-foot panel for
all future measurements with the surface pressure method. The hard
wall pressure measured as a function of incident angle is shown in
Figure 32 for several frequencies with the larger surface. Despite
the fact that the edges have not been treated, the improved performance
for the larger surface, especially at high frequencies, can be seen.

The pressure measured at the surface of an absorbing material as
a function of incident angle shows little evidence of the diffraction
effects that were obtained with hard wall pressure measurements. This
is because the material covering the surface of the rigid panel helps
to eliminate the discontinuity at the edges and will attenuate, a
diffracted wave as it travels across its surface to the microphone.

A further limitation and source of error for measurements with
the surface pressure method are the temperature variations during
testing. Although this variation has a minimal effect on the pressure
levels , it has a direct relationship on the measurement of the phase
angle. A change in temperature will affect the speed of sound and
thus the wavelength. The speed of sound as a function of temperature
is

-40

-20 20
Incident Angle

Figure 32 Hard Wall Pressure as a Function of Incident Angle for Six-Food Square Panel

-j
-j

0= 4^.03^5^.4 + *?)* (5.3)

where

C = speed of sound in ft/sec

°F = temperature in degrees Fahrenheit
Since the wavelength \ is related to the speed of sound,

it is also affected by a temperature change and will influence the
measurement of the phase angle. If the same phase measurement is
made at different temperatures, a phase shift between the two will
be noted as shown in Figure 33. Although there is only a very, small
variation in one wavelength for the temperature change, this variation
is accumulated over a distance of several wavelengths. Therefore,
over a distance of one wavelength, there is less error in the phase
measurement than over a distance of two wavelengths. At high fre- ,
quencies , where there are several wavelengths between the loudspeaker
and microphone, the probability of error in measuring the phase angle
becomes very high. The following expression corrects the phase error
between similar measurements made at different temperatures ~T t and

^ = x£ (-^ ~) (5.4:

V c, c, /

where

"X = the distance between source and microphone
C, = speed of sound at temperature T,
G = speed of sound at temperature T^

T tl o, i \ 1

►«,*-

Figure 33 Phase Variation for Temperature Changes

To control temperature variation in the anechoic chamber, the
thermostat of the temperature control system for the room was set at
a constant level during all measurements. Although the temperature
was constant to within +.0,5 F during each testing period, the actual
temperature levels between different tests could vary as much as 0,5 F
to 1 F. The phase correction for measurements taken over a distance
of 8 '4" between the source and microphone and for temperature varia-
tions of 0.5 and 1.0 F is listed in Table 9 for several frequencies.
In the same manner, a temperature gradient between the source and
microphone would further interfere with an accurate measurement of
phase angle. Therefore, the phase measurement is especially sensitive
to temperature changes.

As mentioned previously, measurements with the surface pressure
method can be made using a free field pressure instead of the hard
wall pressure as the reference measurement. However, there is some
difference between the data obtained using each of these measurements.
Since the same surface pressure was used for each measurement, the
error must be due to the reference measurements at the perfectly
reflecting boundary and for free field conditions. Assuming a
pressure doubling effect for the incident pressure at the perfectly
reflecting surface, the difference between the hard wall pressure and
the free field pressure at the same point in space should be six
decibels. For a perfectly reflecting surface with an infinite
impedance, the phase component of the hard wall pressure should be
the same as the phase components of the free field pressure. There-
fore, the difference between the phase components should be aero.
The pressure ratio and phase difference between these measurements

TABLE 9
Error in Phase Measurements Due to Temperature Variations

Wavelength (cm) Delta Phi (Degrees)

Freq

"T; = 76.0

69.177

T = 76.5
'a.

69.210

T 3 = 77.0

T..T,

T,, T 3

500

69.242

0.6

1.2

1000

34.589

34.605

34.621

1.2

2.5

2000

17.294

17.302

17.310

2.5

4.9

3000

11.530

11.535

11.540

3.7

7.4

4000

8.647

8.651

8.655

4.9

9.9

5000

6.918

6.921

6.924

6.2

12.3

6000

5.765

5.768

5.770

7.4

14.8

8000

4.324

4.326

4.328

9.9

19.7

as a function of incident angle are shown in Figures 34 and 35. The
pressure ratios are on the order of six decibels and are relatively
unchanged for angle of incidence. Only at 1000 Hz, where the length
of a side of the six-foot square surface is approximately six times
the wavelength is the pressure ratio much less than six decibels.
This is because the assumption of an infinite surface is not valid at
this frequency. The phase differences , on the other hand , vary
considerably both with frequency and angle of incidence. Therefore,
the phase angle is responsible for the error between measurements
using a reference pressure at the reflecting surface and for free
field conditions. A 3% error in the phase angle will result in a
10 degree phase shift which will clearly alter the absorption proper-
ties of the material. However, the error limits for absorption
measurements cannot be quantitatively stated in terms of the error in
measuring the phase angle. This is because the absorption properties
are also dependent on the difference in pressure levels at the reflect-
ing surface and material surface for each measurement. Nonetheless,
it can be stated that the surface pressure method strongly depends on
an accurate measurement of the phase angle.

It is obvious that by using a finite sample and reflecting sur-
face, the assumption of an infinite boundary is not valid at low
frequencies where the wavelength is on the order of the dimensions of
the sample. Therefore, the low frequency limit for measurements with
this method must be determined. Measurements by Ingard and Bolt using -
and eight-foot square .panel indicate reasonable data for frequencies
as low as 500 and 700 Hz. Considering Equation 5.2 and comparing our
results relative to those of Ingard and Bolt, reasonable measurements

1000 Hz

B 3000 Hz

A 4000 Hz

V 5000 Hz

Figure 34 Pressure Ratio in Decibels as a Function of Incident
Angle Between the Hard Wall Pressure "?, and the Free
Field Pressure f 3

(Degrees)

<■>

. A

□

1000 Hz

2000 Hz

3000 Hz

4000 Hz

5000 Hz

80 Q

Figure 35 Phase Difference as a Function of Incident Angle Between
the Hard Wall Pressure and the Free Field Pressure

with the six-foot panel used in this study should be obtained for
frequencies as low as 700 and 1000 Hz. However, experimental results
indicate that the low frequency limit for measurements occurs for a
much higher frequency. Measurements at normal incidence with the
surface pressure method were compared with measurements using a
standing wave tube to determine the actual' limits of this method.
The absorption coefficient and impedance at normal incidence of a
one inch thick sample of Owens Corning 705 Fiberglas measured with
a standing wave tube are shown by the curves in Figures 36 and 37
respectively. The data points in these figures are the values at
normal incidence measured by the surface pressure method for a six-
foot square sample of the same material. The imaginary component of
the impedance does not have a negative value until 3000 Hz for these
measurements. At this frequency the length. of the side of the sample
is approximately 15 times the wavelength. It. becomes apparent that
for frequencies below this limit, the wavelength becomes comparable
to the dimensions of the sample and the surface does not behave as an
infinite boundary. Therefore, the ratio of the horizontal dimension
of the sample to the wavelength at the lower limiting frequency
should be at least 15 for the assumption of an infinite boundary to
be valid.

Using the surface pressure method and proceeding as outlined in
Chapter 4.3, the specific normal impedance of a one-inch thick sample
of Owens Corning 705 Fiberglas was measured as a function of incident
angle for several frequencies. The results are shown in Figures 36
to 41, where measurements are compared for both the hard wall pressure
and free field pressure used as a reference. The discrepancy between

1.00 r

,80

.60

.40 -

°.T

— Standing Wave Tube Data
Surface Pressure Method Data
Hard Wall Reference

O Free Field Reference

1 S ■ ' ' 'ilo

J L

J I I U,

Frequency (KHz)

Figure 36 Absorption Coefficient at Normal Incidence Measured by the Standing Wave Tube
and by the Surface Pressure Method for O.C. 705 Fiberglass -1.0"

+2 -

+1 -

-1
-2

-4

-6
-8

— Standing Wave Tube Data

j i \ i i

_i i

,5 1.0

Frequency (KHz)

Figure 37 Specific Normal Impedance Measured by the Standing Wave Tube

and by the Surface Pressure Method for O.C. 705 Fiberglas -1.0"

these measurements is due to the difficulty in measuring the phase
angle as stated previously. Both the real and imaginary components
of the impedance increase from the values at normal incidence as the
incident angle is increased from zero to 90 degrees. Similar results
at each frequency indicate that the glass fiber material behaves as
an extended reacting material.

The absorption coefficients measured as a function of incident
angle for the same sample are shown in Figures 42 to 45. In each of
these figures, data is again compared for measuremerits using both the
hard wall pressure and free field pressure as a reference. Despite
the difference in impedance values for these two measurements , the
variation between absorption coefficient data is small. This is
especially obvious at 3000 Hz, where the large discrepancy between
impedance measurements in Figure 39 makes itself evident in Figure 43
as only a small difference between the absorption coefficients. As
the incident angle increases from zero to 90 degrees, the absorption
coefficient increases from its normal incidence value to a maximum
value and then decreases as the incident angle approaches grazing
incidence. At an oblique angle of approximately 60 degrees, the
absorption coefficient has a maximum value and the material is almost
totally absorbent. The behavior of the absorption coefficient at
grazing incidence is confirmed from our investigation of the pressure
at the surface of the material as a function of incident angle. From
Equation 4=30, the absorption coefficient as a function of incident
angle is

1* I rr ,^^ I*

OC^J^OQSY- -^ (4 . 30 ,

+2 r

o
n

-J—

[]

Hard Wall

Reference

Free Field

Reference

—

Pyett's Theory

.3-

Figure 38 Impedance Vs. Incident Angle for
O.C. 705 Fiberglas -1.0", 2000 Hz

+ 1

o -

Q O

Hard Wall Reference
Q Free Field Reference
— Pyett's Theory

80 Q

Figure 39 Impedance Vs. Incident Angle for
O.C. 705 Fiberglas -1.0", 3000 Hz

+2<

3
2

e> a Q

Hard Wall Reference
Free Field Reference

Pyett's Theory

— O ""

n Q
Q

""T"

— -fi^T'

-K

80 9

Figure 40 Impedance Vs. Incident Angle for
O.C. 705 Fiberglas -1.0", 4000 Hz

-1

Q Hard Wall Reference
H Free Field Reference
— Pyett's Theory

f>0

so e

Figure 41 Impedance Vs. Incident Angle for
O.C. 705 Fiberglas -1.0", 5000 Hz

O Hard Wall Reference
E3 Free Field Reference
— Pyett's Theory

so e

Figure 42 Absorption Coefficient Vs. Incident Angle
for O.C. 705 Fiberglas -1.0", 2000 Hz

Figure 43 Absorption Coefficient Vs. Incident Angle
for O.C. 705 Fiberglas -1.0", 3000 Hz

1.0

ot-

^ @ ^

1K \§

Hard Wall Reference
Free Field Reference
Pyett's Theory

i \ \

Figure 44 Absorption Coefficient Vs. Incident Angle
for O.C. 705 Fiberglas -1.0", 4000 Hz

Hard Wall Reference
S Free Field Reference
Pyett's Theory

Figure 45 Abosrption Coefficient Vs. Incident Angle
for O.C. 705 Fiberglas -1.0", 5000 Hz

Similar results for the absorption coefficient and specific
normal impedance as a function of incident angle were obtained for
other samples of Owens Corning Fiberglass but were not included
since their behavior did not differ markedly from that for the
Owens Corning 705 Fiberglas.

5.4 Theoretical Results

The expressions for the specific normal impedance of a porous
material as derived by Beranek and Ford are quite similar as seen
by Equations 3.50 and 3.84 respectively. Obviously, the main differ-
ence arises from Beranek* s assumption of isothermal conditions for
wave propagation within the material. Another difference between
these two expressions is that the porosity \L does not appear as a
part of the argument of the hyperbolic cotangent function in Ford's
equation. This is because the continuity equation used by Beranek
and Ford respectively differ as shown below.

!s--ft*&

d* d-t

As would be expected, the phase velocities for propagation within the
material also differ and are given by the following expressions

c = __iiii) — (5 . 7)

if
c= ill^ — _ r«i.o,i.2,i-1 (5.8)

\<4]

2 (i- IS*)* '

However, for the materials we will consider, the porosity has a value
of .96 to .99 so that the difference in propagation velocities should
have a minimal effect on the results. Since the glass fiber materials
of interest in this study resemble the mathematical model for a porous
sound absorbing material as used in both Beranek's and Ford's theories,
one should be able to use these theories to calculate their absorption
properties. The results for each of these theories will be presented
separately and compared with a standing wave tube measurement.

Using Beranek's theory, the sound absorbing properties of a one-
inch thick sample of Owens Corning 705 Fiberglas can be calculated
in terms of its physical properties - namely, the porosity and flow
resistance. These parameters which appear in Equation 3.50 are chosen
to correspond to the properties of the material with both the nominal
flow resistance of 78 cgs rayls/inch and the measured flow resistance
of 26.3 cgs rayls/inch used in the calculations. The agreement be-
tween measurement and theory for the impedance and absorption
coefficient at normal incidence for this sample is shown in Figures
46 and 47. As seen by the curves, the theory underpredicts the real
component of the impedance and also the absorption coefficient. Since
only two physical parameters, the porosity and flow resistance,
determine the acoustic behavior of the material, it would seem that
this discrepancy is due to the value of one or both of these proper-
ties. To determine the effect each of these parameters has on the
impedance, .calculations were made for a range of values in both

2 -

-4

Beranek's Theory, T^ = 26.3, D. = .961

Beranek's Theory, 'Rj = 78.0, D- = .961

j i i i i i_i_

J I I I L_l

1.0
Frequency (KHz)

Figure 46 Impedance Measured with Standing Wave Tube and Calculated from
Beranek's Theory for O.C 705 Fiberglas -1.0"

1.00

.80

.60

.40

.20

— Beranek's Theory, ~R & = 26,3, -Q. = .961

- Beranek's Theory, "K s - 78.0, -O- = .961

J 1 I L_l

1.0

J L

J I 1 l_l

Frequency (KHz)

Figure 47 Absorption Coefficient Measured with Standing Wave Tube and Calculated from
Beranek's Theory for 0-C 705 Fiberglas -1.0"

porosity and flow resistance that would be representative of the
manufacturing tolerances for these materials. The impedance calculated
from Beranek's theory for a material with limiting values of .94 and
.99 for porosity and a flow resistance of 26.3 cgs rayls/inch is
shown in Figure 48. Similarly, for the same range of porosity, the
impedance calculated for a flow resistance of 78 cgs rayls/inch is
shown in Figure 49. These results indicate that a variation in
porosity from .94 to .99 will have a negligible effect on the normal
impedance of this material below 5000 Hz. For a constant porosity
of .961, the impedance and absorption coefficient of this material
calculated for several flow resistance values are shown in Figures
50 and 51. The variation in flow resistance affects only the real
component of the impedance, and leaves the imaginary component
relatively unchanged for frequencies below 2000 Hz. Considering
the experimental results in Figure 21, a flow resistance of nearly
140 cgs rayls/inch would be necessary to calculate impedance values
corresponding to those measured with the standing wave tube apparatus
for a one-inch thick sample of Owens Corning 705 Fiberglas. Since the
range of flow resistance values for this material due to manufacturing
specifications is 60 to 99 cgs rayls/inch, the value of 140 cgs rayls/
inch is completely outside of this range. Therefore, the flow
resistance as used in this theory does not account for the total
dissipation within the material and other dissipation mechanisms
must be present. Beranek has remedied this problem by introducing
a "dynamic" flow resistance to compensate for this factor. This
parameter is determined from standing wave tube measurements by
fitting curves for the impedance calculated at different flow

.5 1.0 2
Frequency (KHz)

Figure 43 Specific Normal Impedance Calculated from Beranek's Theory
for -ft 5 = 26,3, XI = ,94 andXl= =94

+2
+1

-1
-2

-4 -

-6
-8 "

Beranek's Theory, "K s = 78.0, O. = .94
Beranek's Theory, ~B. S = 78.0, £\ = .99

J i i i i i i i

J I L

J I I L

.5 1.0
Frequency (KHz)

Figure 49 Specific Normal Impedance Calculated from Beranek's Theory
for ~R S - 78.0, Q= ,94 and £\ = ..99

.5 1.0

Frequency (KHz)

Figure 50 Specific Normal Impedance Calculated from Beranek's Theory

for n

,961 and"R s = 20, 60, 100, 140

.uu

-s*-'^--*-'^ .%

.■sy~ „ ..lT^;>fc^

.80

/ .

.60

#'/

.40

///

/ H* = 20

"K a = 60

T^= 100

.20

_..--R s - 140

j -*-s; r" i i i i i i i

i , ,. i ii

.1 .2 .5 1.0 2 5 10

Frequency (KHz)

Figure 51 Absorption Coefficient Calculated from Beranek's Theory
for XI = .961, ~R S = 20, 60, 100, 140

o
o

101

resistance values to the experimental results. The dynamic flow re-
sistance is different from the measured static flow resistance and is
partially explained by Beranek as being due to the nonisotropic nature
of the materials. However, this new parameter does not really solve
the problem since it bears no direct relationship to the static flow
resistance and can only be determined from impedance measurements.
In Table 10 , the variations between these values as presented by
Beranek are listed, with "dynamic" values being both above and below
the measured static values for different materials. In short, the
use of this new parameter does not seem to logically account for the
dissipation within the materials.

Ford's theory for sound absorption by a porous material is also
dependent on the flow resistance and porosity of the material, but
introduces a new parameter T to account for wave propagation' within
the material under isothermal or adiabatic conditions or any condition
between these two extremes. To determine what effect this parameter
has on the acoustic properties of the material , the value of Y will
be chosen as 1.0 and 1.4 for isothermal and adiabatic conditions
respectively and as 1.2 as an average between these two extremes »
The values of porosity and flow resistance corresponding to the
properties of a one-inch thick sample of Owens Corning 705 Fiberglas
were used to calculate the impedance and absorption coefficient at
normal incidence from Ford's theory. Using the measured flow resist-
ance of 26.3 cgs rayls/inch, the calculated results for the sample
are shown by the curves in Figures 52 and 53. The calculated results
for the same sample using the nominal flow resistance of 78 cgs rayls/
inch are shown in Figures 54 and 55. As seen by these curves, the

102

TABLE 10
Beranek's Flow Resistance Data

Flow Resistance CRs/pc,)

Static Dynamic

10.0

4.5

6.0

10.0

6.0

17.

13.

,5 1.0
Frequency (KHz)

Figure 52 Impedance Calculated from Ford's Theory for

O.C. 705 Fiberglas -1.0", "R s = 26.3, A = .961

f^S

40
20

/ / ■

///

— IT- 1.0

Y = 1.2

Y = 1.4

-*-— =f=^y

i i i

J... _J « 1. 1 j i i

.1 .2 .5 1„0 2 5 10

Frequency (KHz)

Figure 53 Absorption Coefficient Calculated from Ford's Theory for
O.C. 705 Fiberglas -1.0",T\ S = 26.3,11= .961

,5 1.0

Frequency (KHz)

Figure 54 Impedance Calculated from Ford's Theory for
O.Co 705 Fiberglas -1.0",T\ = 78.0, XI = .961

.5 1.0 2 5

Frequency (KHz)

Figure 55 Absorption Coefficient Calculated from Ford's Theory for
O.C. 705 Flberglas -1.0'\"R S = 78.0, fl = .961

o
en

107

value of Tf affects only the imaginary component of the impedance and
leaves the real component unchanged for frequencies below 4000 Hz„
However, comparing these results with Figures 21 and 22, the values
calculated from Ford's theory also underpredict the real component
of the impedance and also the absorption coefficient* A change in
the flow resistance would affect the absorption characteristics in
the same manner as shown in Figures 50 and 51. This is obvious
since both Equation 3.50 of Beranek's theory and Equation 3.84 of
Ford's theory have the same dependence on the flow resistance. Thus,
a flow resistance of 140 cgs rayls/inch would be required for the
real componenent of the impedance to coincide with standing wave
measurements for this material, and as noted previously, this value
would not be consistent with the manufacturer's quoted properties of
the material. For the value of f chosen as 1.0, i.e., isothermal
conditions, the values calculated from Ford's theory are in close
agreement with results from Beranek's theory. This would be expected
since Beranek's theory limits wave propagation within the porous
material to isothermal conditions only.

The investigation of these theories for normal incidence acoustic
absorption has revealed several important results in terms of the
properties of a glass fiber material. These results will be stated
in terms of their effect on the real and imaginary components of
the impedance since the absorption coefficient is dependent on both
of these parameters. The value of flow resistance will affect only
the real component of the impedance, leaving the imaginary component
unchanged. For conditions of wave propagation within the material,
the value of Y will alter only the imaginary component of the

108

impedance. Furthermore, the use of a flow resistance also does not
account for the total dissipation in glass fiber materials and further
dissipation due to viscous or thermal effects may be prominent for
these materials. Due to the discrepancy between measurement and
theory for normal incidence acoustic absorption, the extension of
these theories to oblique incidence behavior was hot included. This
subject will be covered by Pyett's theory.

The oblique incidence acoustic behavior of a material can be
calculated from Pyett's theory in terms of two propagation parameters <,
These parameters are determined from normal impedance measurements
with a standing wave tube for samples of different thicknesses.
Figures 56 and 58 show the attenuation constant and phase constant
for samples of Owens Corning 703, 704, and 705 Fiberglas, as compared
to the phase constant K for wave propagation in air. The impedance
and absorption coefficient calculated at oblique incidence for a one-
inch thick sample of Owens Corning 705 Fiberglas are shown in Figures
38 to 45 together with experimental results. The calculated impedance
increases from its value at normal incidence as the incident angle
increases from zero to 90 degrees. Hence, the glass fiber material
behaves as an extended reacting material. The absorption coefficient
has a maximum value at an oblique angle of incidence of approximately
60 degrees. This agrees favorably with experimental results for this
material. Although the agreement between measurement and theory is
fairly good, there are two disadvantages with this approach. First,
several normal impedance measurements must be made in order to
determine the bulk acoustic parameters for the material; and second,
due to the non-homogeneous nature of the material, it is possible

1.0

s
o

3--<r,^

Frequency (KHz)
Figure 56 Attenuation Constant and Phase Constant for O.C. 703 Fiberglas

1.0 -

•H

•a
3

6
o

.4 "

s^+j*

Frequency (KHz)
Figure 57 Attenuation Constant and Phase Constant for O.C. 704 Fiberglas

■-^

c
ft

1.0

•

*s^

^"■q

*'" Q^-

J^n

fr?'

"*©

) i

. .. i

Frequency (KHz)
Figure 58 Attenuation Constant and Phase Constant for O.C. 705 Fiberglas

112

that the bulk acoustic parameters measured for a single sample may not
be representative of the acoustic parameters for the entire material.

1/3

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

In summary, measurements of the absorption characteristics of a
material at oblique incidence by the surface pressure methods are valid
within a certain frequency range. The low frequency limit is deter-
mined by the size of the sample which must be large enough, relative
to the wavelength, so that the surface behaves as an infinite boundary.
The upper frequency limit is determined by the accuracy in measuring
the phase angle upon which this method depends strongly. However, due
to the limitations of temperature variation and diffraction, the
accurate measurement of the phase angle makes this method very diffi-
cult to perform.

The following recommendations can be made for future measurements
using the surface pressure method.

1. The sample should be as large as possible to insure that the
surface behaves as an infinite boundary. The ratio of the length of
the horizontal dimension of the sample to the wavelength at the lowest
frequency of interest should be at least 15.

2. The sample should be either square or rectangular in shape so
that the geometry of the finite surface does not reinforce the effect
of diffraction.

3. The vertical edges of the perfectly reflecting surface should
be covered with a sound absorbing material to reduce or eliminate
diffraction from the edges.

4. Measurements at very oblique angles are not valid because of
the rapid drop in surface pressure for incident angles greater than
80 degrees.

114

5. Adequate temperature stabilization between measurements must
be assured and temperature gradients prevented so that no additional
phase shift is introduced in the measurement of phase angle.

6. A theoretical analysis of the diffraction due to a finite
sample would provide useful information on the limitations inherent
with this method.

Additional oblique incidence measurements using techniques presented
in Section 1.2 should be performed with the same samples to judge
the validity of oblique incidence data obtained using the surface
pressure method. The interference pattern method presented by Sides
and Mulholland (4) would be preferred for future tests since it
eliminates the problem of measuring the phase angle.

There was reasonably good agreement between oblique incidence
absorption measurements with the surface pressure method and values
calculated from Pyett's theory for the glass fiber materials. This
indicates the oblique incidence acoustic behavior of a material can
be calculated from experimentally determined bulk acoustic parameters.
However, as stated previously, there are two disadvantages with this
method. First, the bulk acoustic parameters for a material can be
determined only from several normal impedance measurements with samples
of different thickness ; and second , if a material is in any way in-
homogeneous, the bulk acoustic parameters measured for one sample may
not be accurate representation of the parameters for the entire
material .

It would be most advantageous to eliminate experimental measure-
ments for an acoustic material and thus be able to predict its behavior
for normal and oblique incidence from a knowledge of its physical

115

properties. The theories of Beranek and Ford have attempted to
provide this type of analytical approach based on the parameters of
porosity and flow resistance. However, considering the results for
glass fiber materials, the agreement between measurement and theory
is poor. Beranek has assumed that the compressions and rarefactions
within the material occur for isothermal conditions rather than for
adiabatic conditions which prevail for wave propagation in free air.
This, he states, is true for many materials, especially below 2000 Hz.
To modify this assumption, Ford has included a parameter "T in his
theory which may have a value of 1.0 for isothermal conditions or 1.4
for adiabatic conditions or any value between these extremes. However,
even for the variation in this parameter, the results calculated from
theory do not predict the increased attenuation measured experimentally
for these materials. Therefore, the effect of the porosity and flow
resistance terms must be investigated* For glass fiber materials , the
range in value for porosity has little effect on the calculated results
as was previously shown. The total sound attenuation within these
materials is accounted for by the flow resistance, which is assumed
to be constant within the range of, sound pressure levels generally
encountered. The dependence of the impedance on flow resistance was
presented in Section 5.4. In order that the impedance calculated for
the glass fiber materials be in reasonable agreement with experimental
values , the value of flow resistance must be greater than the upper
limit of the range of values specified for manufacturing tolerances
within the material. This would indicate other dissipation mechanisms
involved within the material that are not included in the flow resist-
ance term,, Due to the internal structure of glass fiber materials,

116

the attenuation due to viscous and thermal effects for individual
fibers may have a pronounced effect on sound absorption. While the
frame of the material is assumed to remain rigid in these theories ,
movement of individual fibers may also create increased attenuation*.
In short, viscous and thermal interactions within the material on the
level of the micro structure have been neglected by these theories and
the total dissipation is accounted for only by the flow resistance,
a macroscopic property. Therefore , an investigation of the dependence
of sound absorption on the internal microstructure of these materials
would prove quite helpful in predicting their acoustic behavior.
Attenborough (25, 26) has modeled a fibrous absorbent as a collection
of cylindrical scatters and has developed a scattering theory approach
to determine the absorption characteristics in terms of fiber diameter
and fiber spacing. A scattering cross section, which includes viscous
and thermal effects, for a cylindrical obstacle is used in conjunction
with a single scattering theory to determine the sound absorption of
the glass fiber material. Further modifications include a multiple
scattering treatment to account for the interactions among scattered
waves. In this case, then, the dissipation mechanisms are due to:

1. Mode conversion to damped viscous and thermal waves in air
at the fiber boundaries „

2. The energy loss due to formation of the internal incoherent
field due to multiple scattering.

A logical continuation of this work should develop Attenborough ' s
scattering theory to include shear and thermal wave interactions.
Further consideration should include possible structure modifications
of the model to give a more accurate representation of the actual

117

material. In addition, the inhomogeneous nature of these materials
indicates that a statistical approach might also be used to treat
variations within the micros true ture for different samples of the
same material. Beran (27) introduces flow through a porous media
using Darcy's Law and statistics to determine the permeability of the
medium,, Such an analysis based on the micro structure would thus pro-
vide insight into the actual dissipation mechanisms involved within
a material, and as an ultimate goal, would dictate which parameters
are to be controlled in production in order to optimize results.
These topics will be the subject of future research and study.

118
BIBLIOGRAPHY

1. "Impedance and Absorption of Acoustical Materials by the Tube
Method," 1970 Annual Book of ASTM Standards , Part 14, American
Society for Testing and Materials, Philadelphia, pp. 126-138,
(1970).

2. "Sound Absorption of Acoustical Materials in Reverberation Rooms,"
1970 Annual Book of ASTM Standards , Part 14, American Society for
Testing and Materials, Philadelphia, pp. 161-167, (1970).

3. Dubout, P. and Davern, W. , "Calculation of the Statistical Absorp-
tion Coefficient from Acoustic Impedance Tube Measurements,"
Acustica, 9, No. 1, pp. 15-16, (1959).

4. Sides, D. J. and Mulholland, K. A., "The Variation of Normal Layer
Impedance with Angle of Incidence," Journal of Sound and Vibration,
14, pp. 139-142, (1971).

5. Tocci, G. C. , "Noise Propagation in Corridors," M.S. Thesis,
Massachusetts Institute of Technology, (1973).

6. Velizhanina, K. A., Voronina, N. N. , and Kodymskaya, E. S. ,
"Impedance Investigation of Sound -Absorbing Systems in Oblique
Sound Incidence," Soviet Physics - Acoustics, V7_, No. 2, pp. 193-
197, (1971).

7. Pyett, J. S , "The Acoustic Impedance of a Porous Layer at Oblique
Incidence," Acustica, 3, pp. 375-382, (1953),

8. Shaw, E. A= G. , "The Acoustic Wave Guide, I and II," Journal of the
Acoustical Society of America, 2J5, p. 224, (1953).

9. West, M. , "Acoustic Waveguide Having a Variable Section," Journal
of the Acoustical Society of America, 47, p. 12 (1970).

10. Ingard, U„ , and Bolt, R. H. , "A Free Field Method of Measuring the
Absorption Coefficient of Acoustic Materials," Journal of the
Acoustical Society of America, 23_, No. 5, pp. 509-516, (1951).

11. Beranek, Leo L„ , Noise and Vibration Control , McGraw Hill Book
Company, New York, pp. 245-269, (1971).

12. Industrial Insulation 700 Series, Owens Corning Fiberglass
Corporation.

13. Sabine, H„ J., personal communication, November 16, 1972.

14. Sabine, H. J„, personal communication, January 26, 1967.

15. Zwikker, C. , and Kosten, C. W. ,■ Sound Absorbing Materials .
Elsevier Publishing Company, Inc., New York, (1949).

119

16. Beranek, Leo L. , "Acoustic Impedance of Porous Materials," Journal
of the Acoustical Society of America, 13, pp. 248-260, (1942).

17. Ford, R. D. , Landau, B. V., and West, M. , "Reflection of Plane
Oblique Air Waves from Absorbents , " Journal of the Acoustical
Society of America, 44, No. 2, pp. 531-536, (1968).

18. Morse, Philip M. , and Ingard, K. Uno, Theoretical Acoustics ,
McGraw Hill Book Company, New York, pp. 227-270, (1968).

19. Rybner, J., Nomograms of Complex Hyperbolic Functions , Jul.
Gjellerups Forlag, Copenhagen, (1955).

20. McCracken, Daniel D. , A Guide to Fortran IV Programming , John
Wiley and Sons, Inc., New York, pp. 31-35, (1965).

21. Kinsler, L. E. , and Frey, A. R. , Fundamentals of Acoustics , John
Wiley and Sons, Inc., New York, pp. 130-136, (1962).

22. Standing Wave Apparatus Type 4002 - Instructions and Applications,
Bruel and Kjaer, Copenhagen, (1955).

23. "Airflow Resistance of Acoustical Materials," 1970 Annual Book of
ASTM Standards , Part 14, American Society for Testing and Materi-
als, Philadelphia, pp. 223-229, (1970).

24. Hughes, W. J. , personal communication, April 18, 1973.

25. Attenborough , K. , and Walker, L. A., "Scattering Theory for Sound
Absorption in Fibrous Media," Journal of the Acoustical Society
of America, 49, pp. 1331, (1971).

26. Sides, D. J., Attenborough, K. , and Mulholland, K. A., "Application
of a Generalized Acoustic Propagation Theory to Fibrous Absorb-
ents," Journal of Sound and Vibration, 19, pp. 49, (1971).

27. Beran, M. J., Statistical Continuum Theories , Chapter 6 - Flow
Through Porous Media, Interscience Publishers, (1968).

120

APPENDIX

NEWTON-RAPHSON ITERATION METHOD

The solution to an equation may, be found using an iteration
technique such as the Newton-Raphson method. According to this
method, if X; is an approximation to a root of the function,

F(^=0 (A.l)

then a better approximation is given by "*\ + y where

*U, a *i ~ -^^ (A.2)

F (.X\) denote's the derivative of F^O with respect to x evaluated
at K; . The root of Equation A.l can be obtained to the desired
accuracy by iterating successive approximations. The use of an
electronic computer renders this method very simple to perform.
In our case we are interested in solving the equation

cosKUS^-U + iV (A „ 3)

for values of ^5 A <i where U and V are known. Since $ is complex,
this involves finding solutions for the complex argument of a hyper-
bolic cosine function. If we introduce the complex numbers K and Y>
such that

K~ U + j V (A.4)

121

3= 2 5^ =2aK-^iM (a„5)

^>'- ^i 4 jk (A.6)

Equation A. 3 may be rewritten as a function of B

FCb") = cosl>-B--A - O

(A. 7)

The derivative of r(B>) with respect to B is

F'<^= sinnB (A. 8)

Therefore, if "Bj is an approximation to the root of Equation A. 7,
then a better approximation is given by

B- -R- cask's,- A
i+l " D « " . i -, (A. 9)

The root of F (B^ is obtained by taking successive approximations with
the iteration formula of Equation A„9. It must be noted that there
are two solutions to Equation A. 7 since the hyperbolic cosine is an
even function.

3, = \>,-t ^\> x (A.10)

1 -J x (A.ll)

Furthermore, the hyperbolic cosine of a complex argument is invariant
for multiples of 2^ added to or subtracted from the imaginary

122
component. Therefore,

S r - b^ 4 ? ± 2.vk) (Aol2)

3V-V^K±^

(A. 13)

are the set of all solutions. However, considering the physical as-
pects of the problem, we are interested only in roots with a positive
real component. This corresponds to the attenuation constant of the
appropriate wave propagation parameters.

123

APPENDIX - B

INTERFERENCE PATTERN CALCULATION

To predict the location of the pressure maxima and minima, an
incident plane wave is assumed diffracted at the edges of a finite -
sized panel. Three pressures - b, and to , the pressures diffracted
from the edges, and b , the pressure from the incident wave - will
be measured by the microphone located at the center of the panel. The
important consideration is the phase relationships between these pres-
sures as determined by the angle of incidence. Referring to Figure 59,
the phase component of each pressure will be taken relative to the line
X ( X . Therefore,

frD£

Uwttkj + K ft sme ^

(B.2)

(B.3)

The pressure at is the sum of these three pressures. Omitting the

itft
time factor e," , we have

f = fie + A e * + Le (Be4)

124

Furthermore, we will assume that the amplitude of the pressure at each
edge is the same so that "B - C „ Multiplying ^ by its complex conju-
gate, we obtain

lt.f= ^ + ^B 7 'c^(VT 5me ) + ^^^ Co 5^sme\cos( v k^ (B ,

The pressure fluctuations will have maximum and minimum values where
the derivative with respect to Q is zero.

(B.6)

For the derivative to be zero for some angle , the following condi-
tions are found:

I. case =

(B.7)

e =

3 n

2 > Z,

(B.8)

Sine = h — ** = °it,V

125

± *,

Figure 59 Geometry for Surface Pressure Method Tests

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Full text of "Absorption characteristics of glass fiber materials at normal and oblique incidence"

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