HANDBOOK OF ACOUSTIC NOISE CONTROL

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• Document Page Count: 327
• Document Creation Date: December 23, 2016
• Document Release Date: April 2, 2014
• Sequence Number: 5
• Publication Date: April 1, 1955
• Content Type: REPORT

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A. WILSON NOLLE ?
BOLT BERANT.:1? AND NEW:JAN INC.
APRIL 1955
WRIGHT AJP. DE'VELGPM3NT CENTER
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WADC TECHNICAL REPORT
VOLUME I
SUPPLEMENT 1
HAM3rgn:f.?v P.,CCUSTiC NOiSTI CC.CIF:ON.
Volume I. MT:km! Acoustics
Supplomont 1
EDITORS
STEPhEN 1. LUKASIK
A WILSON NoLLe
20 IT B;":.ANEK AND NEWMAN INC
APRIL 1955
WRIGHT AIR DEVELOPMENT CENTER
AIR RESEARCH AND DEVELOPMENT COMMAND
UNITED STATES AIR FORCE
WRIGHT-PATTERSON AIR FORCE BASE, OHIO
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ABSTRACT
Rnnnco:r. of Accazt.... Voico Control to intonded to proviao
ru morall v..:cu of tho problem of Oa control of accuotto =lc?. Sitoo
no publication of tha firat to tho ncixt for thoir roviaion
hro bocc.:o upparont. In ocJ como, matorial boa t,--on cadca to onlarco
002;',3 cl,:ticoo. In othcro, Lootions havo boon oroploto-
ly ro-uritton tu procant tho latcat on.ori:7,atal or thoorotical infor-
LrLtica
lath ovol--inorco-;l33 intoroot rta activity in rocuotic noiro 002-
tzclo publittfA proccflurcc L1t, at nocccaity, 1a3 1::)hica tho ns..mot
tlxiro:11:3 in tha eiold. 'Zisro.aro fou arc:...o of tho noico control problcm
unro tho proaw. casuca aro till obc*ta. Ac tho oporationnl rcvirc_._:nto
for noiro ncat..cui. a.'nrieC:b obanc2 end an nsu or toro pm:?rful coura cow:coo
onl:'or in cAr f.ianclirz bettor arl=?ro uill havo to ho fc-Ind?
prccootin3 tel7o rovicca cootiono, ea atV.:pt to hoinz El_10 to hcoP
ep with cur oxyarlins Imou10.63.
r.211.3 cupplc=nt contalvi ce.AiticaoC DO C13 to Volt=1 I uhich
tzcatcd tho c_11JrationCL ccatvol of Nr:oriono typo of nolco uourcon.
Volu7) I/,uhich or-117=d tho intorootico botroca noico and
toin3 1:7:.oc3 ctW1C to-otL-sr uith tho un-
c.:Vvicao oZ Volt: :3 CE21 :1, pro71C3 a unifica viou of ZOle0
ccntwol pvoblcco..
Futizcaszo mum:
to roposn hao boon rovic.:J.1 in apr...o7eis
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SECTION
4.1
4.2
4.3
6.3a
6.5
11.2
11.3
11.5
12.1
12.2
12.6a
12.9
TABLE OF CONTENTS
PAGE
Introduction xii
Propeller Noise
1
Noise from Aircraft Reciprocating Engines *0
43
Total External Noise from Aircraft with
Reciprocating Engines
45
Noile Generating Mechanisms in Axial
Flow Compressors
53
Ventilating Fans and Ventilating Systems ...
59
Insulation of Airborne Sound by Rigid
Partitions
75
Insulation of Impact Sound
127
Transmission of Sound Through Cylindrical
Shells
147
Specification of Sound Absorptive Properties
161
??
7.ined Due's
217
The Resonator as a Free-Field Sound Absorber
263
Acoustical Shielding by Structures
295
Errata
307
Rcrerczces 49, 73, 12$, 146 ,214 ,261 ,294,
316
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41*
TABLE OF CONTENTS
SECTION PAGE
Introduction xii
4.1 Propeller Noise 1
4.2 Noise from Aircraft Reciprocating Engines 43
4.3 Total External Noise from Aircraft with
Reciprocating Engines 45
6.3a Noi'3e Generating Mechanisms in Axial
Flow Compressors 53
6.5 Ventilating Fans and Ventilating Systems ... 59
11.2 Insulation of Airborne Sound by Rigid
Partitions 75
11.3 Insulation of Impact Sound 127
11.5 Transmission of Sound Through Cylindrical
Shells 147
12.1 Specification of Sound Absorptive Properties 161
12.2 1.1ned Duc'd 217
12.6a The acconator as a Free-Field Sound Absorber 263
12.9 Acoustical Shielding by Structures 295
Errata 307
References 49, 73, 12$. 146 .214 ,261 ,294. 316
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LIST OF ILLUSTRATIONS
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FIGURE
TITLE
PAGE
4.1.1
Coordinate system used in calculation of
noise radiated by a propeller
3
4.1.2
Distribution of fundamental frequency sound.
5
4.1.3
Distribution of second harmonic frequency
6
7
4.1.4
Overall rotational noise, for 1000 HP input
9
4.1.5
Overall rotational noise, for 2000 HP input
10
4.1.6
Overall rotational noise, for 4000 HP input
11
4.1.7
Overall rotational noise, for 6000 HP input
12
4.1.8
Overall rotational noise, for 8000 HP input
13
4.1.9
Overall rotational noise, for 10000 HP input
14
4.1.10
Cancellation of odd harmonics by a two-
bladed propeller
16
4.1.11
Force distribution on propeller and result-
ing sound spectrum
18
4.1.12
Measured and ca)culated polar sound pressure
distributions
23
4.1.13
Polar sound pressure distributions for
various forward speed Mach numbers
24
4.1.14
Acoustic PUL vs blade tip speed and input
HP to blade
31
4.1.15
Directivity for overall SPL for propeller
in a test stand
34
4.1.16
Propeller noise spectra
36
4.1.17
Idealized Karman vortex trail
39
4.3.1
Directivity of airplane noise
46
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List of Illustrations
Figure Title Page
6.5.1 System used to measure ventilating fan PWL 60
6.5.2 Spectra of vaneaxial and centrifugal fans .., 64
6.5.3 Possible positions for ventilating duct open-
ing in a room 66
6.5.4 Directivity for various duct opening positions 68
6.5.5 Room constant vs volume for various type
rooms 70
11.2.1 Shielding of a sound source 76
11.2.2 Shielding of a sound source by a partition 76
11.2.3 Loss or TL in a composite wall 83
11.2.4 Wall area subtended by an obliquely incident
plane wave 86
-
11.2.5 Bending wavelength vs frequency, for steel
and aluminum 95
11.2.6 Bending wavelength vs frequency, for concrete 96
11.2.7 Bending wavelength vs frequency, for plywood 98
11.2.8 Bending Wavelength of the lobest natural
frequency of a rectangular plate supported
at its edges 100
11.2.9 Illustration of the coincidence effect 101
11.2.10 Coincidence effect frequency vs plate thick-
ness for various materials 104
11.2.11 General behavior of TL vs frequency and angle
of incidence 107
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List of il1u2trations
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Figure Title Page
Average TL vs f:equency for a statistical 108
distribution of angles of incidence
Average TL Vs frequency for a plate with
Internal losses 110
TL for two similar panels, differing in stiff-
ness 115
11.3.1 Impedance of an infinite plate driven at a
point 130
11.3.2 Schematic diagram of impact of a mass on a
resilient plate 133
11.3.3 Spectrum of impact-induced vibration 135
11.3.4 Floating floor with sound bridges to ceiling
below 142
11.5.1 Coordinate system used in analysis of
cylindrical shell TL 150
11.5.2 Contours of equal TL for various angles of
incidence; 1)0 = 2 153
11.5.3 Contours of equal TL for various angles of
incident sound ; 7)c = 1/2 156
11.5.4 TL vs frequency for randomly incident sound 157
11.5.5 TL vs frequency for sound randomly incident
on a cylindrical shell having internal
losses 158
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List of Illustrations
Figure Title Page
12.1.1 Statistical abso,lotion coefficient for given
normal specific accrlatic impedance 175
12.1.2 Design chart for perforated facings 183
12.1.3 Sample statistical absorption coefficients
vs frequency 184
12.1.4 Sample normsl specific acoustic impedance 187
12.1.5 Statistical absorption coefficient for
perforated facings with unpartitioned air
backing 191
12.1.6 Statistical absorption coefficient for
perforated facings with partitioned
aAr backing 192
12.1.7 Design chart for perforai,ed facings with air
backing 194
12.1.8 Calculation of effective f1cr4 resistance 196
12.1.9 Specific flow resistance for various materials 198
12.1.10 Specific acoustic impedance for various
mateAals 204
12.1.11 Specific acoustic impedance for various
materials 205
12.1.12 Specific acoustic impedance for various
materials 206
12.1.13 Specific acoustic impedance for various
materials 207
12.1.14. Specific acoustic impedance for various
materials 208
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List of Illustrations
Figure Title
12.1.15 Statistical absorption coefficient for
various matertals
12.1.16 Normal absorption coefficient for various
materials
Page
209
210
Chamber absorpt%c:1 coefficient for 1" tile
on various mounts
211
Chamber absorption coeffir:ient for 3/4" tile
on various mounts
Chamber absorption coefficient for 1/2" tile
on various mounts
12.2.1 Coordinate system used in calculation of
sound propagation in a lined duet
212
213
C.
12.2.2 Design chart for lined duct with t/Ax . 0.2 221
12.2.3 Design chart for lined duct with t/tx . 0.4 222
12.2.4 Design chart for lined duct with t/lx = 0.6 225
12.2.5 Design chart for lined duct with t/Ax 0.8 226
12.2.6 Design chart for lined duct with t/Ax = 1.0 227
12.2.7 Summary of measurements of attenuation of
lined duets 243
12.2.8 Attenuations for ducts in Table 12.2.1 244
12.2.9
12.2.30 Design chart for lined duct attenuation; low
frequencies 253
12.2.11 Design chart for lined duct attenuation;
F = 0.25 254
Attenuations for mufflers ln Table 32.2.2
? ? ?
247
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List of Illustrations
Figure Title
12.2.12 Design chart for lined duct attenuation;
F 0.5
Page
257
12.2.13 Design chart for lined duct attenuation;
F 1.0 258
12.2.14 Design chart for lined duct attenuation;
high frequencies 260
12.6a.1 Sketch of resonator 265
12.6a.2 The hole parameter as a function of hole flow
resistance and hole thickness 274
12.6a.3 The optimum radius for a spherical resonator 276
12.6a.4 Resonance frequency as a function of
resonator size 279
12.6a.5 The Q for an optimum design resonator 283
12.6a.6 The reverberation time for an optimum design
resonator 283
12.6a.7 Departure of absorption, reverberation time
and from optimum design values 284
12.6a.8 The effect of non-linear aperture resistance
on an optimum design resonator 290
12.9.1 Sketch of geometrical situation considered 296
12.9.2 Noise reduction due to a shielding structure 298
12.9.3 The effect of a source above the ground 300
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Figure Title
12.9.4 Correction of noise reduction fsr ground
attenuation .............................. 302
scatterinG by atmospheric turbulence ..... 304
12.9.5 Correction of noise reduction due to
J 12.9.6 Irleasured attenuation near the edge of a
finite obstacle .......................... 3?5
Page
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INTRODUCTION
0
This section briefly describes the changes that have
been made to Volume I WADC TR 52-204, Handbook of Acoustic
Noise Control. The changes are essentially either of two
basic types.'In some cases, new sections have been added
on subjects not covered in Volume I. More et.en, wever,
the new sections reflect changes in theory or practice
which made a reorganization of the material desirable. In
one case, the new material was of a somewhat different na-
ture and was simply appended to the existing section.
These changes are detailed below to aid the reader in recog-
nizing the relative status of the old and new section. It
will be hoted that the revision has poceded on a section-
by-section basis. This has necessitated certain changes in
the figure and equation numbering conventions which are also
indicated below.
All of Chapter 4 has been revised although the bulk
of the chances are in Sec. 4.1 which makes up the mAin part
of the chapter. The discussion of propeller noise has been
reorganized around the existing theory. Both rotational
noise and vortex noise have been treated and N. A. C. A.
charts constructed from the Gutin theory are given. The
design procedure based on the empirical PIM chart is essen-
tially smchanced although its extension to other than three
blade propellers involves a somewhat greater uncertainty
than indicated in the original section. Chiefly, the
empirical chart works in the transonic and supersonic tip
speeds where available theory is not as well developed.
Also, the two spectrum charts have been replaced by a single
curve which is similar to the transonic tip speed case of
the original section.
Section 6.3a adds lc^ the empirical information on
axial flow compressors pmsented in Sec. 6.3 The new
section discusses the physical principles involved in noise
generation by an axial flow compressor. It contains a
short statement of the theoretical results to date and
illustrates them with a calculation of the absolute sound
pressure level for a compressor of given operating condi-
tions. The previous empirical design procedure is still
applicable. Nothing new is presvnted on centrifugal
compressors.
Section 6.5 on ventilating fans and noise from
ventilating systems is new. There is no section in Volume I
to which it corresponds.
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The leetions on wall construction and floating floors
in Volume I have been greatly expanded and reorganizea
around existing theory. However, the original sections are
still correct in what they say and they form a good intro-
duction to the more detailed dis-uf?Rion of he revised
Secs. 11.2 and 11.3. In particu'_r. Sec. 11.3 on the Insula-
tion of Impact Sound corresponds only roughly to the
original Sec. 11.3 dealing with floating floors. The
original section has more architectural details which may be
useful to the reader.
The new section on the transmission of sound through
cylindrical shells is intended to replace completely the
original section in Volume I. Research in this field is
continuing, however, and more experimental and theoretical
information may be expected in the future.
Section 12.1 on the specification of souna absorptive
properties of materials is new. It replaces the very short
introductory section in Vol. I which simply listed several
topics to be discussed in connection with the control of
airborne sound.
The section on the attenuation of sound in lined ducts
(Sec. 12.2) has been greatly expanded. Several different
theoretical procedures for calculating the attenuation, each
of various degrees of accuracy and usefulness are presented,
and all the available empirical information is summarized.
A tabular summary of the various procedures is given. This
revised section is intended to replace the original section
in Volume I completely.
Section 12.6a discusses the use of acoustic resonators
In free space. Since the original section discussed resonators
attached to ducts, the subject matter of the old and new
sections are complementary rather than overlapping.
Finally, Section 12.9 presents a new design procedure
for the prediction of acoustic shielding by an obstacle.
Although it is based on the same diffraction theory as the
original section, several modifications found necessary In
actual practice have been introduced.
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Because the total number of equations, figures, etc.
In each revised section do not, in general, equal the cor-
responding number in the section replaced, a new identifica-
tion scheme has been used. Previously equations, figures,
tables and references were numbered consecutively through
a chapter and were identified by chapter and/or a serial
number. Now all identification lumbers refer to both chap-
ter and section. in addition to a serial number. For
example, the fifth equation in Ch. 12, occurring say in
Sec. 2 is now numbered Eq. (12.2.51 while previously it
would be numbered simply Eq. (12.5" References, instead
of being a single number, such as Ref. (7) now contain a
section identification also; the fourth reference is Sec.11.5
and is now numbered (5.4). Finally, a letter a following
a section designation indicates that the section does not
replace the previous section, but merely supplements it, e.g.,
Sec. 12.6a. 3igure, equation, table and reference numbers
then contain the letter also, e.g., Fig. 12.6a.5.
A list of errata to Volume I is given at the end of
this volume.
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CHAPTER 4
AIRCRAFT PROPELLERS AND RECIPROCATING ENGINES 0
4.1 Propeller Noise
Introduction. The propeller, rather than the engine,
is the?Chief source of noise In the usual reciprocating-
engine aircraft of 200 horsepower or more. For this reason,
considerable work has been done toward explaining the action
of this important noise source. The problem has not as yet
been treated rigorously from a theoretical standpoint, but
the approximate analysis which has been done has proved
satisfactory for engineering purposes in the case of pro-
pellers operating at subsonic blade speeds and not too close
to obstacles. Also, the approximate analysis mows clearly
the role played by the various parameters which are important
In propeller noise generation, including particularly horse-
power, thrust, tip speed, diameter, and number of blades.
The results of this analysis are given here. Measurements
are cited and comparisons between theory and experiment are
shown where possible. Equations and charts for engineering
calculations are given. Their use is explained in a numerical
example at the end of the section.
Gutin's Theory of Rotational Propeller Noise. A rotat-
ing propeller blade ,at constant speed carries with it a
steady pressure distribution. Hence, any non-axial point,
fixed in space with reference to the aircraft, experiences
a period.Lc pressure variation, generally of complex wave
form, always having the blade passage frequency as the funda-
mental. This periodic pressure variation is an acoustic
disturbance, and is known as the rotational noise. For
points lying in, or very nearly in, the volume swept out by
the propeller blades, and for cases where there is negligible
overlap of the pressure distributions of adjacent blades,
the pressure distorbance due to a multiple-blade propeller
can be approximated simply as a repetition, at the appro-
priate frequency, of the disturbance dua to the passage of
an isolated blade. (In other words, for such near points,
the pressure disturbance at a given time is due to the
nearest blade, the influence of the more distant blades be-
ing negligible.) To this approximation, the acoustic
disturbance very near the propeller can be simply expressed,
and the disturbance at more distant points can then be
calculated by integrating the signal propagated from all
regions near the propeller. To facilitate this calculation,
the disturbance is considered to radiate from a zero-thick-
ness disk in the region swept out by the propeller. This
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i3 the basia for Gutin's analysis of the rotational pro-
peller noise 14/. The Gutin analysis does not consider
nonperiodie drs urbances (principally vortex noise), which
are produced by an actual propeller along with the periodic
rotational noise. These will be considered later. The
analysis assumes that the forwart speed of the propeller is
small compared to the speed of sound.
Gutin's analysis proceeds by writing expressions for
the reaction on the air of the time-dependent thrust and
drag forces due to a single rotating propeller blade. These
forces are then expressed as a Fourier series; the funda-
mental frequency is the blade passage ftequency n a/where n
Is the number of blades in the ,ropeller and QC. is the
rotational Irequency in radians/sec. The force exerted on
the air by a rotating blade also depends on the thrust dis-
tribution along the blade. In the Fourier expansion, the
sine function is approximated by its argument mn Or:t where m
is the harmonic number and t is the time. This is justified
provided that the discussion is reatricted to a suitably
small value of the product of number of blades and of har-
monic number, and provided that the portions of the blade
near the hub (which produce a relatively small part of the
air forces) are ignored. Gutin also shows that his
expressions, which are in no case valid for high harmonics,
are correct when the air forces are not uniformly distributed
over the width of the blade.
Expressions for the aerodynamic disturbance in the
propeller disk 1,71v1ng now been established, the next step
is to compute the resultant acoustic effect at extrrnal
points. The coordinates shoun in Fig. 4.1.1 are used.
From hydrodynamics, we can immediately write the velocity
potential g for the resultant sound field from the known
forces acting on the air due to the rotating propeller
blade The sound pres&ure is the time derivitive of
the ve ocity potential. That is, for an air density p,
the sound pressure p is pdP/dt. While this gives the
desired acoustic solution in principle, some simplifica-
tions are desirable for ease in calculation. Gutin
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Coordinate systems used in calculation of noise
radiated by a propeller.
restricts the point of observation to the xy plane, with-
out loss of generality, and also restricts r to values
much greater than the propeller diameter. The latter
stipulation will make the succeeding work inapplicable to
the near field, so that the results under this restriction
will not apply to noise levels within the aircraft itself.
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T i ; d, ,tr.blc to not the rewilt in a form which
1 ,I t.11._.1 1.:1:, .led(2 of the distribution
1 o.? 11.,! , 1 onl; the 1. hide. This ifs achieved
,I Iv ,ni 1 ' .'lr,.., the total thrust P and the
to', ' :I to -?ct t i t 4 fCcttve. ; ean radii 113. and R2
I ,,? ,ure becomes
I 3 In c/) .ricn
,
kkR
sin
0311(22
(4.1.1)
1 -til frquency is col, c is the velocity
1: J I-3 the r2.2Jel ft.nation of order inn and k
_3 ?cyof the m th harmonic of cul. Gutin
t'Ltt, fur the lower harmonics produced by
viv, a number of blades, both R1 and
occx,1 to Fic, the radius corresponding
tr of r,:.sultant thrust for a single blade, which
0 .0 Or 0.7 or 0.8 of the propeller radiusgo ?
to the final niruplified result,
r , P ncM - P cos IP- F J 2 mn (kR sin 2)-)1
c
co1Rc
(4.1.2)
:his expression is a sum of two terms, the first of which
4.3 the thrust term, and the second of which is the torque
Iterm. The torque is proportional to the input power, W,
through the relation
-
W = M (, ()..1.3)
/
11.
ti
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FROM GUTIN
THEORY EQ.(4.1.2)
130=0.75 R0
150
1200
goo
KEMP'S MEASUREMENT
(FUNDAMENTAL)
60?
FROM GUTIN
THEORY EQ.(4.1.2)
Rc = 0.7 R0
1000
30?
Figure 4.2.1
t>- 0?
DIRECTION OF
FLIGHT
Calculated and measured distributions of
fundamental-frequency sound pressure from a
propeller. The measurements are by Kemp 1.14/,
The calculations are from the Gutin equation
(4.1.2), for values of Re c;ual to 0 7R and
o
to 0.75 Ro.
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FROM G UTI N
THEORY EQ. (4.1.2)
Calculated and C:;30ured dletributiono of cocond-
harmaic coznd preocuro frfs a proDoXlor. The
Ezaourazanto are by Kc-1D 1.12/. The calculationo
are fma tho Cutin cqyation, (4.1.2), uith no = 0.75 Ro
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0
vs. The thrust P is related to the input power by an
aerodynamic relation which Gutin gives in the form
P = (200112)313 (4.1.4)
where S is the area of the propeller disk and 11 is an
efficiency factor estimated to equal about 0.75.
Cutin calculated the expected polar distribution of
radiated sound for the first two harmonics, for the follow-
ing situation: Two-bade propeller, radius 2.25 meters,
1690 kg thrust, 515 kgm torque, 13.9 rev/sec. The results
were compared with experimental data for this situation
as taken by Paris 1.1/ and by Kemp 1.iy, with values of
both 0.7 and 0.75 Fang tried for R77110. The comparison
with the Kemp results is shown in Figs. 4.1.2 and 4.1.3.
The agreement is fair for the fundamental, but appears to
deteriorate for higher harmonics. This would be expected
from the nature of the assumptions made in the derivation.
Fortunately, the fundamental usually constitutes the
greatest single contribution to the sound output. Gutin's
calculations showed slightly better agreement with the
Paris data (fundamental only).
The general features of the polar patterns in Figs.
4.1.2 and 4.1.3 are found in virtually all eases of noise
generation by a propeller free of obstacles. The torque
term results in an aeoustic pressure pattern which is zero
on the propeller axis and maximum in the propeller plane.
The thrust term results in an acoustic preSsure which is
somewhat smaller than the maximum torque contribution (this
need not always be true), and which is zero in the plane
va. of the propeller as well as on the axis. The two contri-
butions are out of phase for positions in front of the
propeller, but in phase for positions to the rear. The
combined effect of the two terms is a radiation pattern
having symmetry of rotation, which is zero on the propeller
axis and which is maximum at a position some 150 behind
the propeller plane.
N.A.C.A. Propeller Noise Charts Based on Gutin's B.g.tl-
tion. No propeller noise analysis is available which does
not Includeat least some of the approximations made by
Gutin. Fortunately, the simplified Gutin relation,
7
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1
? ? e
' a I
; - : 1 ? .2 Ivr :.c. Aten
; :tt 1:. .11;tMt
1.3 .?t
. I : -01,31 .3 I. 3..t. 3c. tux,: 1.) 1?
1.1 :(:?:_ 21.I. ? LI, )--2.3?:c(r, to, ,n? of 1 lades,
3. ? ( ..1. _??1 I-to I..? . i.r.c b..tc.'n 1.p. The
iC c :o?? : _ level at
-? ' 5?..1 .1.. :".L ?1.0;1 rt..(31-37ed 11,01;3
t; L7,1. ?lition of
1-,_ , 13. tr: 0?? 313) . The sound
?.. . (t(?.:?3 .13 n t the A, ? h.. ?l1e3 have
0:1 ? . .1 1 ) Lzive t1U. ii' t; hence,
? -. .31 .(?( 3011.,3tI.V of overall
. ? .. ..,r.? W 1C.f. Cr:v.:1211'f ly tI contributions
?al. 3I tr3 41.3..3.3b cirap 0 T
; 3. )3' 31 t'adiation pattern
3 1,..t Imunct un-L3 ',. ? ?1. 1,3 t,:??t ?,?.;tien of
? ' ? t.: ? 1-, c?:-..tvergze
it' ice, L r1lnul.:1 Toh'_:ac..,;30. from tho chart
.o 3.131Ca? 13ouncl pi' sure level
, ?:*)1(..tht. c 31,Y) r1r 55 (lb to the 11.A.C.A.
r 3.1),..-ruz?;,;.ttcly the pol.:er leel of the
nil a 1'O .0 30'.11,(10
':ne :-.:3?Iult J3 found in the N.A.C.A. publica-
lnd others in the fora ;?nd symbols
" 11:3 iz-.; lapted to simple ensineering
on.
1(9.337,13Diit
T cos p Juo ( 0 . 8mB It sin /3).
(4.1.5)
8
't t-
? %." ?
.
,
STAT
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bes
vas.
Eq. (4.1.2), seems to give the maximum overall sound
pressure in the far field of a propeller to an accuracy
sufficient for the usual requirements of noise-control
engineering, at least for those propellers operating at
subsonic tip speeds which are currently in use.
A convenient set of propeller-noise charts has been
computed from the Gutin relation by Hubbard 1.5/ under the
auspices of the N.A.C.A. These are reproduced in part
in Figs. 4.1.4 through 4.1.9. The independent variables
are input horsepower, propeller diameter, number of blades,
and rate of rotation or Mach number of the blade tip. The
result is read from the charts as sound pressure level at
a distance of 300 feet, at a position 1050 removed from
the forward propeller axis (approximately the position of
maximum sound pressure in ordinary cases). The sound
pressure contributions from the first four harmonics have
been added on an energy basis to give this result; hence,
the values obtained are closely representative of overall
sound pressure level, since ordinarily the contributions
of the higher harmonics drop off rapidly.
Analysis of a typical propeller radiation pattern
shows that the sound pressure level in the direction of
maximum output is about five db above the space-average
value. Hence, 5 db should be subtracted from the chart
values to obtain the space-average sound pressure level
at a distance of 300 ft. Adding 55 db to the N.A.C.A.
chart values gives approximately the power level of the
propeller as a noise source.
The GutIn result is found in the N.A.C.A. publica-
tions by Hubbard 125/ and others in the form and symbols
of Eq. (4.1.5). his is adapted to simple engineering
computation.
169.3thBDMt PH
p - c(0.8M
2sA 2 T cos p JmB(0.8tBMt
0
(4.1.5)
8
? 404;i ;;
STAT
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140
130
1
,
?
WO
1.10
103
...,
93
6
70
6
SO
t .
3
4
-
--r
---
....-
--
.......
...?
Yort?S
-
-
...,-
.......-
' .......E?e2,...7,'
.
140
130
? A .6 h a
WWUCO
13C0 16.13
I. rp?
D = 8 ft
.s
D = 16 ft
1.0
2203 2500 sans
-II
IL
lib
hiliPPLA
ILIFF0411
Pip"
la
is
70
II
4111
aaa
UU
A
17
Vt
76 ?
116 13
a. no
Propeller Diameter
1)0
130
103
of'
3.11.
Vi' 93
60
70
60
50
A A me .7 .t .9 1.0
7 9/3? UW
11.1
FIGURE 4.1.4
""-?!!!--""
D = 12 ft
1500 ..7S3 1b3
....
.
' ...
I
1
4
.4.
...
uo.
oo
....
.
ii,
1
1.
Pepa"Pled
1.0.
1
dUIRI
911
00
Amp.
KaitArlid
_I-.
rouri
-e
1l
,
s_,,,
Lel
I
91C0
.1. .5 .6 h .7
40%r?it* CIO70 FC0
A 1.0
906 106 116
Overall rotational propeller sound pressure level, at 1000 horsepower input,
as a function of tip Mach number and number of blades, for various propeller
diameters (solid-line curves). The values are for 300 ft distance, in a
direction 105? from the forward axis. To obtain the approximate acoustic
power level of the propeller add 55 db to the result. The broken-line curves
are estimated levels due to vortex noise. From Ref. (1.5).
9
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milimmummom
mmilmmin
ilibmom Immo
mommumm mum
mommommimmomm
um implimwrom
el
um madapolumm
nommeAmmum
....sOmmandomm am
Vimilmirm
r ? MIRO
OM
Ik1411 11111
II SM.
MA IR MMEMIMME
IIIIIIIPIIM
MI MO
IMMOMMEMINIMMUM
MMIMMIMMUNIMM
MIUMMOSEMINAM
EMMIMEMMUMEMMOM
1111110111=0101111WME
INIMMEMOMM
mummemmagotamm
immoseassamm
_Aims mmgmmom
itniNOM
imams mmusems
imommffigammomm
mommumminumgma
mamismompummu
mmumilimmumm
mum mum
momm? mum
mummum? ?
momingsmomm
? maimmismon
immumgmum
um ? mmmumgral
?
pp moms=
Amiummorm
'orA ommoun
mipmplir um
wimp MMONIMM
MEM
M INIMMEM
ir MUMMOMUMM
MEMMEMMINIMMEMM
MAIMMEMMMINIM
MRSSMINIMEMM
FIGURE 4.1.5
Same as Fig. 4.1.4, but for 2000 horsepower input.
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MIMMIUMMI7MMERM
MMINIMMIm MUM
IIMMUMMEM mown
mommummumprogmm
mmummormoodum
ummilmalgmumm
Wil
lwailormA ? mm mp imams
1112 mmAtim slam
mllymmmm mom
um Amommilm mum
m mamma maim
Gomm *mum
F MENEM MEM
imMEMMOMMINOM
IMMINIMOMEMMUUMS
MIMI& I 1
1.0
3 ?4 ?7
1,173
23,20 15W
D = 8 ft
D = 16 ft
Propeller Diameter
D = 12 ft
D = 20 ft
KuuRE 4.1.6
Same as Fig. 4.1.4, but for 4000 horsepower input.
11
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I
.9 .0 ?? 6gt .1 .1 ?9 1.0
UW 19W 11t0 2Q 700 900 1107 1.700 /505 1700 1953
1417.
1.rpn
Propeller Diameter
= 16 ft D = 20 ft
mirmlimmumum
? mom ommumm?
m um maw
ommuummampo
mummompag
mimmowdessmil
minspfammmm
monsieftlimm no
mmilMISMIMIUMli MN
HOFIIAMMEMMEMIMINI
WAMMMEMEEMM
IMEMOMMEMOMMM
MWEIMMEMINIMMI
MmENIMMEMMUMMINIE
NaMMILIMEMMEMMOIM
MEMMENIUMMOMEMM
MMIIMMOMMUMMI
Mm1111 MEM
? erti
STAT
50
? . 6 .7 .1 9 1.0 .1 ?
WO 9co 3100coo 500
14.7,
FIGURE 4.1.7
Same as Fig. 4.1.4, but for 6000 horsepower input.
- - -
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A
U-13 /320 15C0
I.
WOO UW 1LW 19M
3, 11.
Proreller Diameter
= ft
11111MISIMMIIMINUMMINI
111111.111111111Mallinall
11111111111111111011117-21111
4 =,Co
"
leirdid9131136111.11
CO
A Will1111111111
IIIr
NM IIIIIIII
al1111111111111
111111111111111111111111
OHM 1111111111111
1111111111111111E+31
CO3 7t0 P73 9001e11/0
FIGURE 4.1.8
Same as Fig. 4.1.4, but for 8000 horsepower input.
13
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1.0
17t0 140
Propdler Diameter
D =. 1.2 ft
1111111MMINIMM MIME
OMMEMEMMOMMMOMMmEM
0 WM IUMMEmprooMM 13
MN
Mall MEMPPOOgilm L'O
MIIMMINAVgAMM a
T-411g111111"11
OMOOMMIN =II
Ord111111111
Ili Vommunsms ?
"M112111111111.
Illiiimmommial
Hill:111M
,...molumm ?
a .e a
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?
Here m is the harmonic number; B, number of blades;
D, propeller diameter, ft; Mt, tip Mach number; s, dis-
tance from propeller hub to ebserver, ft; A, propeller
disk area, sq ft; PH, input horsepower; T, thrust in
pounds; p , angle between forward propeller axis and line
of observations. The effective radius has been taken as
0.8 of the total.
In Hubbard's calculations, the thrust is derived
from the input horsepower by a relation equivalent to the
one used by Chitin, Eq. (4.1.4), except that a revised
value of the constant gives thrust values which are 0.78
of those computed by Gutin's procedure. The procedure
used by Hubbard is said to be approximately correct for
propellers operating near the stall condition.
The sound pressure levels given in the N. A. C. A.
charts include an estimated contribution from the non-
periodic vortex noise, which ordinarily constitutes a
small portion of the k,^tal propeller noise power. The
basis for calculation of the vortex noise will be discussed
later. The broken lines in the charts indicate the es-
timated levels of vortex noise only.
Effect of Number and Shape of Blades on the Rotational
NoiseT-M.W-OT the most important parameters which can be
altered in the propeller with a certain amount of flexi-
bility are the number and shape of the blades. It is
readily visualized that the number of the blades determines
the frequency of the fundamental blade passage tone. On
the other hand, it can be shown that the intensity of the
sound will decrease as the number of blades is increased.
A qualitatIve explanation for the reduction of sound
output by an increase of the number of blades can be given
on the basis of the phase: cancellation of the several com-
ponent forces. A simple example is given by the Generation
15
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ONE
"4"-- REVOLUTION
,?11
PRESSURE IMPULSES FROM BLADE
FUNDAt:FNTAL
SECOND WARMONIC
P\IV\P
THIRD HARMONIC
??????
????????????
ONE
REVOLUTION
WWI
PRESSURE IMPULSES FROM BLADES
%
00".4%
%
CANCELLATION OF FUNDAMENTAL
I //-\\ S.
/
SECOND HARMONIC (NEW FUNDAMENTAL)
r%f?-v-v-Nr\f-v-\x,-.
CANCELLATION OF THIRD HARMONIC
(a) ONE-OLADE PROPELLER (b) TWO-DLADE PROPELLER
121017.3 4.1.10
Illuctyation of tn acor5t1a procGuro ce7Donenta
dpvoloz:d by a c=1-b1ed3 aLd of tb..1 canc.:311a-
tica of tta odl hammica of th3 oricAnal aicpal than
a coecadblado lo added.
16
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1
of sound by a propeller consisting of one blade only. The
corresponding aerodynamil force is shown in Fig. 4.1.10.
In that figure the Fourier components have also been repre-
sented (not to scale). In Figure 4.1.10 the case of a two-
blade propeller is considered. The Fourier components of
the force shown in this figure indicate that the odd harmonics
(with reference to the original one-blade propeller) cancel,
while the even harmonics are reinforced. A quantitative
calculation shows that the net effect, however, is an overall
decrease in the sound intensity. For the special case in
which the tip speed, the thrust, and the input horsepower
are kept constant, while the blades are redesigned and in-
crear.ed in number, the acoustic effect can be seen directly
from Eq. (4.1.9. The quantity which varies is mB (JmB
(0.8mBMt sin p ]. Exanination of tables of Bessel functions
shows that, for typical values of the variables, this quan-
tity decreases rapidly as mB increases.
The effect of the blade width can be particularly
Important for the higher harmonics. In the nutin approxi-
mation, the force produced in the propeller plane by the
passage of an individual blade is treated as an impulse.
This is equivalent to assigning the propeller blade a
negligible width. Regier 1.6,/ has evaluated the spectrum
distribution corresponding to several more nearly realistic
force-time characteristics, as shown in Fig. 4.1.11. All of
these distributions have equal areas under the curves, and
thus exert equal forces on the propeller. The horizontal
line for the zero-width blade corresponds to the uniform
Fourier amplitudes in the Gutin approximation; the other
curves show the new distributions which replace this one in
the case of finite blade width. It is apparent that increas-
ing the width of the blade, while the thrust is kept constant,
decreases the intensity of the radiated sound through reduc-
tions in the amplitudes of the higher harmonics.
The role played by the number and kind of blades
in the total noise radiated by a propeller is illustrated
in a series of experiments by Beranek, Elwell, Roberts,
aid Taylor 1.7/. The experiments consisted in measuring
the noise radiated in flight, by certain aircraft of less
than 200 horsepower, for propellers of two, three, four,
and six blades. The propellers exerted approximately
17
STAT
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4 8 2 16 20 24 28
HARMONIC NUMBER
Bffect of tho ct^..7o of the force aictributiee
arer_ol a pre;oller blneo eo tho tar=oic eaatent
of pre:oller retaticnll nolco (for palate ncnr tte
pre;oller plane). nil dictribetionn have the
c=o arca. a3 nur':or refer? to the Curatioo of
to ra/ca co a porcaltaro of tho taco for a fell
revolution.
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equal thrusts and were of nearly the same diameter. The
results may be summarized approximately by the statement
that the intensity if lowered 6 db for each doubling of
the number of blades in the propeller, the input power
and the speed of rotation remaining fixed.
Hicks and Hubbard 1.18_,/ measured the noise from
small propellers of two, fur, and seven blades under
controlled conditions, and compared the measured sound
levels with calculations from the Gutin equation. A
selection of typical results is given in Table 4.1.1.
The sound pressure levels refer to a point 30 ft from the
propeller hub, in open air, in a direction 1050 from the
forward propeller axis. The blade angle is 16.50.
TABLE 4.1.1
MEASURED SOUND PRESSURE LEVELS FROA 4-FOCT DIA/STER PROPELLERS AND
CALCULATED LEVELS FROM TEE GUTIN EQUATION - REFERENCE 1.8
Overall SPL of
SPL by SPL by notational
Input Wave Wide-Band Noise, from
Number of Tip Mach Horse- Analyzer Measure- Gutin
Blades Number poster Method ment Equation
db db db
2
0.3
3.5
T9.6
85.8
83.8
.5
20.5
95-9
95-9
98.0
-7
65.8
111.4
110.4
111.1
-9
1148.2
123.4
121.6
123.0
4
0.3
6.0
75-8
81.9
65.8
-5
34.2
94.3
96.9
90.9
.7
110.0
110.6
111.5
110.5
.8
167.8
116.8
116.4
7
0.3
10.7
68.8
78.3
38.4
-5
53.0
85.0
89-9
80-9
STAT
.64
124.0
99.2
loom
98.6
19
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The results by the wave analyzer method refer to the
square root of the sum of the squares of the amplitudes
for the first five harmonics of the blade passage fre-
quency. This method therefore measures the level of
the periodic rotational noise, provided that the effect
of other noise components falling within the pass band
of the wave analyzer (25 cps) is negligible. The
calculated values represent the square root of the sum
of the squares of the individual calculated amplitudes
for the first five harmonics.
For each propeller, the SPL measured by the wave
analyzer method and that measured by the wide-band
method in the range of Mach numbers above about 0.6,
are both closely equal to the value predicted by the
Gutin theory. This means that the noise at the higher
Mach numbers is almost cntirely of the rotational type,
and that its overall level under these conditions is
adequately predicted by Gutin's equation. Thus, as far
as operation at the higher Mach numbers Is concerned,
theory and experiment agree m to the amount of reduction
in noise level which is obtained by increasing tha num-
ber of propeller blades and reducing the tip speed. For
example, in Ref. 1.8 it is found that for a tip Mach
number of 0.7, 66 horsepower can be absorbed by the 2-
blade propeller with a 16.50 attack angle, and 76 horse-
power by the 7-blade propeller with a 100 attack angle.
Although the horsepower is nearly the same, the second
configuration gives a wide-band sound pressure level of
101 db, as compared to 110 db for the first. The calcu-
lated values are 100 db and 111 db.
In the results for each propeller configuration in
Table 4.1.1, the overall SPL at the lower Mach numbers
is greater than the SPL by the wave analyzer method,
which is in turn greater than the calculated value from
the Gutin equation. These effects are explained at
least partially by the additional observation that the
sound at the lower Mach numbers consists mostly of
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nonperiodic vortex noise rather than periodic rota-
tional noise. In the theory of vortex noise, which is
discussed at the end of this section, it is shown that
this should occur, because vortex noise decreases less
rapidly than rotational noise as the tip speed is reduced.
The data in Ref. 1.8 do not show conclusively whether or
not the Gutin theory remains approximately correct for
rotational noise alone at the lower Mach numbers, since
it is not certain at what point the wave analyzer results
begin to represent vortex noise. These experiments seem
to show, however, that the Gutin equation predicts
overall propeller noise to adequate engineering accuracy
under those operating conditions where rotational noise
is dominant.
Deming's Extension of the Gutin Theory.. Deming 1.9/
attempted?to improve upon the Gutin approximations by
including the finite thickness of the propeller blades
in the analysis, and by introducing the concept of distri-
buted aerodynamic forces, instead of assuming the force
concentrated at one value of the radius. It was hoped
that considering the finite thickness of the blades would
Improve the accuracy of the calculations for the higher
harmonics, for which the assumption that the propeller
thickness is much less than the wavelength of the radiated
sound is not justified. Deming also performed a careful
series of experiments. It was found that the particular
Improvements which he had made in the Gutin theory did
not Veld results appreciably different from Gutin's,
but that the experimental work showed a greater disagree-
ment with the theory than Gutin had originally suggested.
Figure 4.1.13 shows a comparison between Gutin's and Deming's
calculations, together with Deming's measurements.
The Effect of Forward Speed upon Propeller Rotational
Noise. The Uutin equation must be modified, when it is
desired to find the noise radiated by a propeller moving
forward in the air, to take into account the fact that
the forward speed. alters the effective acoustic path length
from an element in the propeller disk to the point of
observation. Garrick and Watkins 1.10/ have worked out
the necessary changes in the theory. Their result for the
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far field is given in Eq. (4.1.6). The point of observa-
tion remains in a fixed position relative to the moving
propeller.
Q(17= t 42
inm
2.mcVx2-1-p2y2
E ( / 2)
-I-
x 2 iRc+13 y;1
2 2 f (x.1./g2y2
\ mw1yRc
111441 x B C
pr
(4.1.6)
In this equa ion, m is the harmonic number; col, funda-
mental frequency in radians/sec; c, speed of sound; p
denotes //a - m2; M, Mach number for forward speed; T,
thrust; Q, torque; B, number of blades; Re, effective blade
radius; x,y, coordinates as in Fig. 4.1.1. Setting p equal
to unity gives a result equivalent to Eq. (4.1.2) or
Eq. (4.1.5) for a statically operate. ft propeller..
It is found from Eq. (4.1.6) that the effect of in-
creasing the forward speed, for a propeller operating at
constant thrust, is to increase the noise output and to
alter the directional distribution in a somewhat compli-
cated fashion. Garrick and Watkins also give equations for
computing the near field of the propeller with forward
speed.
The effect of increasing the forward speed under condi-
tions of constant thrust corresponds to a hypothetical case
which is of less practical interest than the effect of
increasing the forward speed and allowing the thrust to
decrease in the manner of an actual propeller. Apparently
this decrease of thrust will usually cause the noise of
an actual propeller to decrease with increasing forward
MIME 4.1.12
Comaricon of obcerved coundproosure distribution
around a prvreilcr vith Gutinto and De:pinata theories.
rzacured distribution, ; Gutin's prediction
Dening's ce,dified result, _ Part A, funda-
E3nta1 frequ:ncy; Part B, second harmonic; Part C,
third har=onic; Fart A, fourth harconic .
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4D ?
180?
- MEASURED DISTRIBUTION
GUTIN'S PREDICTION
--- DEMING'S MODIFIED RESULT
90?
>00 ISO'
DIRECTION
OF FLIGHT
270?
FIRST HARMONIC
(A)
90?
0?
cj U
DIRECTION
2700 FLIGHT
IRECTION
2700
THIRD HARMONIC
(C)
90?
00
DIRECTION
OF FLIGHT
270?
SECOND HARMONIC
(B)
180?
90?
00
DIRECTION
OF FLIGHT
270?
FOURTH HARMONIC
(D)
23
I????
'410' I
TAT
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0
co ("qua' (c) !, ? T?113S0b.
M ? 0.21 T?1600 lb.
c'''"*/""I (c) M ? 0.4i T?875 b.
0. le&
(d) 1:1 ? 0.61 T?580
FIlotit dkce4-1.
120.
1Cf* jCP
m4=4/?"? (I) M ? OA T'205 b.
-LM y ? 20 feet
Circb 16?20 tat
FIGURE 4.1.13
Polar diagrams of the distribution of rms Elmnd pressure for a 2-blade,
10-foot diameter propeller, for various values of forward-speed Mach
number, M. Solid lines, values along a line 20 ft from the axis and
parallel to it. Broken lines, true polar patterns at constant radial
distance of 20 ft. The blade angle is always adjusted so that the input
is 815 horsepower at a tOrg:110 of 2680 lb-ft. The thrust values are
shown in the figure. From Ref. 1.10.
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speed up to Mach numbers of. about 0.4. Garrick and Watkins
have calculated the,, noise output of a two-blade propeller
for various forward speeds, with the thrust values taken
from actual aerodynamic measurements. The results are
shown in Fig. 4.1.13. The initial drop of noise output as
the forward speed increases is confirmed in a measurement
by Regier 1.11/, who found that the overall noise developed
by a light trainer airplane in normal flight is 6 db less
than that produced by the same airplane in static ground
operation.
As a practical matter, the distinction between the
Gutin relation and the modified equation for the case of
forward flight, Eq. (4.1.6), may be neglected for forward
speeds up to M = 0.3. At this speed, the value of p has
dropped only to 0.95, from the value 1.00 corresponding to
static operation. Therefore, within this range, the effect
of forward speed may be represented adequately by making
the appropriate changes in the thrust value used in the
original Gutin approximation.
Noise Levels Very Near a Propeller. Calculation of
the noise levels near a propeller by Gutin's method requires
that some of the convenient geometric approximations be
omitted and that more complicated integrations be carried
out. These calculations have been done by Hubbard and
Regier 1.11/ for several cases. The work of Garrick and
Watkins on the moving propeller, described above, also per-
tains largely to the near field.
Hubbard aid Regier found that near-field calculated
sound pressures, for the first few harmonics, were in good
agreement with experiments performed with model propellers
of diameters 48 to 85 inches, the range of propeller-tip
Mach numbers being 0.45 to 1.00. The observed pressure
increases very rapidly as the measuring point is brought
close to the propeller tips; this behavior corresponds
closely to what would be observed if the propeller tip
were the effective noise source in the very near field.
The distribution of sound pressure in the propeller plane
can be expressed conveniently in terms of dpb, where d is
distance from the propellcr tips, and D is the propeller
diameter, for a given propeller shape and given rotational
speed. On this basis, good agreement was obtained between STAT
observations taken near the full-sized propellers, and
extrapolated results of the model studies.
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? VS.
The sound pressure ahead of the propeller plane is
? out of phase with that behind the propeller plane in most
cases where the near field was investigated. A plane wall
(simulating a fuselage) placed just behind the microphone,
parallel to the propeller axis, and 0.083 of a propeller
diameter from the tips, doubles the pressure reading for
a given location by reflection, but does not seem to react
on the acoustic behavior of the propeller. (This conclu-
sion might not hold if the wall were brought much closer
to the propeller tips.)
Input power a-id tip speed are of primary importance
in determining the near field. At the lower tip-speed
Mach numbers, the sound pressure for given tip speed and
input power is reduced by using a propeller with a greater
number of blades, but this difference virtually disappears
at Mach 1.0. At constant power, the pressure amplitudes
of the lower harmonics tend to decrease, and of the higher
harmonics to increase, as the tip speed is increased. The
difference in sound pressure produced by square and rounded
tips is found to be very slight, with the square tips pro-
ducing about 1.0 db higher SPL than the round, in a very
restricted region near the propeller plane. Also, blade
width is found to have no Important effect.
,
0
Further, Hubbard and Regier compared their more
accurate near-field calculations with the results obtained
by using the Gutin equation for the near field, in the
plane of the propeller. It is found that the Gutin equa-
tion under-estimates the SPL in this situation. Lpparently
the discrepancy becomes less than 2 db when the distance
from the propeller tips is greater than one propeller dia-
meter, so that the Gutin equation is sufficiently accurate
for many purposes at distances greater than this.
There it is desired to know the overall sound pres-
sure level of propeller noise immediately within an air-
plane cabin, at a location near the propeller tips, the
experimental findings of Rudmose and Beranek 1.13/ may be
used. They analyzed data taken within some 50 types of
aircraft of the period 1941-1945; in seven types, a
systematic study of the parameters which influence the low-
frequency propeller noise was made.
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The following generalizations were made:
(a) The SPL increases by about 2.7 db for each
increase of 100 ft/sec in propeller tip speed.
(b) The SPL increases by approximately 5.5 db for
each doubling of the horsepower per engine.
(c) The SPL increases rapidly as the clearance
between the propeller tips and the fuselage
Is decreased below 8 inches, but becomes rela-
) tively independent of this clearance when the
value is above 20 inches.
(d) Propellers with blunt tips produce more noise
by several db than propellers with fine pointed
tips. The results are summarized in Eq. (4.1.7).
/13 HP
SPL = 102+ 24a-- - ?6-4r18.3 log -6-0-6+0.027(Vo - 700) ?
(4.1.7)
Here d is the minimum propeller-fuselage distance in
inches, HP is the horsepower delivered to each propeller,
and Vo is the propeller-tip speed in ft/sec. This equa-
tion is intended to give the SPL in each octave band
below 150 cps, existing within a typical cabin, at about
2 ft from the wall, in a section of the airplane within
6 ft of the plane of the near propellers, there being no
bulkhead between the observation point and the propeller
plane. The relation represents data for two- and four-
engine aircraft, and refers primarily to 3-blade pro-
pellers. Subsonic tip speeds are assumed. The authors
found that approximate noise levels for 4-blade and
2-blade propellers could be obtained from the same equa-
tion by multiplying the actual horsepower per engine by
3/4 and 3/2, respectively, before inserting the horsepower
value in the equation. The amount by which the overall
propeller SPL in the cabin exceeds the above octave-band
value seems to be at least 3 db in all cases, and more
usually of the order of 5 db. This figure will increase
with increasing tip speed because of the rising pre- SINT
ponderance of high harmonics, mentioned by Hubbard and
Re ier.
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The Rudmose-Beranek experimental results can be
reconciled fairly effectively with the theoretical analy-
sis. The increase of SPL by 5.5 db for each doubling of
input power agrees closely with the predictions of the
propeller charts, Figs. 4.1.4 - 4.1.9, which show that
this effect is generally 5 to 6 db per power doubling.
The increase of SPL at the rate of 2.7 db per 100 ft/sec.
increase of tip speed, as reported by Rudmose and Beranek
for the low frequencies, is somewhat less than that pre-
dicted in Figs. 4.1.4 - 4.1.9, where the effect is about
20 to 30 percent greater than this, for three-blade
propellers. This discrepancy is qualitatively reasonable,
however, because the charts include the combined effect of
four harmonics, and it is known that the effect of tip
speed goes up with increasing harmonic number. The
critical effect of clearance between the propeller tip and
the fuselage is predicted in the analysis and measurements
by Hubbard and Regier 1.12/. The final observation of
Rudmose and Beranek, that propellers with fine pointed tips
produce a lower cabin sound level, is superficially in
contradiction to the findings of Hubbard and Regier, but
can probably be interpreted to mean that an extreme change
of blade shape, in this sense, causes the effective sound
source for fine tip blades to be located further in from
the tip of the propeller. The absolute levels given by
Eq. (4.1.7) are considerably lower than those given by
free-space propeller theory, since Eq. (4.1.7) includes
the noise reduction afforded by a typical cabin.
Dual-Rotatino; Propellers. Hubbard .1.14/ has applied
Gutin's analysis to dual-rotating propellers, and has
found reasonably good agreement w_th the results of experi-
ments on a model unit comprised of two, two-blade, 4-ft
diameter propellers. The sound field no longer has
circular symmetry about the propeller axis, but instead has
maxima in the directions of blade overlap. These maxima
of sound pressure correspond closely to the amplitude
which would be produced by a single propeller having the
same number of blades as the total in the tandem unit.
The intervening pressure minima have amplitudes correspond-
ing closely to the output of one of the dual propellers
only. If the two propellers rotate at slightly different
speeds, the pattern of maxima and minima then rotates,
and the sound reaching the observer is consequently
amplitude modulated. When the number of blades is not
the same in the front and rear units, this modulation Is
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found only for harmonics which are integral multiples
of both fundamental frequencies; for example, the lowest
modulated harmonic of a three-blade, two-blade dual-rotat-
ing propeller is the sixth. The case of tandem propellers
operating side by side was also investigated, and similar
phenomena were found. The results thus far mentioned are
not critically affected by the separation of the propellers.
An additional signal, the "mutual interference noise",
is developed when the spacing of the dual-rotating elements
is made small. This noise component appears to be a maxi-
mum on the forward axis of rotation, where the rotational
noise is small, and has a fundamental frequency equal to
the blade passage frequency. The mutual interference
noise is undetectable at positions near the propeller plane,
where the rotational noise is strong, and apparently
consitutes only a small fraction of the total power
radiated by the propeller. The pressure amplitude of this
additional noise component varies as the propeller power
and as the cube of the tip speed, according to measure-
ments on the axis. The effect of spacing is critical; in
Hubbard's experiment, the mutual interference noise is
the predominant signal on the forward axis at a spacing of
6 3/4", but is not detectable with certainty at a spacing
of 12".
The Effect of Struts on_Enpeller Noise. Uhile no
theoretical analysis has be,'In mace of the effect of a
strut near the propeller plane, the experimental evidence
indicates that a much more serious disturbance is produced
by a strut ahead of the propeller than by one behind.
This question was exnmined in the work on dual-rotating
propellers described above. No strut effect was reported
for the tractor propeller, which was supported by a strut
placed behind. The pusher propeller (supported by a strut
ahead) was found to give 3 db higher overall SPL than the
tractor when the pusher strut clearance was 11.75 inches,
and about 7 db higher SPL than the tractor when this
clearance was 5.75 inches. The effect is nearly independent
of tip speed.
An increase of noise resulting from a strut ahead of
the propeller was also reported by Roberts and Beranek
1.1V in a series of experiments on quieting of a pusher STAT
amphibian. The total noise power radiated by this air-
plane was greater than that from a tractor airplane operat..
ina nt reater power and tip speed. The sound level
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IIImeasured from the pusher did not drop off sharply to the
1 rear as it does for a tractor airplane, and as the Gutin
; theory predicts. Whereas the noise omput of a tractor
, airplane for specified power and tip speed can be de-
creased by increasing the number of propeller blades,
t in at least qualitative agreement with the Gutin theory,
1
the pusher airplane was found to become noisier as the
number of blades was increased above four.
q1"^"4C Tit) Speeds and Ernirical Propeller Noise
Chart. The Gutin theory ol'FiTational noise and its
various modifications are all restricted to subsonic tip
speeds. At present, the knowledge of propeller noise
generation for supersonic tip speeds is restricted to
experimental findings. In general, the experimental data
show that there is no discontinuous change in noise out-
put as the propeller goes into the supersonic range. At
or near the beginning of the supersonic range, however,
the noise pouer output becomes nearly independent of tip
speed, as ahown in N. A. C. A. experiments 1.16/ on a
model propeller, the sound output of which was in good
agreement with the Gutin theory in the subsonic range.
A less extensive series of measurements by a commercial
laboratory (unpublished), on full-scale propellers, seems
to indicate that the noise output for supersonic tip
speeds also becomes relatively independent of input power.
This statement is based upon observations of 10- and 16-ft
diameter propellers in the range 800 to 2000 horsepower.
In the absence of a suitable theory of noise genera-
tion in the range of supersonic tip speeds, the empirical
chart in Fig. 4.1.14 has been prepared as an approximate
FTGURE 4.1.111
Propeller neice c1.rt conctructed rraa emerimsatal data,
chouing the cpurorivate aconGtic pacer level for tip cpeeds
into the crziarconic raa5e. The chart coplics to 3-blade
pre:7211er?, ef dicc:eter cp:Irocir_ctely 12 ft. Pozer levels
for 2- end 4-blade propelleTs lie cppro.linatoly 2 db above
cad belou the chart values, recpcctively. For operating
coalitions to the twor richt of the broken line, propeller
noice usually er.ceedn the eahauGt noise from a reciprocating
eardne, but for operating conditions to the lower left,
ezhaust noica ray preec-linate (cee Gee. 4.3).
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Z0/170/17 I- OZ -1A-09
1,-9000L0000170n1?1701-0-1-8dCll-V10
eseeiej -104 panaiddv Ado paz!4!ueS u! PeWsseloaCI
1,-9000L0000170n1?1701-0-1-8dCll-V10
eseeiej -104 panaiddv Ado paz!4!ue3 u! PeWsseloaCI
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summary of existing information. This chart gives the
overall power level of the propeller when the input horse-
power and the tip speed are known. The information for
tip speeds of 1000 ft/sec and greater was taken from the
twonurces mentioned above. The subsonic portion of the
chart is arbitrarily drawn to have the dependence on tip
speed and input power which was reported by Rudmose and
Beranek for low frequencies, as shoim in Eq. (4.1.7); on
the basis of propeller noise theory, slightly greater
effect of tip speed might be argued. The absolute
values indicated by the subsonic curves are determined
in part by the low-speed portions of the data on large
propellers mentioned above, and in part by several mea-
surements of ground and flight operation of actual aircraft
under known conditions. Where measurements were taken with
a microphone very near the ground and within 50 ft of the
source, pressure doubling at the microphones was assumed,
and 6 db was subtracted from the SPL reading. Where the
microphone was 200 ft or more from the source, so that
ground attenuation might be more important, this reflection
correction was arbitrarily reduced to 3 db. To get the
power level for an outdoor propeller from the SPL measured
in one direction, use was made of the typical propeller
directivity curve shown in Fig. 4.1.15. The individual data
points used to make the chart are generally consistent
with the final chart values within 4 db. The extension
of the curves into the supersonic range is determined by
very few measurements and is therefore tentative.
The chart in Fig. 4.1.14 does not show the effect
of propeller diameter or of number of blades. The chart
is an approximate average of data for propellers of two,
three, and four blades, and is most nearly correct for
three blades. Very roughly, values for propellers of two
and four blades lie 2 db above and below the chart values,
respectively. The chart is most nearly correct for pro-
pellers of diameter 12 ft; for 3-blade, 12-ft propellers,
the subsonic portions of this chart are generally in agree-
ment with the charts based on Gutin's equation, Figs. 4.1.4
through 4.1.9, within 3 db. For propellers of about this
size, the empirical chart in Fig. 4.1.14 may be used in
lieu of the detailed charts for engineering predictions.
Either this chart or the detailed charts, properly applied,
should predict overall static propeller noise within ? 5 db
in most instances.
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???? ?
Parkins and Purvis 1.17/ have measured maximum
sound levels beneath a number of types of 2- and 4-
engine aircraft immediately after takeoff, and have
reduced their results to a standard distance. If it is
assumed that the aircraft as a whole has approximately the
same directivity as a propeller*, so that the maximum SPL
is approximately 5 db abdve the space-average value, and
if it is assumed that the noise powers from the propellers
on a given airplane are additive, these data can be reduced
to give the power level of a single propeller under take-
off conditions. It is foundthat the power levels obtained
in this way are typically 8 db lower than those predicted
by the chart in Fig. 4.1.14. Therefore, 8 db shculd be
subtracted from the chart values to obtain power levels
for flight conditions following takeoff. This correction
is in the expected direction, inasmuch as the chart refers
to static operation, for which noise generation 13 greatest.
The Spectrum of Propeller Noise. The theories of
propeller noise do not give a generally successful treat-
ment of the frequency distribution of the sound energy.
The success of the theories in predicting overall sound
power is attributable partly to the fact that a large part
of the energy radiated is found in the first few harmonics
of rotational noise. The theoretical calculations of rota-
tional noise generally underestimate the amplitudes of the
higher harmonics. noreover, a large part of the high-
frequency energy often comes from vortex noise, the ampli-
tude of which is not rigorously predictable at present.
Theoretical considerations of both rotational and vortex
noise agree qualitatively, however, that the high-frequency
energy increases relative to the low-frequency energy as
tne propeller tip speed is increased (at least, in the
subsonic range).
Same unntbliched r_csour=enta of the polar cound dintribu-
ticn for an airplane onerating on the ground chow that this
ancu-ption is rcaccnable. The obcorved distribution is
oir_ilar to that in Yin. 4.1.13, tfaich is for a propeller on
a teat stand, except that the cound lovela bahind the actual
atrplane do not fall off as rapidly for points tovard the
front of the plane.
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? Z0/170/171-0Z -1A-09
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tTE
OVERALL SOUND PRESSURE LEVEL IN GIVEN DIRECTION RELATIVE
TO SPACE AVERAGE,DECIDELS
"
AHEAD
-,?,?,......_
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The octave-band spectra measured immediately beneath
several types of transport airplanes shortly after takeoff,
presumably under full-power operation, are shown in
Fig. 4.1.16. The information is from Ref. 1.17. There is
a remarkable similarity in the results for the several air-
planes, except that the two-engine airplane (of considerably
lower horsepower than the others) gives relatively less
noise in the two highest octave bands. The arbitrary curve
drawn in this figure is a suggested design curve for
engineering prediction of the propeller noise spectrum under
takeoff conditions, for transport airplanes. It is assumed
that the observed noise from a propeller-driven aircraft
at takeoff is due to the propellers. The results shown
here will be duplicated only in measurements taken fairly
near the aircraft and over a hard surface. Because atmos-
pheric and terrain attenuation of sound rise with increasing
frequency, spectra measured over absorbing terrain, or at a
distance of the order of thousands of feet, will have
-ppreciably lower relative levels in the higheat bands than
'aose shown. The relative high-frequency content of pro-
peller sound also decreases upon change from takeoff to
cruising operating conditions, but data are not available
to show precisely the extent of the effect.
Vortex Noise. It has been generally assumed that
the nol"--75e-a?'ocifc part of the propeller noise (ordinarily
less than the periodic part) is associated with the
shedding of vortices (eddies) in the wake of the moving
riGura 4.1.15
Directivity rattern cc .ted frcm overall...c7vL for a
prorollor on an cArtdoor test stand. Tao directivity
is the cliffercnce in a botv-zen olasorved Erb in a
Given airection cud the E1., 1:13.ich reulrl be obcerved
b-ith non-airectienal ro-alation of the er..c2 total
sound. poner. Coz..puted frci Iota in ref. 1.16.
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1,-9000L0000170n1?1701-0-1-8dCll-V10
aseala -104 panaiddv Ado paz!4!ueS u! PeWsseloaCI
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-
1,-9000L0000170n1?1701-0-1-8dCll-V10
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blade. These vortices are a normal consequency of the
instability of fluid flow past an object of more or less
cylindrical shape. Under idealized conditions, the
vortices form and tear away from the obstacle in regular
fashion, to form a Karman vortex trail 2.418/ as shown in
Fig. 4.1.18. While pressure fluctuations are registered
by a detector placed in the trail, it can be proved that
the vortices in the trail cannot radiate sound; their
pressure distributions fall off very rapidly with distance.
The sound radiated by the vortex shedding process must arise
from the immediate vicinity of the obstacle, as in the
region AWOHB, and must be the result of the pressure im-
pulses which occur whenever the flow system of a vortex ie
suddenly torn from the obstacle.
Some idea of the process is given by dimensional
analysis. The intensity of an acoustic wave is given by
I = p2/pc
(4.1.8)
where p is the fluid density, and c the speed of sound.
Let the acoustic pressure p be measured in units of
1/2 (pu2), where u is the flow velocity past the obstacle,
which can be expressed in terms of the Mach number,
M = u/c. Then the intensity is
I . BPI4
(4.1.9)
where B is a coefficient which may be a function of the
Reynolds number, Re . pui /p of the Mach number M, orl /r,
where is some dimension of the body and r the distance
to the point of observation, and also of e,gc the
azimuth and zenith angles of the point of observation
with respect to some reference axes. The symbol p denotes
the viscosity coefficient of air.
FIGURE 4.1.16
Propeller nolo? spectra =assured beneath several types
of 2- and 4-engine airplane= i=ediately after takeoff.
Data from Ref. 1.17. The Chart &awls the amunt by
uhiCh the porar level for each octave band differs from
the overall power level. The curve is a sugsested bias
for encineering estimates of the.spectran for transport
airplanes under takeoff conditions.
37
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For large distances, the law of conservation of
energy will require that the intensity fall orf with the
square of the distance, as expressed by the next rela-
tion
B(Re, M, 0, pr) ?2 B' (Re, M, 0,
r (4.1.10)
Furthermore, the Mach number effect must occur as a
multiplier, since the sound intensity must vanish for
Incompressible fluids (c Thus, the preceding
equation may be rewritten as
12
B = (2_ mn ) B" (Re,
r n
(4.1.11)
where the Mach number effect has been generalized as a
power series in M. An approximate solution will be
sought by retaining one term of the series. It can be
shown that the exponent n 1 corresponds to a simple
source, and n = 2 to a dipole. The simple source may be
ruled out on the basis that the observed radiation is
directional, or through a theoretical argument which
shows it to be inconsistent with the aerodynamic flow
situation. With the exponent n = ?, it is evident that
the sound intensity will vary as u?. When the direc-
tional function for a dipole is inserted, the final
expression for the intensity is
2n 6
. (Re) -g2-2-2--8 Ap
(4.1.12)
Here A, the projected area of the obstacle in the
38
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dIrection of fluid flow, has been written instead of
The coefficient 0:(R) cannot be determined from
dimensional analysis alone. In the case of a propeller
blade, it is found that the dipole radiation pattern has
its maxima on the propeller axis.
While the noise generated by vortex shedding is
not periodic in ordinary practical situations, the rate
of shedding vortices is in principle a constant in the
case of steady flow around a uniform cylinder. Strouhal
argued by dimensional analysis that the frequency of
vortex shedding from a cylinder is
f = K
(4.1.13)
where d is the diameter. He found an experimental
value of K of about 0.185. This quantity is actually
a function of the Reynolds number, plad is 0.18 for
Reynolds numbers from 103 to 3 x.104.
5"U
OBSTACLE
FIGURE 4.1.17
Idealized Kaman's vortex trail.
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Evaluation of Vortex Noise Intensity. The knowledge
of vortex noise is not yet entirely satisfactory from a
quantitative standpoint. Stowell and Deming 1.12/ experi-
mented with a device in which circular rods, rather than
blades, projected from a rotating hub, and found the inten-
sity of the radiated sound to be proportional to the
projected area A and to the sixth power of the velocity,
as Predicted by Eq. (4.1.12). In a later N.A.C.A. experi-
ment .1.20/, the constant of proportionality was evaluated
from measurements on a helicopter blade. On this basis,
Hubbard adopted the engineering equation below to give the
overall intensity level (essentially equal to SPL) of vor-
tex noise at a distance of 300 ft from a propeller, presum-
ably for those directions where the sound is strongest.
6
kA,V0.7
IL = 10 log10
'
10-16
(4.1.14)
The value of k is given by 3.8 x 10-27. The symbol V0.7
denotes section velocity at 0.7 of. full radius, in ft/sec;
AB denotes total plan area of blades, which is roughly
proportional to the area A of Eq. (4.1.12) if consistent
operating conditions somewhat below stall are assumed.
The relation Eq. (4.:i.19) is the basis for the broken-line
curves showing vortex noise in Figs. 4.1.4-4.1.9. Hubbard
estimates these tentative results as being correct within
10 db for conditions below stall, and points out that
the vortex noise may increase by 10 db when the propeller
Is operated under stalled conditions.
The uncertainty in the present evaluation of vortex
noise may be explained in part by recalling that the
coefficient in Eq. (4.1.12) is a function of the Reynolds
number. Evaluations currently available were made at
Reynolds numbers much smaller than those found in pro-
peller applications. Experiments at high Reynolds numbers
necessarily bring in rotational noise and are therefore
more difficult in that the rotational and vortex noise
contributions must be separated. Moreover, propeller
blades may operate at Reynolds numbers greatly exceeding
105, the value at which laminar flow in the boundary
layer is replaced by turbulent flow. Completely turbulent
flow generates broad-band noise through mechanisms other
than vortex shedding, and the vortex noise analysis does
not apply rigorously.
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AO
?
f'0.4V
The Spectrum of Vortex Noise. The rotating-rod
experiments of Stowell and Deming ,1.19/ and others
give spectra in Which most of the noise energy is lo-
cated in the frequency range given by the Strouhal for-
mula, Eq. (4.1.13). (The formula gives a range of
values rather than a single value in the case of a
rotating rod, since the section velocity varies con-
tinuously from the hub to the tip.) Noise energy is
observed over the entire audible range, however, as
illustrated by oscillograms given in Ref. 1.5. There
Is sone evidence of peaks la the spectrum at harmonics
of the Strouhal frequencies. The spectral distribution
of the noise needs further investigation.
Practical Importance of Vortex Noise. It appear&
that vortex noise never constitutes a significant portion
of the distant sound produced by heavily loaded propellers,
operating at tip speeds of 900 ft/sec or mure. Thus, it
is not necessary to consider vortex noise in connection
with takeoff operation of transport airplanes, and it is
unlikely that vortex noise is important even in the sound
Produced by transports under cruising conditions.
The intensity of rotational noise is much more
sensitive to tip speed and to blade loading (angle of
attack) than that of vortex noise. Consequently, it is
always possible, by reducing the tip speed and possibly
the angle of attack, to reach a condition where the
propeller sound consists largely of vortex noise rather
than rotational noise. Vortex noise t!--- becomes the
limiting factor when an attempt is mwle to reduce pro-
peller noise by reducing the tip specd and increasing the
number of blades. This point was discussed in an earlier
paragraph.
An Example of Calculatins Propeller Noise. Given
the following propeller data, it is desired to estimate
(a) the SPL near the ground (hard surface)
at 500 ft distance;
(b) the SPL at that point in the 600-1200 cps
band: Four propeller blades; tip speed 900 ft/sec STAT
(approximately Mach 0.9); 2000 horsepower input.
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From Fig. 4.1.5, the power level is 112 + 55 =
167 db. The direction-averaged SPL at 500 ft distance,
in free space, would be the power level less 10 log
[4v(500)2], which gives 102 db. Near the hard ground,
pressure doubling raises the SPL by 6 db to give 108 db,
still on a direction-averaged basis. If the typical
dtrectionl distribution of Fig. 4.1.15 is assumed, the
SPL in the propeller plane (900) is 1 db less than the
direction-average value, which yields loy db. This is
answer (a).
The given conditions resemble takeoff operation
for a large airplane. Therefore the spectral distribu-
tion in Fig. 4.1.16 should apply. According to this
figure, the SPL in the 600-1200 cps band is approximately
9 db below the overall SPL, which gives 99 db as answer
(b).
Sometimes it is necessary to estimate sound pressure
levels external to a test cell, with the propeller operat-
ing inside. For a cell which has no sound-absorbing
treatment, and which has openings looking out in a hori-
zontal direction front and rear, a first approximation
to low-frequency sound levels is obtained by making a
calculation as given above, and using the space-averaged
value, since the cell disturbs the normal directionality
of the propeller. For higher frequencies, the cell
openings must be assigned the directionality of a stack
opening, and in general a proper allowance must be
Introduced for sound-absorbing treatment. These topics
are reserved for later chapters.
The calculations above could also have been started
by reference to the empirical propeller-noise chart,
Fig. 4.1.14, which is approximately correct for large
propellers of two to four blades. This chart gives a
power level of 167.5 db, from which about 2 db should be
subtracted to correct from three to four blades, giving
a power level of approximately 166 db. All results would
then be less by one db than those obtained above.
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1
?
4.2 Noise from Aircraft Reciprocating Engines
Reciprocating engine noise has been studied less
extensively than propeller noise, because the maximum
noise levels produced by propeller-driven aircraft, under
full-throttle conditions, are usually attributable to
the propeller. The tentative generalizations given be-
low concerning engine noise are made on the basis of a
few observations (Refs. 1.13 and 1.7); also a ground
airplane test; and unpublished results of tests on an
800 horsepower engine in a dynamometer test cell).
1. The noise developed by a reciprocating engine
is produced almost exclusively by the exhaust,
with possible exceptions in case where
unusually effective mufflers are used.
2. The noise energy of the lowest-frequency
exhaust component of a reciprocating engine
is approximntely proportional to the total
power developed. Quantitatively, the power
level of this exhaust component for an engine
without exhaust mufflers is not less than
Power level of lowest frequency component
122 4- 10 log10 (horsepower).
(4.2.1)
On theoretical grounds, the horsepower value
used in Eq. (4.2.1) should include mechanical
losses in the engine. However, these are
usually not known. In cases where the mechan-
ical losses are large, they must be included.
3. The lowest-frequency exhaust component of
importance usually has a frequency equal to the
number of exhaust discharges per second (two
discharges occurring simultaneously are counted
as one). This frequency is usually below 300 cps.
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4. Usually the spectral distribution of noise
energy is approximately as follows: The power
level in the octave band containing the lowest-
frequency exhsust component ltes about 3 db
below the overall power level. The levels in
octave bands above this one decrease at about
3 db per octave of increasing frequency. No
significant noise is produced in octave bands
below the one containing the lowest-frequency
exhaust component. These conditions may be
typical of engines operated at cruising condi-
tions, and of small engines (150 horsepower
and less).
5. In the case of an engine of 800 horsepower
operated at full throttle, a uniform octave-
band spectrum has been observed (equal poi:er
levels in the octave band containing the lowest-
frequency exhaust component and all higher
octave bands). This may be typical of larger
engines under full-power conditions. In this
case the overall power level is about 8 db
larger than that of the lowest frequency
exhaust component.
6. Directional effects are much craller for engine
noise than for propeller noise. The total
variation in SPL with direction is about 6 db
for the lower-frequency coLmonents of engine
noise. This statement probably holds for high
frequencies also in the case of an isolated
engine, but no detailed reasuremmts for high
frequcIncies are available. In the case of an
engine mounted on an airplane, the high fre-
quency directivity will be affected by shadow-
ing produced by the airplane structure.
Simple relations for the overall power level of an
engine without mufflers are obtained by co:Ibining state-
ments 2, 4, and 5. Por the case of small engines (150
horsepower or less), or engines operated under cruising
conditions, the relation is
Overall power level = 125 10 logio (horsepower).
(4.2.2)
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? e.I1
For the case of a large engine operated at full load,
the relation if
Overall power level = 130 + 10 log10 (horsepower).
(4.2.3)
For example, according to Eq. (4.2.2), the overall
power level is 152 db for ermines delivering 500 horse-
power under cruising conditions. According to Eq. (4.2.3),
the overall power level is 160 db for an engine deliver-
ing 1000 horsepower at full load.
4.3 Total External Noise of Aircraft with Reciprocating
Lngines
According to Secs. 4.1 and 4.2, the overall noise
level of a propeller increases by approximately 5.5 db per
horsepower doubling (plus 2.7 db or more for each in-
crease of 100 ft/sec in tip speed), whereas the overall
noise level of an engine increases at approximately 3 db
per horsepower doubling. It follows from these principles
that the predominant noise source in a propeller-driven
aircraft with very large engine power will be the pro-
peller, but that engine noise will predominate when the
power is low.
This expectation appears to be borne out in the
results of a survey 3.1/ of take-off noise level of vari-
ous airplanes ranging from 65 to 5800 horsepower. In
this survey the microphone was located in the propeller
plane at a distance of 500 ft from the center of the
runway. At this microphene pceition the sound received
from both engine and propeller has approximately the
space-average value, so that directional effects may be
neglected. It is found that the observed sound levels
for aircraft with more than 150 horsepower agree with
values predicted from the empirical propeller chart,
Fig. 4.1.14, to the accuracy of the chart. For airplanes
of 150 horsepower and less, the overall noise levels
exceed those predicted from the propeller chart, but are.
in approxt_rate agreement with levels for engine noise
as given by Eq. (4.2.2). There are, however, other take- SI-AT
off noise data 1.7/ for aircraft with less than 200 horse-
power which are in agreement with propeller noise figures
rather than with estimated noise figures. The reason
for the discrepancy is not known.
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7
30?
60?
r
120
"
150
?
rjr"44/?)0
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7
Excluding grazing incidence, we may also derive this
result from Rayleigh's more general solution for the
transmission through a sheet of non-rigid medium having
high specific mass and mail compressibility 2.1/. This
shows that we do not have to discuss transmission through
a wall in terms of its movement of an inert mass and as
a result cf longitudinal waves excited inside the wall.
The first kind of motion is only a special limit of the
second.
To compare Eq. (11.2.25) with experimental results
found for walls between reverberant rooms, we have to
average r over cos2 /according to Eq. (11.2.22). Doing
this, we get
= (2 F. c.A.an) 2
or*
in [1 (am/2 f,
c)2)
(11.2.26)
[TL] - 10 log (0.23 [TL) )
ITL1random =
(11.2.27)
where [T.L10 is the transmission loss for perpendicular
incidence. This result sometimes fits the experimental
results quite well because it gives values of TL lower
than [TL]0 and also a less rapid increase of TL with
surface mass and frequency.
However, this last equation can hardly be regarded
as the real interpretation of what happens because plotting
T against cos2 2J4for high values of on/2p c, we get a
very sharp peak at grazing incidence where Y-becomes one.
This dependence sometimes is called the random-
incidence mass law.
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) ?????????
?1.
TABLE 11.2.1
SURFACE WEIGHT OF COMMON BUILDING MATERIALS
lb/sq ft/inch of thickness
Aluminum 14
10-32
Brick
Concrete
Dense
Cinder
Cinder Fill
Glass
Lead
Plaster
Gypsum
Lime
Plexiglas
Sand
Dry loose
Dry packed
Wet
Steel
Teansite
Wood
89
12
8
5
13
65
5
lo
6
7-8
9-10
lo
ho
9
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This dependence always has to be expected when Z-E- is
independent of the angle of incidence or is nearly in-
dependent in the region of grazing incidence. We may
call this the "component effect" because it has as its
basis the fact that the normal velocity of a wall is only
a component of the resultant of the velocities of the
source side and the back side. We must realize, however,
that the limit 7'= 1 for 0t. 900 infers an infinite wall
and infinite plane waves and, therefore, cannot be realized
in practice. Furthermore, we know from wave acoustics
that in rooms, sound propagation exactly parallel to a
boundary plane can never occur.
Therefore, it seems reasonable to exclud.. angles
1 for which Eq. (11.2.28) does not hold
(Zr cos 2.512pc)2 >, "7 1. (11.2.28)
By integrating only to a Limiting angle /P, we find for
1
= 1 2 (2pcAcm cos 29)2d(cos20-).(2pc/uxu)21n cos2
cos LA'
corresponding to
[T4 = [TL]0 - 10 log ln l/cos2 114
(11.2.29)
(11.2.30)
Now we have the difficulty that the result depends
on the choice of the limit angle 0". Taking a, = 82.50
as a value which guarantees that Eq. (11.2.28) is satisfied
for (TL]o > 24 db, we get
(TL.1082.5? (TIdo - 6 db. (11.2.30a)
The same result is obtained if we calculate the TL for an
angle 04= 600 only, so we also may write
[TL]600 = (TL)0 - 6 db (11.2.30b)
STAT
90
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We call this the "60? - mass law". It involves an
essential simplification for it replaces the averaging
over all angles of incidence by using a mean angle. In
the present case where the -Cis monotonically increasing
with 2,-, the choice of 2/*= 60? is reasonable.
It would seem better to choose IA= 450 because this
angle is in the middle of the zA-range and has the highest
weightf.ng factor (cos 45o sin 450 = 1/2). In the present
case we get
[TL1450 = [TL]0 -3 20 log G - 20 log f - 31 db
(11.2.31)
which also corresponds to the average `r value if we
restrict the VLregion from 00 to nearly 700. This "450-
mass law" fits the experimental results for light construc-
tions quite well. By averaging over the frequency region
from 100 to 3200 cps (which means replacing f by the
geometric mean of 100 and 3200 cps),
[TL]145,m = 20 log G - 24 db. (11.2.32)
where the second term agrees with that in the empirically
determined Eq. (11.2.23).
For higher values of G, all formulas which we have
derived from the assumption of Eq. (11.2.24) gives TL's
which are much too high. Therefore, we have to look for
other reasons to explain this discrepancy.
3. The Influence of Stiffness. It seems likely that
stiffness may be of importance. If we try to move the wall
very slowly, we feel its stiffness only as the reaction to
the driving force. This stiffness is given by the support-
ing or damping of the wall at the edges and also will be of
importance if a very low frequency sound pressure is driving
the wall. However, several authors have observed higher TL's
at low frequencies than those corresponding to mass law .2.2,2.3/.
Although this problem has not been solved theoretically,
It seems probable that such deviations may be accounted STAT
93.
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3
We call this the "600 - mass law". It involves an
essential simplification for it replaces the averaging
over all angles of incidence by using a mean angle. In
the present case where the is monotonically increasing
with 2,-, the choice of 2I-= 60? is reasonable.
It would seem better to choose Lit". 450 because this
angle is in the middle og the Vi-range and has the highest
weighting factor (cos 45 sin 450 = 1/2). In the present
case we get
[TL]450 = [TT.]0 -3 20 log G - 20 log f - 31 db
(11.2.31)
which also corresponds to the average `I. value if we
restrict the 21-region from 00 to nearly 700. This "450-
mass law" fits the experimental results for light construc-
tions quite well. By averaging over the frequency region
from 100 to 3200 cps (which means replacing f by the
geometric mean of 100 and 3200 cps),
[T1,145,m = 20 log G - 24 db. (11.2.32)
where the second term agrees with that in the empirically
determined Eq. (11.2.23).
For higher values of G, all formulas which we have
derived from the assumption of Eq. (11.2.24) gives TL's
which are much too high. Therefore, we have to look for
other reasons to explain this discrepancy.
3. The Influence of Stiffness. It seems likely that
stiffness may be of importance. If we try to move the wall
very slowly, we feel its stiffness only as the reaction to
the driving force. This stiffness is given by the support-
ing or damping of the wall at the edges and also will be of
Importance if a very low frequency sound pressure is driving
the wall. However, several authors have observed higher TL's
at low frequencies than those corresponding to mass law .2.2/2.3/.
Although this problem has not been solved theoretically,
It seems probable that such deviations may be accounted STAT
-
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1.44.
for by stiffness. But cases where stiffness gives an
increase in the insulation power must be regarded as ex-
ceptions for the present. Usually stiffness in a die
advantage because the reactive forces due to stiffness
and those due to mass are not added. However, because
of their opposite phases they compensate for each other.
(a) Resonence. There are two kinds of effects
where this happens. The first is well-known in acoustics
as resonance, and means that the periodicity in time of
the driving forces equals the periodicity in time of a
free motion, i.e., a motion possible without external
forces. If we have a bar of the length 2 supported at
both ends, the resonance is given by the condition
= AB/2 (11.2.33)
where 1113 is the wave length of the bending wave correspond-
ing to the same frequency. Formally we have the same
condition for an organ pipe open at both ends or for a
tube closed at both ends
= /10/2 (11.2.34)
where is the wavelength in air. But there is an
essential difference betueen the two cases: in the case
of the propagation of the longitudinal waves in a tube,
the wave length is inversely proportional to the frequency
0 = cif
(11.2.35)
whereas in the case of a bending wave, it is inversely
proportional to the square root of the wave length
4 ,
AB = plB/m k/7? (11.2.36)
or, the phase velocity of bending waves is proportional
to the square root of frequency
14
CB = y/B/in
92
(11.2.37)
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t
r.
????
In these equations, B is the bending stiffness. For a
rectangular bar with Young's modulus E, height h and the
breadth b,
B = E b 113/12 . (11.2.38)
If we substitute in = pbh and introduce the velocity for
longitudinal waves
CL = IrVia75 (11.2.39)
we can write instead of Eq. (11.2.36)
= .
(11.2.40)
In the case of plates, Eqs. (11.2.38), (11.2.39) and
(11.2.40) should be modified because of the hindered
lateral contraction in one direction. Taking this into
account, we have
B1 Ebb3/12(1- ?2)
c L
B
/E//i7 (1.112)
8 cIt h/f
(11.2.38a)
(11.2.39a)
(11.2.40a)
where ? is PoissoAs ratio. Since this number is 0.3 in
most cases, the differences between these two groups of
equations, especially between Eqs. (11.2.40) and (11.2.40a)
become so small that we may neglect them and speak simply
of B, CL and AB only*. Furthermore, these values may
depend much more on the individual variation of samples
In the available handbook tables of sound ve-
locities, it is not even stated whether the
longitudinal velocity in a bar, a plate or an
=lastic medium is meant.
93
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:3
of the same material for such things as concrete, brick,
vfnd timber. For exact studies, it is recommended that
CL be evaluated by measuring the lowest natural frequency
fi of a bar. Then CL is given by
2 f1 (11.2.41) = .45 cL
which follows from Eqs. (11.2.33) and (11.2.40). For
rough evaluations,the data in Table 11.2.2 may be used.
TABLE 11.2.2
SOUND VELOCITIES FOR
Glass
Steel
Aluminum
Timber (fir, length-
wise)
Concrete
Bricks with mortar
Plywood
Asphalt
Porous Concrete
Air (20?C)
LONGITUDINAL WAVES
18,000 ft/sec
17,000 ft/sec
17,000 ft/sec
16,000 ft/sec
12,000 - 15,000 it/sec
8,000 - 15,000 it/sec
10,000 ft/sec
7,00.0 ft/sec
4,000 ft/sec
1,130 ft/sec
FIGURE 11.
The bending wavelength A as
f (in kc/sec) for plates of
These curves apply to steel
CL = 17,000 ft/sec.
94
2.5
a function of frequency
thickness h (in inches).
and aluminum, for which
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eseeiej Jo panaidd /Woo pazwues - 'Jed pawssepaa
1,-9000L0000170n1?1701-0-1-8dCll-V10
eseeiej -104 panaiddv Ado paz!4!ueS -1-led u! PeWsseloaCI
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401
61. )
Furthermore, Figs. 11.2.5-11.2.7 contain graphs for the
dependence of length of a bending wave on frequency for
plates of different thicknesses for steel and aluminum,
concrete and plywood. From these graphs te also Noy find
the natural frequency of a rectan^ular bar supported at
the ends or a plate supported on an opposite pair of edges
if we remmber that the wavelength is the double of the
length .
For the case of plates of length itx and breadthv
usually the four edges are supported. Than Eq. (11.2.33)
must be changed to read
B = 241/..1 x) 2 y)2
From Fig. 11.2.8, the value )1B
of lengths and breadths between.
lowest natural frequency may be
the graphs in Figs. 11.2.5-11.2.
formula
(11.2.42)
may be found for plates
0.2 and 20 feet. The
found either frcm this and
7 or by tilling diruetly the
f11 =0.145 CL II [(1/Ax)2+ (1/4) )2].
(11.2.43)
FIGURE 11.2.6
The bending wavelength )1 as a function of frequency
f (in kc/sec) for plates of thickness h (in inches).
These curves apply to concrete, for which CL =
12,000 ft/sec.
97
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r L '13 100k at two examples: for a steel plate of
fx = 6 ft, I, = 3 ft and h = in. (which
_ r rl? ,..hinehocT), we find with CL = 17,000 ft/sec,
_y .1,,a the region of audibility and far below
Al of 100-3200 cps. But if we take a common con-
11 of 4 in. tnIckness with Px= 72 ft and = 8 ft,
Y
- f p or - 12,000 ft/sec, we find a frequency o +0 cps.
'y .11s and plates are seldom only supported at
_ ilch mans that only the transverse motion 13
1 1 0, not the slope at the boundary. If we assume
'cl.(2 at the boundary Is also hindered, that the
! ; 11y clamped, we have to expect natural fre-
I! 3 MGra than an octave higher. However, clamp-
, occurs very seldom. Usually, the boundary
correspond more to supporting than to clamping.
? he lo-;est natural frequencies are in the low frequency
an octave below this natural frequency we may
,C2 stiffness alone controls the transmissivity of
tic 111.
On the other hand, we cannot conclude that above this
natural frequency the wall is mass controlled. This
?Juld be the case if only this lowest type of natural
L,Sia existed. But since a plate is a two-dimensional
continuum, we have to consider a doubly infinite number of
natural frequencies given by
fn,m = 0.45 cLh [n/1x) (m/_,(.2y I
(11.2.44)
FIGURE 11.2.7
The bending wavelength A as a function of
frequency f (in kc/sec) for plates of thick-
ness h (In inches). These curves apply to
plywood, for which CI, = 10,000 ft/sec.
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Let us look at two examples: for a steel plate of
dimensions x = 6 ft, iv = 3 ft and h = 1/8 in. (which
may occur in machinehood), we find with cL = 17,000 ft/sec,
a frequency below the region of audibility and far below
the region of 100-3200 cps. But if we take a common con-
crete wall of 4 in. thickness with 1,2c = 72 ft and = 8 ft,
assuming CL = 12,000 ft/sec, we find a frequency of )0 cps.
Certainly walls and plates are seldom only supported at
the edges which means that only the transverse motion is
hindered but not the slope at the boundary. If we assume
that the slope at the boundary is also hindered, that the
plate is really clamped, we have to expect natural fre-
quency tones more than an octave higher. However, clamp-
ing actually occurs very seldom. Usually, the boundary
conditions correspond more to supporting than to clamping.
Then the lowest natural frequencies are in the low frequency
range and an octave below this natural frequency we may
say the stiffness alone controls the transmissivity of
the wall.
1
On the other hand, we cannot conclude that above this
lowest natural frequency the wall is mass controlled. This
would be the case if only this lowest type of natural
mode existed. But since a plate is a two-dimensional
continuum, we have to consider a doubly infinite number of
natural frequencies given by
?2,
fn,m = 0.45 cip In/Ix) +
(11.2.44)
FIGURE 11.2.7
The bending wavelength A as a function of
frequency f (in kc/sec) for plates of thick-
ness h (in inches). These curves apply to
plywood, for which CL= 10,000 ft/sec.
99
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0.2'4 nit
2 5
(IN FEET)
0.2
FIGURE
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FIGURE 11.2.9
Sketch showing hou the coincidence effect operates
when a sound uave in air, whose wavelength is X
impinges on a plate at the angle P . 1111..an A/sin
is equal to the wavelength of a 'bending wave in tho
plate, the TL becom2s quite mall.
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,
r.
'New
and so we would have to expect the occurrence of
resonances in higher frequency regions too. Indeed, for
very undamped systems like a bell, this is the case.
However, if there are energy losses either in the
plate or at the boundaries, we know by experience, or on
the basis of an asymptotic law derived by Schoch 2Lry,
that the higher natural modes have only a small influence.
Then a plate on which a sound wave impinges perpendicularly
acts like an inert mass the higher the frequency of sound
is compared to the lowest natural frequency of the plate.
Summarizing, stiffness is desirable only if the
lowest natural frequency is above the frequency region in
which we are interested. This condition is usually diffi-
cult to fulfill. Thus, we must make the natural frequencies
of walls as low asrossible. This means we should construct
walls of small stiffness but heavy mass.
(b) Trace Natchinm, (Coincidence Effect). The same
rule as above applies because of analler effEct, where
inertia and stiffness also work against one another and
which seems to be of greater importance since it gay
happen in the middle of our frequency region. If a plane
sound wave impinges on a wall at oblique incidence then
the pressure is working with opposite phases in the dis-
tance of half the "trace-wavelength A0/2 sin V. So
the plate is forced to be deformed with the same periodi-
city as shown in Fig. 11.2.9. For any observer moving
with the trace velocity co/siniA along the plate, the
deformation appears the same as we get if the plate is
periodically supported at distances of sin 2)4. If
this periodicity in space of the driving forces agrees
with what the plate would present without forces, i.e.,
if
Vz. AB , (11.2.45)
we have to expect total transmissivity just as in the case
of resonance. Now by putting this into Eqs. (11.2.36) and
(11.2.40), we find that this "coincidence" or, as we may
say more precisely, this "trace matching", happens for
special combinations of frequency and angles of incidence
given by
f = (c02,727r sin2 Ifi7 (11.2.46)
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and f = 0.56 co2/cip sir2 1 (11.2.46a)
Furthermore, since sin varies between zero and one,
we may find these "trace matchings" only above a critical
frequency given by
fc02/270 71171i: (11.2.47)
or(11.2.47a)
fc 56 c02/cLh
In Fig. 11.2.10 these frequencies are plotted as a
function of the thickness for different materials. The
region where ti,ace matching is possible is to the right
of these lines. We sea that it is possible over the whole
frequency range for thick walls and that it is impossible
only in thin plates.
The question arises as to how this statement can be
in agreement with the general dependence on surface e,.!ght
found empirically. To discuss this problem more quantita-
tively, we will again consider the tvansmissian impedance
which can be defined for a wall of infinite length on which
an infinite plane sound wave is incident. In this case
we get 2.5/
1 4
jwm - jB sin221- mC0
(11.2.48)
= j 2r fm (1-f2 sin4A/fc2) (11.2.48a)
The first term gives the inertia reactance and is pre-
dominant below the frequency of trace matching. The
second term gives the reactance of the bending stiffness;
this term increases with the angle of incidence, being
zero at perpendicular incidence, and is proportional to
the third power of the frequency. From simple resonance
phenomena we are accustomed to a stiffness reactance
inversely proportional to the frequency. But this is
the case here. The f3 dependence is overcompen-
sated by the fact that the stiffness of a beam supported
at its ends is inversely proportional to the fourth
power of the length of the bea.?and this length is given
by c0/2f sin /A; hence B --1/L4 --1/f4. MAT
????????????????????????......
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and f = 0.56 c02/cip sir2 2)4 (11.2.46a)
Furthermore, since sin varies between zero and one,
we may find these "trace matchings" only above a critical
frequency given by
f = (c 2/2v) Vii0T
c o (11.2.47)
or(11.2.47a)
fc 56 c02/cLh
In Fig. 11.2.10 these frequencies are plotted as a
function of the thickness for different materials. The
region where trace matching is possible is to the right
of these lines. We see that it is possible over the whole
frequency range for thick walls and that it is impossible
only in thin plates.
The question arises as to how this statement can be
in agreement with the general dependence on surface e,.!ght
found empirically. To discuss this problem more quantita-
tively, we will again consider the t,:ansmission impedance
which can be defined for a wall of infinite length on which
an infinite plane sound wave is incident. In this case
we get 2.5/
= jwm - JB sin2ii-to3/c04 (11.2.48)
j 2v fm (1-f2 sin41elfc2) (11.2.48a)
The first term gives the inertia reactance and is pre-
dominant below the frequency of trace matching. The
second term gives the reactance of the bending stiffness;
this term increases with the angle of incidence, being
zero at perpendicular incidence, and is proportional to
the third power of the frequency. From simple resonance
phenomena we are accustomed to a stiffness reactance
inversely proportional to the frequency. But this is
the case here. The f3 dependence is overcompen-
sated by the fact that the stiffness of a beam supported
at its ends is inversely proportional to the fourth
power of the length of the bea and this length is given
by c0/2f sin bg; hence B --1/f4. TAT
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7
i
0.1
1
I STEEL , ALUM 'NUM
2 CONCRETE (MEAN VALUE)
L. 3 PLYWOOD
F 4 ASPHALT
5 POROUS CONCRETE
se-
77
-1;4
114
I
I
A A
pinI/V
4
I
- /
M
'" /
f
,
III
,
Lz.
v
II
I
UI
L
/
,
Nu
i
mi
1.111
MN
--Li
) "---f =IMM111.1
0.1
1.0
(KC /SEC)
104
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From Eq. (11.2.48a) we get for the TL
= 10 log [1 4- (irfm cos 29/pc0)2(1-f2 sinit Z9/f)2].
(11.2.49)
In Fig. 11.2.11, a map is given showing contours
of equal TL over a [log f c052 29 plane. Dark regions
indicate good sound insulation; light regions, poor sound
insulation. For 2st = 00 we have a monotonic increase of
TL corresponding to the 00 mass law. There is, in general,
a decrease from bottom to top due to the component effect.
The trace matching effect cuts a deep valley beginning at
the point (fc,0) and curving asymptotically to the (cos22)t= 1)
line. At the left of this valley the wall is mass controlled
while at the right it is stiffness controlled.
Since for a given material and a homogeneous wall,
stiffness also increases with thickness with the third
power, we see that the heavier wall insulates better also
in the region where stiffness predominates. Since the
specific material constant, i.e., the longitudinal sound
velocity cid only varies between 10,000 and 18,000 ft/sec
for most materials in which we are interested, it
has been very difficult to decide if the empirical
dependence on weight means a dependence on mass only or
if stiffness is a factor too. From Figs. 11.2.10 and
11.2.11 we conclude that in most cases of walls in build-
ings, stiffness must be predominant except at perpendicular
or near perpendicular incidence. The special values for
FIGURE 11.2.10
The critical frequency f, plotted as a
function of the plate thickness h (in
inches) for which the coincidence effect
is possible. At this frequency, the TL
is quite small.
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which Fig. 11.2.11 has been calculated corresponds to a
plywood panel of 0.8 in. thickness. But the type of de-
pendence may be regarded as general.
For comparison with measurements and also for most
of the practical applications, we are interested in the
average value for a statistical distribution of angles
of incidence. This requires putting Eq. (11.2.48a)
into Eq. (11.2.20) and then int('srating over sin2 2Acor-
responding to Eq. (11.2.22a). This integration has
been carried out omitting only a small region above
f = fc. The results are given in Fig. 11.2.12 using the
dimensionless parameters
= f/fc (11.2.50)
/4-c.= rfcm/Pco (11.2.51)
The last parameter determines the TL for the critical
frequency and perpendicular incidence
ITL]oc = 10 log (1 + CLc2)=-- 20 log 0:c
(11.2.52)
The results are shown in Fig. 11.2.12 and can be
usea to give a general idea of dhat can be expected for
very large, undamped walls. The experimental results
never show such a pronounced valley just above fc. This
may be (lastly understood if we plot Y as a function of
FIGURE 11.2.11
Contours of equal TL on a cos2 2k - frequency
plane. The "valley" at the right is a result
of the coincidence effect.
1
106
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1,-9000L0000170n1?1701-0-1-8dCll-V10
eseeiej -104 panaiddv Ado paz!4!ueS u! PeWsseloaCI
ANGLE OF INCIDENCE
1,-9000L0000170n1?1701-0-1-8dCll-V10
eseeiej -104 panaiddv Ado paz!4!ueS u! PeWsseloaCI
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03
50
A n
3
TL
2
11141/11
_
Aida -5
idii
Fil
1
P0:.
6c?
20044

k
NI
1
111
ili
,00
ad- 50
ill
i
.1
0.1
FIGURE 11.2.12
TL vs. the frequency parameter e = f/fc, for various values of
ac = gfcm/Peo'
io8
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sin2 o. and see that this curve again has a sharp peak at
the angle of trace matching. We may write the expression
for as
341 (0)2) (11.2.53)
where 6 is the relative variation of the abscissa with
sin2 29-o 1/4 (11.2.54)
E= sin2 - 1/4 (11.2.55)
and 2 6 is the bandwidth, which in the present case is
6 = V12 CCeN 1-Vt). (11.2.56)
It is not important that the dependence on c: given by
Eq. (11.2.53) only holds for a small region because the
integration we have to execute gives
62
= [ d E
1 + E2/62
-e
1
6[tan-162/6 + tan-1 6-1/
(11.2.57)
or with sufficient accuracy
(11.2.57a)
as long as the limits 62 and 61 are greater than 3 .6 .
But these restrictions are possible only for higher fre-
quencies where trace matching no longer occurs at
grazing incidence. And in this region we would have to
expect
[TL] . 10 log (2 Olice I 1 - /r)
=:(172) [TL]oc 4- 20 log (f/fc) - 2.
109
(11.2.58)
STAT
(11.2.58a)
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3
40
T1.
20
20
0. -100
'rot
/*OS
I
I- oos
1-0 i-o
1
.."/"..........
i
F
-1-
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sO
21.
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:o
to
S. *COO
1111
I
i'll
1,
,.e,..,
?.
1
I
(7-0.
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7...
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pPr
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110
OS
CO
40
T1.
SO
20
10
CO
40
ft
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ao - WO
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ac - GOO
/* 01
1 "MN
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I
t
tO
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C.
30
43
TI.
20
1t1
.....0
'
I /V-
rol
ri-o3
lip
pp
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00
STAT
_
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Arot
It seems plausible that such sudden changes of transmis-
sivity with the angle of incidence will not really occur
and the assumption that under the conditions of trace
matching total transmissivity will be reached must be
violated; the conditions for trace matching must be also
violated if we Lake into account the finite length of
the wall or any kind of losses either in the wall itself
or at its edges.
1
We can treat the inner losses by introducing a
complex Youngs modulus instead of the usual real modulus
= E(1 + Yt) (11.2.59)
where )1, the loss factor, characterizes the phase shift
between strain and stress and from experiment may be
regarded as independent of frequency.
With this complex modulus the transmission impedance
becomes a complex quantity also, given by
3 4_e_ 4 3 4 21_ 4
ZT= YiBw sin 1,-/co + j[wm - Bw sin /co 1.
(11.2.60)
Putting this into Eas. (11.2.20) and (11.2.22a), we again
may find rm and finally TL. The results of this even more
troublesome calculation are given in Fig. 11.2.13 for the
case of- c = 100, 200, 400 and 800 corresponding to [TL]00
= 40, 46, 52 and 58 db.
The behavior at high frequencies again may be under-
stood by looking at the neighborhood of the peak only.
Here first the peak itself is lowered to the value
`C max = )1[1 + 10:c - 12 . (11.2.61)
FIGURE 11.2.13
TL vs. for various values of ac when inner
losses are introduced into the plate.
111
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If we now write
T ??-Cmax /1 + c2/62]
41.2.62)
and change 6 to
8 + g / (2 cce g2
(11.2.63)
W2 get for high frequencies
7 1 /7 .702 a.c 2 /1-1A + 2 "re( 3_e2)]
(11.2.64)
or
AM: [m] = 10 log [2 cc g2 1(i + gh-1/)/7)
(11.2.65)
For 1. 0 tht5 is identical to Eq. (11.2.58); for
yi(cce i V1-1/4 > 4, Eq. (11.2.65) becomes
[TL].[TL]oc + 30 1og(f/fc)-10 log f/(fc-f)-10 log(l/rP
(11.2.65a)
For a rouch evaamation, the third tem may be neglected.
Equation (11.2.65) also vanishes asymptotically but the
first order theory of bending waves which have been used
is valid only as long as )1B This will be the case
if g . 3.24 CL h/co 2B)u m,, these frequencies are given
by
fn = nc(1 + pd/w2n2m2)/2d cos V-
or approximately
fn = ne/2d cos
119
(11.2.77)
(11.2.77a)
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The difference between Eqs. (11.2.77) and (11.2.77a) is
of interest only if thq minimum transmission coefficient
must be calculated. Equation (11.2.77a) is true when
the sound pressure in the air space at the opposite points
of the walls is either in phase or 1800 out of phase.
For Pl. = 0, the condition becomes
fn = nc/2d (11.2.770
which is the well known formula for the eigenfrequencies
of one dimensional sound motion between parallel rigid
walls. Thefefore we may call the fn the resonance fre-
quencies of the air space and fo V-e resonance frequency
of the double wall.
With increasing separation of the two partitions,
the lowest of these resonance frequencies decreases.
This is the reason why increasing this distance may not
always be helpful. If, for example, we want to avoid
the case where fl becomes smaller than about 1000 cps,
we should keep
d 7 in. (11.2.78)
Again Eq. (11.2.77b) has to be used instead of
Eq. (11.2.77a) if the lateral coupling in the air space
is hindered. If this is done by a porous material, the
difference between minima and maxima of TL in this
frequency region will decrease.
Influence of Absorbing Material in the Air Space.
As has been shown by London, in most cases we would
not expect any improvement in sound insulation by an
additional partition without introducing any absorption2
The reason is that for each frequency above fno, (n + 1)
angles of incidence exist for which total transmission
occurs. By averaging over all angles, the sound trans-
mission in the neighborhood of these angles predominates
and results in an average transmission coefficient that
is higher than that for the single wall. This may be
substantiated in a manner similar to that shown for the
problem of transmissivity in the case of trace matching.
As in that case, it may be shown that the results for
the average transmission coefficient are influenced
strongly by any kind of energy losses.
120
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The difference between'Eqs, (11.2.77)-and1(11.2.77a) Is
of interest only if the. MiniireUM-tranSMISSion coefficient
must be calculated. Equation-(11.2.77a) is true when
the sound pressure in the air space at the opposite points
of the walls is either in phase or 1800 out of phase.
For Pz- = 0, the condition becOmes
which is the well known formula for the eigenfrequencies
of one dimensional sound motion between parallel rigid
walls. Therefore we may call the fn the resonance fre-
quencies of the air space and 1'0 t,e resonance frequency
of the double wall.
With increasing separation of the two partitions,
the lowest of these resonance frequencies decreases.
This is the reason why increasing this distance may not
always be helpful. If, for example, we want to avoid
the case where fl becomes smaller than about 1000 cps,
we should keep
d t, 7 in.
Again Eq. (11.2.77b) has to be used instead of
Eq. (11.2.77a) if the lateral coupling in the air space
is hindered. If this is done by a porous material, the
difference between minima and maxima of TL in this
frequency region will decrease.
Influence of Absorbing Material in the Air Space.
As has been shown by London, in most cases we-wouZ)f
not expect any improvement in sound insulation,by,,aa:
additional partition without introducing any absQiJ
The reason is that for each frequency above fno, J
angles of incidence exist for which total transssoi-
occurs. By averaging over all angles,the sound
mission in the neighborhood of these'arigles
and results in an average transmission?copffir
Is higher than that for the Single
substantiated in a manner 'similar t?that S.110 71'
problem of transmissivity in the case of
As in that case, it may be shown that the 'i'
the average average transmission -::-efficient .are
strongly by any kind r)f:r loss-s.
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4
Ii
To get agreement between the experimental data
obtained with reverberant rooms and theoretical calcu-
lations, London introduced external friction terms in
the impedances of the partition walls which he assumed
to be inversely proportional to cos V2-. There is no
physical evidence for such resistance terms and, there-
fore, the physical properties cf walls offer no data for
the evaluation of these resistances.
Another possible way to introduce energy losses
is by means of a complex Young's modulus, as was done In
the case of trace matching effects. But here also an
adequate value of the loss factor can only be found by
experiment and must be assigned a much higher value than
the loss factor corresponding to the material alone.
The only kind of energy losses which we are able
to calculate from measurable physical data are those
which occur when the space between double walls is filled
with porous material. The theory of those materials has
been developed to such a degree (see Sec. 12.1) that
sufficient agreement between theory and experiment has
been achieved. We are Interested only in porous materials
that do not make an elastic connection between the walls
by virtue of their skeleton. Under this assumption, we
need only two quantities to characterize the porous ma-
terial. The first is the propagation coefficient for
propagation perpendicular to the walls inside the porous
material, which is assumed to be a complex quantity
kx = kx - j gx . (11.2.79)
The second 13 the characteristic impedance of the porous
material, which is defined by the ratio of sound pressure
to the component of the velocity perpendicular to the walls
for a propagating wave
Zx = p/vx (11.2.80)
This also is a complex quantity. With these definitions,
we find for the transmission coefficient
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121
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??????
3
tc=1[1?(Z1+Z2)cosW2pc]cosh(jkxd)+[(21-EZ2)/44Zx costY2pc
pc/Ax cos ?-I- Z1Z2 cosOkpc] sinh(jcd)I-2
(11.2.81)
Fortunately for most practical applications the last term
prPdominates so that we may simplify the cumbersomc ex-
pression in Eq. (11.2.81) to
A
2Z ro , A s
X I
sinh(j kxd)
Z1 Z2 cos g?
2
(11.2.81a)
and write for the transmission loss of the whole con-
struction
[TL] =
=
20 log
20 log
Z12 cos41- ,
k I
_d)
, A j
-;?-t: sinh(j kxd).
Zx
2Zx fc
Zi cos
sinh(j
4- 20 log
2 r c ?
(11.2.82)
Finally, we get for the improvement of the transmission
loss given by the second partition and the air space
filled with absorbing material
[TL] = 20 log
A
Z2 Binh (j kxd)
122
(11.2.83)
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Ii
3
To simplify the theory, it seems reasonable for the
present problem to neglect the small vibrations of the
fibers. These vibrations are of importance at low fre-
quencies only. Then we may write
iccx = w I (7e.. _ Bin2 IA) _ Jr crAn p 11/2 /c
and
A
zx 21--20
MPH
../.
MPH
TIO
1
xl title Fig. 3.5 read Q