The diffraction of sound by barriers with surfaces of large, but finite, acoustic impedance was analyzed. Idealized source-barrier-receiver configurations in which the barriers may be considered as semi-infinite wedges are discussed. Particular attention is given to situations in which the source and receiver are at large distances from the tip of the wedge. The expression for the acoustic pressure in this limiting case is compared with the results of Pierce's analysis of diffraction by a rigid wedge. An expression for the insertion loss of a finite impedance barrier is compared with insertion loss formulas which are used extensively in selecting or designing barriers for noise control

Hadden, W. J., Jr.

Pierce, A. D.

English

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DIFFRACTION OF SOUND BY NEARLY R I G I D  BARRIERS 
W .  James Hadden, J r . , and  Allan D .  Pierce 
Georgia I n s t i t u t e  of Technology 
SUMMARY 
An ana lys i s  is presented of the  d i f f r a c t i o n  of  sound by b a r r i e r s ,  
whose surfaces a r e  charac te r ized  by la rge ,  bu t  f i n i t e ,  acous t ic  
impedance. The discussion is l imi ted  t o  idea l i zed  source-bar r ie r -  
r ece ive r  configurations i n  which the  b a r r i e r s  may be considered as  
s emi - in f in i t e  wedges. P a r t i c u l a r  a t t e n t i o n  is  given t o  s i t u a t i o n s  
i n  which t h e  source and rece iver  a r e  a t  l a rge  d is tances  from t h e  
t i p  of t h e  wedge. 
l i m i t i n g  case is compared with t h e  r e s u l t s  of  P i e rce ' s  ana lys i s  
of d i f f r a c t i o n  by a r i g i d  wedge. An expression f o r  t h e  i n s e r t i o n  
lo s s  of a f i n i t e  impedance b a r r i e r  i s  compared with i n s e r t i o n  lo s s  
formulas which are used ex tens ive ly  i n  s e l e c t i n g  o r  designing 
b a r r i e r s  f o r  no ise  cont ro l .  
The expression f o r  t he  acous t i c  pressure  i n  t h i s  
INTRODUCTION 
I 
The d e s i r e  f o r  e f f e c t i v e  measures t o  p r o t e c t  r e s i d e n t i a l  
areas from noise  assoc ia ted  with various modes o f  t r anspor t a t ion  
has l e d  t o  a resurgence of  i n t e r e s t  i n  t he  problem of sound 
d i f f r a c t i q n  by b a r r i e r s .  
i n t e r e s t  because of t h e i r  ubiquity both as physical e n t i t i e s  and 
as sub jec t s  o f  s c i e n t i f i c  i nves t iga t ions .  L i t t l e  a t t e n t i o n  has 
been given, however, t o  t he  e f f e c t  of t h e  f i n i t e  . acous t ic  imped- 
ance of  a b a r r i e r ' s  surfaces on i t s  performance as a noise  
s h i e l d .  The inc lus ion  of  t h e  effect of la rge ,  bu t  f inite; '  
a cous t i c  impedance i n  ca l cu la t ions  of t h e  i n s e r t i o n  l o s s  f o r  a 
b a r r i e r  is  cons is ten t  with cur ren t  i n t e r e s t  i n  b e t t e r  b a r r i e r  
design and s e l e c t i o n  procedures. This paper describes some of  
t h e  r e s u l t s  of  a t h e o r e t i c a l  study of d i f f r a c t i o n  by hard  wedges 
and suggests a method of adapting these r e s u l t s  t o  widely used 
formulas ( r e f s .  1 and 2) based on r i g i d  wedges. 
Wedge-shaped b a r r i e r s  are of p a r t i c u l a r  
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https://ntrs.nasa.gov/search.jsp?R=19770003380 2020-03-20T06:47:14+00:00Z
SYMBOLS 
Values are given i n  dimensionless form. 
P 
r, 6, z 
w 
C 
k 
Q 
8 
L 
R 
Y 
a 
I L  
acoust ic  pressure 
coordinate axes i n  cy l indr ica l  coordinates 
angular frequency 
acoust ic  wave speed 
acoust ic  wave number 
normalized acoust ic  impedance (eq. (2)) 
e x t e r i o r  angle of wedge 
modified spreading dis tance of d i f f rac ted  ray (eq. (4 ) )  
spherical  spreading dis tance (eq. (14)) 
complex angle . (eq.  (5)) 
angle between source-receiver path and wedge ver tex 
i n s e r t i o n  loss  (eq. (12)) 
(eq. (6)) 
PROBLEM STATEMENT 
We s h a l l  r e s t r i c t  our a t ten t ion  t o  an idea l ized  case i n  which 
t h e  source may be ideal ized as a point  source and the  b a r r i e r  as a 
wedge whose faces occupy t h e  planes 8=0 and e=8 .  
configuration of source, wedge and receiver  i s  depicted i n  f igure  1. 
The acoustic pressure f i e l d  must s a t i s f y  the  reduced wave equation 
(exp (-iwt) time dependence suppressed throughout; k=w/c) 
The geometrical 
k 2  sin2 ‘I * = O [S + 7 5  + 7 ae” i a  i a  + 
i n  the  region O:< e < @  ( R  <#32 ZIT). 
a t  the surfaces 8=0,8 may be expressed as 
The impedance boundary conditions 
where the upper s ign  i s  talien f o r  8=0 and Q i s  a dimensionless 
impedance. 
condition (outgoing waves from t h e  wedge). 
In addition, the  pressure f i e l d  must obey a rad ia t ion  
1010 
The so lu t ion  f o r  t h e  po in t  source case has been obtained from 
the  exact so lu t ion  f o r  t h e  d i f f r a c t i o n  of  p lane  electromagnetic 
waves, p a r t i c u l a r l y  use fu l  versions of which have been given by 
Williams (ref. 3) and Malyuzhinets ( r e f .  4) .  The d e t a i l s  of t h i s  
ana lys i s  are presented i n  reference 5. 
PRESSURE FIELD I N  THE SHADOW ZONE 
The plane wave so lu t ion  can be modified, following Keller's 
geometrical theory of d i f f r a c t i o n  ( r e f .  6 ) ,  t o  y e i l d  a so lu t ion  
f o r  t h e  pressure  f i e l d  due t o  a poin t  source. For s i t u a t i o n s  i n  
which both the  source and rece iver  are many wavelengths away from 
t h e  t i p  of t h e  wedge, with t h e  rece iver  loca ted  within t h e  acous t ic  
shadow o f  t h e  wedge, t h e  acous t ic  pressure a t  a po in t  ( r , e ,z )  due 
t o  a source a t  ( r O , e O , z O )  may be approximated as 
which contains t h e  modified spreading f a c t o r  L ,  
which i s  in t e rp re t ed  as t h e  n e t  distance a wave t r a v e l s  along a 
l i n e  from t h e  source t o  t h e  wedge t i p  and then along a d i f f r a c t e d  
ray  t o  t h e  rece iver .  An addi t iona l  condition f o r  t h e  v a l i d i t y  of  
equation (3) i s  t h a t  t h e  quan t i ty  (krro/Lrr) be much l a r g e r  than 
unity.  The function G ( 8 , 8 0 , a )  i n  equation (3)  descr ibes  t h e  
v a r i a t i o n  of t h e  s t r eng th  o f  t he  d i f f r a c t e d  pressure  f i e l d ,  with 
respec t  t o  source and rece iver  angles 8,80, and with respec t  t o  
t h e  impedance through the  parameter a which i s  defined by 
-1 
cos a = (n s i n  a )  
where y i s  the  angle t h e  inc ident  ray makes with t h e  wedge ax i s  
cos y = (z-zo)/L ; s i n  y=(r+rg)/L (6 )  
1011 
The functional form of G(8,80,a) f o r  a r b i t r a r y  values o f  t he  
impedance is  qui te  complicated; it is discussed f u l l y  i n  ref- 
erence 5. 
t i on  i n  the  case of  a hard wedge. 
It i s  possible  t o  effect a considerable simplifica- 
NEARLY R I G I D  WEDGE 
For a wedge with very large impedance q, t h e  sca t te r ing  
function G(8,80,a) can be simplified t o  the  form 
i n  which w e  have used 
and 
The function Q (-e) takes on a ra ther  simple form f o r  wedge angles 
given by $ = pg/Zq, with p an odd integer  and q and p r e l a t i v e  
primes. In such a case, w e  may use 
2 
n= 1 
p-l 
fl-1 0 0) 
-v sin(v.rr) 
Q B ( - ~ )  = C s i n  v(2nx-e) s in~v(2n- l )n-e l  
If we neglect t he  term involving SB(8,eo) i n  equation (7), we 
recover the  f a r - f i e l d  l i m i t  of Pierce 's  expression f o r  t he  
d i f f rac ted  pressure f i e l d  of a r i g i d  wedge (ref. 7 ) .  
10 12 
PRACTICAL APPLICATIONS 
The combination of equations (3)  and (7) l eads  t o  an expression 
f o r  t he  acoustic pressure i n  t h e  shadow zone a t  la rge  dis tances  
from the  t i p  of t he  wedge which included a f i r s t -o rde r  correction 
f o r  f i n i t e  wedge impedance: 
A pa r t i cu la r ly  useful  measure of a b a r r i e r ' s  shielding e f f ec t  is  
the inser t ion  l o s s ,  defined as 
I L  f 20 log,, IPNo Barrier I 
:]'Barrier I 
No Barrier For the  present case we may take a s  p 
ikR e - -  
~ N . B .  - R 
with 
Thus we may express t h e  inser t ion  loss i n  terns of t ha t  f o r  a 
r i g i d  wedge 
= 20 loglo (L/R) + 1 0  logi&2nkrro/L) ILRigid 
1013 
and the  f i n i t e  impedance correction 
The r e s t r i c t i o n  t o  b a r r i e r  angles of the form prr/2q presents no 
r e a l  problem: 
c lose ly  enough by su i tab le  choices of p and q, o r  one may in t e r -  
po la te  f o r  design purposes between the  inser t ion  losses  f o r  wedge 
angles with values of p,q which a re  convenient f o r  computations. 
A s  an a id  t o  the  inser t ion  l o s s  computation values o f  t he  func- 
t i o n  Q (-e), given by equation ( l o ) ,  may be computed f o r  several  
wedge lng les  6 and then p lo t ted  t o  provide the  des i re  i n t e r -  
polation. 
f igure  2 .  
t o  the  inser t ion  lo s s  f o r  an obtuse @edge with the  i n t e r i o r  
angle 120 
i n  f igure  3 f o r  several  combinations o f  source and receiver  
locations.  
ance i s  stronger f o r  source and/or receiver  locations nearer the  
surface of the  wedge. 
a desired barrier angle may be  a p p r o x i m a t e d  
A se lec t ion  of t he  resu l t ing  curves i s  presented i n  
Numerical values f o r  the  finite-impedance correction 
and surface admittance q-l = 0.1 - i0.05 a re  presented 
As might be expected, the  e f f ec t  o f  the  f i n i t e  imped- 
0 
ADAPTATION TO CONVENTIONAL DESIGN PROCEDURES 
The most widely used b a r r i e r  design char t s  ( ref .  1 , 2  and 8) 
consider only r i g i d  ba r r i e r s  and generally deal only with the  
e f f ec t ive  path difference,  L-R, i n  the form of the  Fresnel 
number N = Z(L-R) /h .  In reference 9,  Pierce has shown that  i n  
general t he  inser t ion  l o s s  formula thus obtained ( r e f .  1, equa- 
t i o n  7.15), 
= 20 logla [*] anh 2rrN + 5dB ILRig i d  
i s  va l id  pr imari ly  near t he  edge of the  shadow boundary, which 
corresponds t o  having one o f  the  functions (eq. (8)) very smal l .  
1014 
In such cases it would seem t o  be an acceptable p rac t i ce  t o  add 
the  correction term, equation (163, t o  t he  r i g i d  wedge inser t ion  
lo s s  computed from equation (17). 
CONCLUDING REMARKS 
The r e s u l t s  of a theore t ica l  study of t h e  d i f f r ac t ion  of 
sound in to  the  shadow of a wedge with large but f i n i t e  acoust ic  
impedance have been presented. The finite-impedance correction 
f o r  the  inser t ion  lo s s  of t he  wedge is  cast i n  a’form which 
is  amenable f o r  some wedge angles t o  calculat ions using modern 
desk calculators .  The inser t ion  lo s s  correction can be used 
i n  conjunction with other  calculat ions f o r  r i g i d  ba r r i e r s ,  
although 
is of greater  u t i l i t y  and involves l i t t l e  addi t ional  computa- 
t i ona l  e f f o r t .  
the  r i g i d  wedge inser t ion  loss formula obtained here 
10 15 
REFERENCES 
1. Kurze, U .  J. and Beranek, L .  L . :  Sound Propagation Outdoors. 
Noise and Vibra t ion  Control ,  ed. L .  L .  Beranek (McGraw- 
H i l l ,  New York, 1971), pp. 174-178. 
2 .  Kurze, U .  J .:  Noise Reduction by Barriers. J .  Acoust. SOC. 
Am., v o l .  55, no. 3, March 1974, pp. 504-518. 
3 .  Williams, W .  E . :  D i f f r a c t i o n  of an E-polarized Plane Wave 
by an Imperfect ly  Conducting Wedge.Proc. Roy. Soc. London, 
series A,  v o l .  252, 1959, pp. 376-393. 
4. Malyuzhinets, G .  D . :  The Radiat ion of  Sound by t h e  Vibra t ing  
Boundaries of  an Arb i t r a ry  Wedge. 
vo l .  1, 1955, pp. 152-174 and 240-248. 
Sovie t  Physics-Acoustics,  
5. Pierce, A .  D .  and Hadden, W. J . ,  Jr.: Theory of  Sound 
Di f f r ac t ion  around absorbing Barriers. Proc. I n t ' l .  Conf. 
on Acoustic Pro tec t ion  of  Res ident ia l  Areas by Bar r i e r s .  
Centre  National de l a  Recherche Sc ien t i f ique ,  Marseilles 
1975. 
6. Keller, J .  B . :  Geometrical Theory of  D i f f r ac t ion .  J .  Opt. 
SOC. Am., v o l .  52, no. 2 ,  1962, pp. 116-130. 
7. Pierce, A .  D .  and Hadden, W. J . ,  Jr.: Noise D i f f r a c t i o n  
around Barriers o f  F i n i t e  Acoustic Impedance. Proc. 3rd 
Interagency Symposium on Univers i ty  Research i n  
Transpor ta t ion  Noise. U .  S. Dept. o f  Transpor ta t ion ,  
November 1975, pp. 50-58. 
8 .  Kurze, U .  J .  and Anderson, G .  S . :  Sound Attenuat ion by 
Barriers, Applied Acoustics,  vo l .  4 ,  no. 1, 1971, pp. 35-53. 
9. Pierce, A. D.: D i f f r a c t i o n  of Sound Around Corners and Over 
Wide Barriers. J. Acoust. SOC. Am., vol. 55, no. 5, 1974, 
pp. 941-955. 
1016 
Figure 1.- Source-wedge-receiver configuration. 
100 
8.0 
6.0 
QP 
4.0 
2.c 
0.G 
THETA 
150° 
180° --- 
I 
210 
240° 
270 
-.- 
- ..- 
--- 
0 oo -- 
I I I I, 
I80 210 240 270 300 330 3 
WEDGE ANGLE, 
Figure 2.- Curves of Qg (-0).  
1017 
/ 
/ 
/ 
/ 
/ 
I ’ ,’ 
4 ’  
SOURCE 
CONFIGURAT IO 
RECEIVER 
/ ./’ 314 
--- 1,2 
N I 2 3 4 
A IL (DB) 6.1 4.6 1-8 6 -9 
Figure 3.- Finite-impedance correction to insertion loss. 
Surface admittance = 0.1 - 0.05i. 
1018 


Diffraction of sound by nearly rigid barriers

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