Franken, Peter A

English

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NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 
TECHNICAL NOTE 3557 
A THEORETICAL ANALYSIS O F  THE FIELD O F  A RANDOM NOISE 
SOURCE ABOVE AN INFINITE PLANE 
By Peter A. Franken 
Massachuset ts  Institute of Technology 
Washington 
November 1955 
https://ntrs.nasa.gov/search.jsp?R=19930084244 2020-06-17T17:07:52+00:00Z
NATIONAL ADVISORY C O M M I m  FOR AEBONAUTICS 
TECHNICAL NOTE 3557 
A !JXEORETICAL ANALYSIS OF THE FIEGD OF A RANDOM N O I S E  
SOURCE ABOVE AN I N F I N I T E  PLANE 
By Peter A. Franken 
SUMMARY 
A theore t ica l  study of the sound f i e l d  from a random noise source 
above ground as measured by a receiver w i t h  f i n i t e  band width is  presented 
herein. 
i n  atmospheric acoustics. 
considered. The special  case of a perfectly ref lect ing plane is  discussed 
f i r s t ,  and nondimensional curves are  given of sound pressure l eve l  versus 
distance f o r  two different  receiver band widths. The analysis is  then 
extended t o  the case of a plane of a rb i t ra ry  impedance, and curves of 
pressure leve l  versus distance are given f o r  typ ica l  f i e l d  operating con- 
dit ions.  The sound f i e l d  consists of two major regions. In  the f i r s t  
the sound pressure leve l  f luctuates  about an average curve sloping approx- 
imately 6 decibels per doubling of distance. 
from the source the leve l  decreases monotonically 12 decibels per doubling 
of distance. 
and on the ground impedance. With, f o r  example, an octave band of 1,000 
t o  2,000 cycles and the receiver 10 f e e t  above a ground of normal imped- 
ance pc, the maximum pressure-level f luctuat ion i s  about 2 decibels and 
occurs around 300 f e e t  from the source, and the t rans i t ion  between the 
6-decibel-slope region and the 12-decibel-slope region occurs around 
TOO f e e t  from the source. 
This study represents one phase of a general program of research 
For simplicity, only the far f i e l d  has been 
Beyond a cer ta in  distance 
The fluctuations depend on the band width of the receiver 
INTRODUCTION 
The sound f i e l d  from a point source emitting a pure tone above a 
plane boundary, which has been studied by several  investigators (e.g., 
r e f .  l), can be divided essent ia l ly  into two parts,  one i n  which there 
i s  a marked space variation of sound pressure due t o  interference between 
the d i rec t  and the reflected sound and one, beyond a cer ta in  distance 
from the source, i n  which the sound pressure decreases monotonically with 
distance. In studies of sound propagation i n  the outdoor atmosphere there  
i s  generally more in te res t  i n  the average rate of decay of the sound pres- 
sure w i t h  distance from the sound source and less i n  leve l  f luctuations 
i n  space due t o  interference. In fac t ,  these fluctuations w i l l  generally 
INCORPORATED 
DALLAS, TEXAS 
2 
The present investigation w a s  conducted a t  the Acoustics Labor 
of the Massachusetts In s t i t u t e  of Technology under the sponsorship 
with the f inancial  assistance of the National Advisory Committee for  
Aeronautics . 
SYMBOLS 
C velocity of sound above plane 
d mean source-to-receiver distance 
f frequency 
f m  mean frequency of band pass f i l t e r  
H height of receiver above plane 
h height of source above plane 
P2 resul tant  mean square pressure 
- 
P sound pressure 
Q image source strength 
R r a t i o  of mean square pressure w i t h  plane present t o  free spa 
mean square 
plane wave r e f l ec t  ion coef f i RO 
ratio ass-filter end points 
S 
X 
80 
A 
\ 
A, 
PC 
T 
\Ir 
horizontal  source-to-receiver distance 
admittance r a t i o  of ref lect ing plane 
path-length difference 
grazing angle of ref lected ray 
wavelength of sound 
wavelength corresponding t o  fm 
character is t ic  impedance of air  
delay time 
correlation function 
ANALYSIS 
The sound f i e l d  about a point source above an in f in i t e  plane can be 
obtained as the sum of the d i rec t  wave and the wave ref lected from an 
image source below the plane (ref. 1). 
figure 1, the expression f o r  the path difference between the d i rec t  and 
the reflected wave i s  
With the geometry defined i n  
A = 2hH/d (1) 
Limiting the discussion t o  r-f i e l d  points imposes the r e s t r i c -  
on A << d o r  
2hH/d2 << 
4 
I n  the discussion which fo l lows  it is  also assume 
of the same order of magnitude so. that  d = x; ho 
not necessarily res t r ic ted  t o  these it ions. 
Suppose that the  source e m i t s  a random noise and tha t  the 
sponding sound pressure of the d i r ec t  wave a t  the 
i s  the random function of time p ( t ) .  The pressure spectrum S(f) 
of p ( t )  i s  defined (ref. 2) as 
J O  
and the power spectrum w ( f )  is  then 
(3) 
Since p ( t )  
terms, t h i s  def ini t ion of w ( f )  is  adequate. The correlation func- 
t ion  $(T) of p ( t )  i s  defined as 
is  assumed t o  be perfectly random and t o  contain no periodic 
i 
= T l im p ( t ) p ( t  + T) d t  
- m  T ( 5 )  
where p ( t  + T) i s  the value of the pressure at  a time T later a t  the 
same point. The power spectrum and correlation function are related by 
Fourier formulas 
$(I-) = Jd” W ( f )  COS 2 d T  df (7) 
Now, consider a receiver with a band pass f i l t e r  with a range from 
t o  fb. 
comes 
fa 
I n  such a case, using equation (7 ) ,  the correlation function 
6 
where the bars indicate t i m e  averages. Only f a r - f i e ld  points are  con- 
sidered, t ha t  is, points where 
difference between the d i rec t  and reflected rays 
pared with the source-to-receiver distance for e i ther  ray. Also, 
p ( t ) 2  - p ( t  + T ) ~  or  the path-length 
A(= cz)  i s  small com- 
In the case of f r ee  space, 
The r a t i o  of mean square pressure when the plane i s  present t o  free-space 
mean square pressure is, combining the r e su l t s  of equations (9) t o  (12), 
Util iz ing a trigonometric identity,  equation (13) becomes 
R = 2 + 2  
R = 2 + 2 COS 25rfm~ (15) 
which is the correct expression for a pure-tone source. 
Equation (14) expresses R as a function of the two dimensionless 
2 
- 
variables r and fm-r. The free-space mean square sound pressure PF 
is inversely proportional to x2. 
sound pressure level 10 log PF2R versus the "numerical distance" &x/bH, 
with r as a parameter, since, using equation (2), 
Plots can be made of the recorded 
= wo (17) 
where wo i s  the constant value of the power spectrum (eq. ( 4 ) )  i n  
Dividing equation (19) by equation (12), the measured free-space mean 
square pressure, yields  
(1 - Ro)( =iRo + 7) R = (1 -I- R0)' i- 
n 
The f irst  term i n  t h i s  equation is the ray-acoustics expression f o r  plane 
wave reflection, and the second term i s  a small correction fac tor  which 
reduces monotonically with distance. 
With the source above the plane, the pressure spectrum f o r  the image 
ray i s  
m-> 
10 NACA 3557 
1 + 2Re(Q) cos 2xf.t - 2Im(Q) s i n  2rtf-c + 
fa  
Following the same procedure used t o  evaluate equation (l7), the r a t i o  
of mean square pressure t o  free-space mean square pressure becomes i n  
this case 
R = 1 + Ro2 + 2Ro *(T) + 
wo(fb - fa) C 
- Ro)(5Ro 7) + 
4C2faf, 
3(1 - R ~ )  COS 2rtf,~ cos 2XfbT 
- + ( 2 m )  
2 ~ 2  fa ( fb  - fa )  fb(fb - fa )  i (23) 1 
For 
the plane. 
ref lect ing plane. 
T = 0, t h i s  expression reduces t o  equation (20) f o r  the source i n  
Ro = 1, it reduces t o  equatiop (13) for the perfectly For 
Numerical Examples 
In  order t o  evaluate 
h = H = 10 f e e t .  
R 
In  t h i s  case the terms i n  equation (23) con- 
i n  equation (23) an octave-band pass f i l t e r  
is  chosen with 
heights 
ta ining a frequency squared i n  the denominator o r  
are negligible i n  the f a r  f i e ld ,  o r  
fa  = 1,000 cps, f b  = 2,000 cps, and source and receiver 
T i n  the numerator 
2 I4-d 
+ W 0 ( f b  - f a )  R = l + R o  
Figures 3(e)  and 3(f)  are similar plots  f o r  p = 1, h = H 
and octave bands 75 t o  l5O cps and 300 t o  600 cps. 
terms i n  equation (23) containing 
always negligible. 
For these bands the 
T i n  the numerator are small but not 
It can be seen from f igures  3(b) t o  3(f)  t h a t  above a plane of f i n i t e  
impedance the sound pressure l eve l  measured near the source f luctuates  
about an average curve sloping approximately 6 tle 
distance and tha t  a t  large distances from the sou 
pressure leve l  approaches a decrease of 12 decibels per doubling of dis-  
tance. In particular,  f igure 3(d) indicates t ha t  fo r  an octave band 
of 1,000 t o  2,000 cycles, with the  source and receiver 10 f e e t  above a 
ground of normal impedance pc, the maximum pressure-level f luctuation 
is  about 6 decibels and occurs around 300 f e e t  from the source and t h a t  
the t rans i t ion  between the 6-decibel-slope region and the 12-decibel- 
slope region occurs around TOO f e e t  from the  source. 
slope a t  large distances can be predicted from the following expressions 
f o r  the image source Q and f o r  the resul tant  pressure P in the  far 
f i e l d  : 
e l s  per doubling of 
the measured sound 
The 12-decibel 
12 
REFERENCES 
1. Ingard, Uno: On the Reflection of a Spherical Sound Wave From an 
Infinite Plane. Jour. Acous. SOC. Am., vol. 23, no. 3, May 1951, 
PP. 329-335. 
2. Rice, S. 0.: Mathematical Analysis of Random Noise. The Bell System 
Tech. Jour., vol .  23, no. 3, July 1944, pp. 282-332. 
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A theoretical analysis of the field of random noise source above an infinite plane

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