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Sampling statistics of sound fields in reverberation rooms: pure tone
excitation
SER
Ti41
ISSN 0701 -5232Ottawa, A p r i l 1985
National Research Conseil national
1
Council Canada de recherches CanadaSAMPLING STATISTICS OF SOUND FIELDS I N REVERBERATION ROOMS: PURE TONE EXCITATION
W.T. Chu
ABSTRACT
The sampling s t a t i s t i c s of sound f i e l d s i n r e v e r b e r a t i o n rooms have been s t u d i e d by means of a computer experiment based on t h e normal mode model. The f o r m u l a t i o n by t h i s model e x c e l s p r e v i o u s a n a l y s e s by t h e s t a t i s t i c a l model i n i t s a b i l i t y t o show t h e e f f e c t of room a b s o r p t i o n e x p l i c i t l y . The v a l i d i t y of t h e a s s u m p t i o n s u s e d by t h e s t a t i s t i c a l model h a s a l s o been examined i n t h i s r e p o r t .
INTRODUCTION
When a h i g h l y r e v e r b e r a n t room i s e x c i t e d by a s t e a d y h i g h - f r e q u e n c y t o n e , a v e r y complex p a t t e r n of s t a n d i n g waves w i l l be produced. A s t h e f r e q u e n c y of t h e s o u r c e c h a n g e s , one w i l l f i n d enormous v a r i a t i o n s i n t h e a c o u s t i c e n e r g y d e n s i t y a t any f i x e d l o c a t i o n i n t h e room. I n o t h e r words, t h e f r e q u e n c y r e s p o n s e c u r v e a t any f i x e d l o c a t i o n shows l a r g e f l u c t u a t i o n s . On t h e o t h e r hand, i f t h e s o u r c e f r e q u e n c y i s f i x e d , one w i l l f i n d t h a t t h e s p a t i a l v a r i a t i o n of t h e a c o u s t i c e n e r g y d e n s i t y i s e q u a l l y l a r g e . The sampling s t a t i s t i c s of t h e f i r s t problem have been c o n s i d e r e d by S c h r o e d e r , w h i l e t h o s e of t h e l a t t e r c a s e were i n v e s t i g a t e d by w a t e r h o u s e 2 and ~ u b m a n . ~ Although t h e a p p r o a c h e s t a k e n by them seem t o be d i f f e r e n t , t h e b a s i c i d e a and a s s u m p t i o n s i n v o l v e d a r e e s s e n t i a l l y t h e same. The e f f e c t of room a b s o r p t i o n i s a b s e n t from t h e i r a n a l y s e s , e x c e p t t h a t i t i s used t o d e f i n e t h e Schroeder c r i t i c a l f r e q u e n c y l i m i t ( f c ) . I t i s g e n e r a l l y b e l i e v e d t h a t t h e i r p r e d i c t i o n s a r e v a l i d f o r f r e q u e n c i e s above f c . So f a r , t h e few e x p e r i m e n t a l r e s u l t s 2 s 4 9 9 6, t h a t s u p p o r t t h e s e p r e d i c t i o n s were n o t d e s i g n e d t o show any of t h e i r l i m i t a t i o n s , s u c h a s t h e f a i l u r e t o c o n s i d e r room a b s o r p t i o n . It was t h i s s i t u a t i o n t h a t l e d t o t h e p r e s e n t i n v e s t i g a t i o n .
A s i m i l a r a n a l y s i s u s i n g t h e normal mode model c a n b e employed t o e x t e n d s u c h s t u d i e s t o i n c l u d e room a b s o r p t i o n a s a n i m p o r t a n t parameter. With t h e normal mode model, v a l i d i t y of t h e a s s u m p t i o n s u s e d i n t h e p r e v i o u s work by S c h r o e d e r , Waterhouse and Lubman, can be examined by means of a computer e x p e r i m e n t . The e f f e c t of t h e d e v i a t i o n s of t h e assumptions on t h e f i n a l r e s u l t s of t h e sampling s t a t i s t i c s h a s a l s o been s t u d i e d
a n a l y t i c a l l y .
REVIEW OF THE STATISTICAL MODEL
The a p p r o a c h e s t a k e n by S c h r o e d e r , Waterhouse, and Lubman a r e b a s i c a l l y s i m i l a r , b e l o n g i n g t o a model of room a c o i ~ s t i c s c a l l e d t h e s t a t i s t i c a l
model. I n t h i s model, t h e r e v e r b e r a n t sound f i e l d i n a room u n d e r pure-tone e x c i t a t i o n i s assumed t o c o n s i s t of t h e phasor sum of many modes1 o r
-
The mean-square p r e s s u r e ( p 2 ) a t any p o i n t i n s i d e t h e room c a n be w r i t t e n a s 2
-
s c h r o e d e r l s t u d i e d t h e v a r i a t i o n of p2 a s a f u n c t i o n of f r e q u e n c y , k e e p i n g t h e s o u r c e and t h e r e c e i v e r p o s f t i o n s f i x e d , whereas w a t e r h o u s e 2 and ~ u b m a n ~
-
s t u d i e d t h e s p a t i a l v a r i a t i o n of p 2 , k e e p i n g t h e f r e q u e n c y and t h e s o u r c e p o s i t i o n f i x e d . The g e n e r a l c o n s e n s u s i s t h a t i fl e i }
i s u n i f o r m l y d i s t r i b u t e d f r o m 0 t o 2 a , and i f m i s s u f f i c i e n t l y l a r g e , t h e f o l l o w i n g f o u r a s s u m p t i o n s can be p o s t u l a t e d : ( 1 ) z e r o mean v a l u e : <C1> = < C 2 > = 0 ; 2 2 ( 2 ) e q u a l mean-square v a l u e : <C1> = < C 2 > ; ( 3 ) z e r o c o r r e l a t i o n : <C 1 2 C > = 0 ; ( 4 ) C1, C2 a r e G a u s s i a n d i s t r i b u t e d . The symbol <> d e n o t e s s p a t i a l o r f r e q u e n c y a v e r a g i n g . A s a r e s u l t , t h e p r o b a b i l i t y d e n s i t y f u n c t i o n ( ~ ( 1 ) )- -
of t h e normalized mean-square p r e s s u r e [ I ( I = p 2 / < p 2 > ) ] w i l l h a v e a gamma d i s t r i b u t i o n :and a s t a n d a r d d e v i a t i o n of one. The c o r r e s p o n d i n g p r o b a b i l i t y d e n s i t y f u n c t i o n ( P ( ~ ) ) of t h e sound p r e s s u r e l e v e l ( y ) i s where: y = 10 l o g ( 1 ) = R n ( I ) / k ; k = Rn(10)/10. The s t a n d a r d d e v i a t i o n f o r t h i s c a s e i s 5.57 d B . l I n t h e few e x p e r i m e n t a l d a t a 2 t h a t s u p p o r t t h i s t h e o r y , t h e e f f e c t of room a b s o r p t i o n i s u s e d o n l y i n d e f i n i n g t h e Schroeder c r i t i c a l f r e q u e n c y l i m i t (f c): 1
where V i s t h e room volume i n c u b i c m e t r e s and TsO i s i t s r e v e r b e r a t i o n
t i m e , which i s governed by t h e room a b s o r p t i o n . A t f r e q u e n c i e s above f c ,
t h e s t a t i s t i c a l model s h o u l d a p p l y . A s i m i l a r a n a l y s i s u s i n g t h e normal mode model w i l l show t h e e f f e c t of room a b s o r p t i o n e x p l i c i t l y .
NORMAL
MODE
ANALYSISFor a r e c t a n g u l a r room of s i d e s L,, L
,
L, and volume V = LxLyLZ, t h e mean-square sound p r e s s u r e a t t h e p o i n t-+
3
( x , y , z ) d u e t o a s i m p l e s o u r c ew h e r e t h e o r i g i n o f c o o r d i n a t e s i s a t a c o r n e r o f t h e chamber, N s t a n d s f o r t h e t r i o of numbers R , m, n a n d :
AN = ~ / E ~ E ~c O = E ~ 1 and , E~ = 2 f o r i f 0, q N ( x , y , z ) = c o s k x x c o s k
Y
y c o s kz
z,I n t h i s e q u a t i o n k = w/c i s t h e wavenumber, p i s t h e d e n s i t y of t h e medium, and B i s t h e s p e c i f i c normal a d m i t t a n c e of t h e chamber s u r f a c e , assumed t o be r e a l and s m a l l . For c o m p a r i s o n w i t h t h e s t a t i s t i c a l m o d e l , E q . ( 5 ) h a s been r e c a s t i n t o a s i m i l a r form t o Eq. ( 1 ) . T h a t i s :
where:
(N
= t a n - I [2kbN/ (k2-ki)] ;S i n c e $N i s o n l y a f u n c t i o n of t h e wave number, and n o t a f u n c t i o n o f s p a c e , t h e model u s e d by Waterhouse and Lubman i s n o t e x a c t l y t h e same a s t h e o n e p r e s e n t e d i n t h i s a n a l y s i s . However, t h e g e n e r a l f o r m a t of t r e a t i n g t h e mean-square p r e s s u r e a s t h e sum of two components i s s t i l l a v a l i d one. An i m p o r t a n t a s p e c t of Eq. ( 6 ) i s t h a t t h e room boundary a b s o r p t i o n h a s b e e n i n c o r p o r a t e d as a p a r a m e t e r i n t h i s model. A s 6N+0 ( f o r p e r f e c t l y
-
r e f l e c t i n g b o u n d a r i e s ) , +N+O and t h e s e c o n d t e r m of Eq. ( 6 ) d r o p s o u t a n d p2 h a s o n l y one component. A l l t h e r e s u l t s p r e d i c t e d by t h e s t a t i s t i c a l modelw i l l b e i n v a l i d s i n c e i t assumes two t e r m s . The e x t e n t t o which t h e
boundary a b s o r p t i o n w i l l a f f e c t t h e r e s u l t s and t h e a s s u m p t i o n s used i n t h e s t a t i s t i c a l model w i l l be t h e s u b j e c t of t h e p r e s e n t i n v e s t i g a t i o n .
-
Only t h e c a s e of s p a t i a l s a m p l i n g o f p2 w i l l be c o n s i d e r e d h e r e . No a n a l y t i c a l s o l u t i o n s c a n be o b t a i n e d when t h e s p a t i a l s a m p l i n g i s c o n f i n e d t o t h e c e n t r a l p o r t i o n of t h e room. Thus, a computer e x p e r i m e n t b a s e d on Eq. ( 6 )i s u s e d .
COMPUTER EXPERIMENTS
E q u a t i o n ( 6 ) was t r e a t e d a s t h e c o m b i n a t i o n of two components, C 1 and C 2 , s i m i l a r t o Eq. ( 1 ) . Keeping
t o ,
k , and SN f i x e d , a l a r g e number of4
+
v a l u e s of C 1 and C2 were computed by v a r y i n g r. A l i s t i n g of t h e computer
program u s e d i s shown i n Appendix A . The c o m p u t a t i o n s were p e r f o r m e d f o r t h e f o l l o w i n g c o n d i t i o n s .
( 1 ) The room d i n e n s i o n u s e d was 8 x 6.5 x 4.9 m.
( 2 ) Seven 6 v a l u e s were u s e d r a n g i n g from 0.001 t o 0.01. These l i m i t s o f t h e r a n g e a r e a p p r o x i m a t e l y e q u a l t o t h e s u r f a c e a b s o r p t i o n o f p a i n t e d c o n c r e t e , and t h e a l l o w a b l e w a l l a b s o r p t i o n l i m i t s e t by t h e ASTM s t a n d a r d s , r e s p e c t i v e l y . ( 3 ) Nine 1 / 3 o c t a v e - b a n d c e n t e r f r e q u e n c i e s r a n g i n g f r o m 160 Hz t o 1
kHz
w e r e u s e d . ( 4 ) Four d i f f e r e n t s o u r c e p o s i t i o n s were u s e d . ( 5 ) To r e d u c e s t a t i s t i c a l f l u c t u a t i o n s , a p p r o x i m a t e l y 1600 s p a t i a l s a m p l e s w e r e used f o r e a c h c a s e e v e n t h o u g h t h e y w i l l n o t a l l be i n d e p e n d e n t . These s a m p l e s w e r e t a k e n i n t h e c e n t r a l r e g i o n o f t h eroom
a t
l e a s t h a l f a w a v e l e n g t h from t h e room b o u n d a r i e s . To a p p r o x i m a t e random s a m p l i n g , t h e f o l l o w i n g scheme was u s e d . Ar e g u l a r l y s p a c e d g r i d was f i r s t e s t a b l i s h e d and t h e n e a c h g r i d p o i n t was p e r t u r b e d by a random d i s t a n c e l e s s t h a n a g r i d s p a c i n g i n a random d i r e c t i o n g e n e r a t e d by random numbers. No p o i n t c l o s e r t h a n a miniplum d i s t a n c e (dm) t o t h e s o u r c e was u s e d ; dm i s e q u a l t o 0 . 1 6 ( V / T 6 0 ) 2 a s g i v e n by Ref. 10. T h i s i s i m p o r t a n t b e c a n s e t h e summation i n v o l v e d i n one o f t h e components of Eq. ( 6 ) d i v e r g e s when t h e f i e l d p o i n t c o i n c i d e s w i t h t h e s o u r c e p o s i t i o n .
( 6 ) The t o t a l number o f modes summed s h o u l d be governed by t h e f i n a l
v a l u e s r e a c h e d by b o t h components of Eq. ( 6 ) a s N g o e s t o
However, i t i s i m p r a c t i c a l t o a p p l y t h i s c r i t e r i o n t o e v e r y s p a t i a l p o i n t u s e d . Thus t h e u p p e r mode numbers w e r e d e t e r m i n e d by a
-
p r e l i m i n a r y i n v e s t i g a t i o n of t h e r o o n a v e r a g e d v a l u e s of p 2 a s a f u n c t i o n o f f r e q u e n c y . The s e t of v a l u e s c h o s e n t h a t can be a p p l i e d t o a l l t h e c a s e s c o n s i d e r e d i s l i s t e d a s f o l l o w s :
f ( H z ) 160 200 250
315
400 500 630 800 1000 U p p e r M o d e ft 3 0 30 30 30 35 4 0 40 45 55.A few s p o t - c h e c k s showed t h a t no changes w e r e p r o d u c e d i n t h e t h i r d s i g n i f i c a n t f i g u r e s o f t h e i n d i v i d u a l sum of Eq. ( 6 ) i f more modes w e r e added a f t e r t h e c h o s e n l i m i t s .
The s t a t i s t i c a l q u a n t i t i e s i n v e s t i g a t e d a r e :
il
( 1 )
<cl>/<C:>'
and( 4 ) p r o b a b i l i t y d e n s i t y o f C 1 a n d C2;
( 5 ) s t a n d a r d d e v i a t i o n s o f t h e n o r m a l i z e d mean-square p r e s s u r e s and sound p r e s s u r e l e v e l s .
The r e s u l t s a r e t a b u l a t e d i n T a b l e s I t o
VIII,
where t h e s o l i d l i n e sd i f f e r e n t b e t a v a l u e s u s e d (Appendix
B).
B e f o r e one c a n d i s c u s s t h es i g n i f i c a n c e of t h e s e r e s u l t s , i t i s n e c e s s a r y t o e s t a b l i s h how d e p a r t u r e s f r o m t h e f i r s t t h r e e a s s u m p t i o n s ( z e r o mean v a l u e , e q u a l mean-square v a l u e , and z e r o c o r r e l a t i o n ) used i n t h e s t a t i s t i c a l model w i l l a f f e c t t h e computed s t a n d a r d d e v i a t i o n s .
THEORETICAL PREDICTIONS
E f f e c t of Non-zero Mean V a l u e s of t h e Components
A l t h o u g h t h e e f f e c t of a non-zero mean v a l u e o n t h e s t a n d a r d d e v i a t i o n of t h e n o r m a l i z e d mean-square p r e s s u r e s c a n be p r e d i c t e d a n a l y t i c a l l y , no c l o s e d f o r m s o l u t i o n h a s b e e n f o u n d f o r i t s e f f e c t on t h e s t a n d a r d d e v i a t i o n of t h e c o r r e s p o n d i n g sound p r e s s u r e l e v e l s . Thus i t
i s
n e c e s s a r y t o s t u d y t h i s e f f e c t n u m e r i c a l l y . A computer s u b r o u t i n e c a p a b l e of g e n e r a t i n g G a u s s i a n d i s t r i b u t e d random numbers w i t h a r b i t r a r y mean and s t a n d a r d d e v i a t i o n was u s e d . Employing d i f f e r e n t s t a r t i n g p o i n t s , two s e t s-
o f u n c o r r e l a t e d random numbers c o u l d b e g e n e r a t e d t o s i m u l a t e C 1 and C2 of p2 i n Eq. ( 1 ). F o r a r a n g e of<c>/<c2>'
between 20.2 and a r a t i o of t h e components' mean-square v a l u e s of up t o 1 0 , 0 0 0 , t h e change i n t h e s t a n d a r d d e v i a t i o n o f t h e n o r m a l i z e d mean-square p r e s s u r e s i s l e s s t h a n 0.7%, w h i l e t h a t f o r t h e sound p r e s s u r e l e v e l s i sl e s s t h a n 3.3%.
I n t h e f o l l o w i n g s e c t i o n s , t h e e f f e c t of non-zero c o r r e l a t i o n i s
t r e a t e d t o g e t h e r w i t h t h e non-equal mean-square v a l u e s of t h e components a s a s i n g l e problem. Thus t h e e f f e c t of non-zero mean v a l u e s i s n o t i m p o r t a n t , i r r e s p e c t i v e of t h e o t h e r two a s s u m p t i o n s .
E f f e c t o f Non-zero C o r r e l a t i o n between t h e Components
Assuming t h a t t h e two components h a v e a b i v a r i a t e G a u s s i a n p r o b a b i l i t y d e n s i t y f u n c t i o n , t h e y c a n be t r a n s f o r m e d i n t o two i n d e p e n d e n t and t h u s u n c o r r e l a t e d G a u s s i a n random v a r i a b l e s by a r o t a t i o n o f c o - o r d i n a t e s . l 1 T h a t i s , g i v e n t h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n o f t h e t w o components: 2 2 2 2 1 u2C1
-
2R12C1C2+
0 C P(C1,C2) =2
2
3
Exp (-+ 2 2 2 2,(7
2 "l(72-R12 2 where: u1 = <C1> w i t h <C1> = 0 , 2 2 a 2 = <C2> w i t h <C2> = 0, one c a n d e f i n e and l e t t h e t r a n s f o r m e d v a r i a b l e s b e :Then t h e r a t i o of t h e mean-square v a l u e s ( h e r e t h e same a s t h e v a r i a n c e ) o f t h e t r a n s f ormed v a r i a b l e s i s : L L where: A = a l / 0 2 , S i n c e t h e c o r r e l a t i o n c o e f f i c i e n t c a n be i n c o r p o r a t e d i n t o t h e d e f i n i t i o n of t h e r a t i o o f t h e mean-square v a l u e s of t h e new v a r i a b l e s , t h e e f f e c t of non-zero c o r r e l a t i o n between t h e components c a n b e t r e a t e d w i t h non-equal mean-square v a l u e s of t h e components as a s i n g l e problem.
E f f e c t of Non-equal Mean-square Values of t h e Components
A c c e p t i n g t h e a s s u m p t i o n s t h a t t h e components a r e G a u s s i a n d i s t r i b u t e d w i t h z e r o mean v a l u e s , and a r e u n c o r r e l a t e d , t h e p r o b a b i l i t y d e n s i t y
f u n c t i o n s ( ~ ( 1 ) and ' ~ ( y ) ) c a n be r e d e r i v e d f o r b o t h t h e n o r m a l i z e d
mean-square p r e s s u r e
(I)
and t h e sound p r e s s u r e l e v e l ( y ) a s a f u n c t i o n of t h e r a t i o o f t h e components' mean-square v a l u e s (A). T h i s i s done i n Appendix C and t h e r e s u l t s a r e q u o t e d h e r e .- -
where: I = p 2 / < p 2 > , a n dI O
i s t h e m o d i f i e d B e s s e l f u n c t i o n . The s t a n d a r d d e v i a t i o n ( ~ ( 1 ) ) i s e q u a l t o : w h i c h i s e q u a l t o a w h e n o n e of t h e components v a n i s h e s . T h i s c o r r e s p o n d s t o t h e c a s e where t h e w a l l s are p e r f e c t l y r e f l e c t i v e . A l s o , when A = 1 , t h e p r o b a b i l i t y d e n s i t y f u n c t i o n r e d u c e s t o :and a ( I ) = 1
as
o b t a i n e d byater rho use.^
The c o r r e s p o n d i n g p r o b a b i l i t y d e n s i t y f u n c t i o n f o r t h e sound p r e s s u r e l e v e l s i s :where: y = 10 l o g ( 1 ) = R n ( I ) / k ; k Rn(10)/10.
However, no a n a l y t i c a l s o l u t i o n h a s b e e n f o u n d f o r t h e s t a n d a r d d e v i a t i o n o f t h e sound p r e s s u r e l e v e l s f o r a r b i t r a r y v a l u e s of A . N u m e r i c a l c o m p u t a t i o n o f u s i n g :
i s n e c e s s a r y . When A = 1, Eq. ( 1 1 ) r e d u c e s t o : P , ( y ) = k Exp[ky
-
E x p ( k y ) lwhich i s t h e same r e s u l t a s d e r i v e d by s c h r o e d e r l and a l s o by Bowers a n d ~ u b m a n . ' ~ The t h e o r e t i c a l v a l u e o f t h e s t a n d a r d d e v i a t i o n i s 5.57 dB. le F o r t h e e x t r e m e c a s e where o n e of t h e components d i s a p p e a r s , t h e
p r o b a b i l i t y d e n s i t y f u n c t i o n becomes:
1 and t h e t h e o r e t i c a l v a l u e of t h e s t a n d a r d d e v i a t i o n r i s e s t o Appendix C ) . 9.65 dB ( s e e I n summary, t h e e f f e c t of non-zero m e a n - v a l u e s of t h e components i s n o t i m p o r t a n t . The e f f e c t of non-zero c o r r e l a t i o n and non-equal mean-square v a l u e s c a n b e t r e a t e d t o g e t h e r a n a l y t i c a l l y . Knowing t h e s e q u a n t i t i e s , i t i s p o s s i b l e t o p r e d i c t t h e s t a n d a r d d e v i a t i o n s of b o t h t h e n o r m a l i z e d
mean-square p r e s s u r e s and t h e sound p r e s s u r e l e v e l s , assuming t h e components a r e G a u s s i a n d i s t r i b u t e d . Some o f t h e s e p r e d i c t e d v a l u e s a r e t a b u l a t e d i n T a b l e I X . On t h e o t h e r h a n d , i f t h e computed s t a n d a r d d e v i a t i o n s f r o m t h e sampled d a t a d i d n o t a g r e e w i t h t h e p r e d i c t e d v a l u e s b a s e d on t h e known v a l u e s o f t h e c o r r e l a t i o n c o e f f i c i e n t and t h e r a t i o o f t h e mean-square v a l u e s o f t h e components, t h e a s s u m p t i o n of G a u s s i a n d i s t r i b u t i o n must be v i o l a t e d . DISCUSSION To m i n i m i z e s t a t i s t i c a l f l u c t u a t i o n s , a v e r y l a r g e number of s a m p l e s h a v e been t a k e n which a r e n o t n e c e s s a r i l y i n d e p e n d e n t . Thus i t i s n o t f e a s i b l e t o e s t a b l i s h some k i n d o f c o n f i d e n c e l i m i t s f o r t h e computed s t a n d a r d d e v i a t i o n s t o be u s e d a s a g u i d e l i n e f o r s e t t i n g t h e a l l o w a b l e t o l e r a n c e s f o r t h e d i f f e r e n t s t a t i s t i c a l q u a n t i t i e s i n v e s t i g a t e d . S u b s e q u e n t l y , t h e s e t o l e r a n c e s had t o b e c h o s e n a r b i t r a r i l y , b u t t h e y were c a r e f u l l y p i c k e d t o p r o v i d e a c o n s i s t e n t p i c t u r e . G a u s s i a n D i s t r i b u t i o n Assumption S i n c e t h e s a m p l e s a r e n o t c o m p l e t e l y i n d e p e n d e n t , i t i s n o t m e a n i n g f u l t o a p p l y t h e C h i - s q u a r e t e s t f o r c h e c k i n g t h e c l o s e n e s s o f f i t of t h e
p r o b a b i l i t y d i s t r i b u t i o n of t h e two components. I n s t e a d , a mean-square d e v i a t i o n ( D ) of t h e computed p r o b a b i l i t y d e n s i t y f u n c t i o n o f t h e
s t a n d a r d i z e d random v a r i a b l e s [ z = (c-<C>)/U] f r o m t h e s t a n d a r d G a u s s i a n d e n s i t y i s d e t e r m i n e d a s a n i n d i c a t o r f o r t h i s t e s t . S i x t y i n t e r v a l s of e q u a l w i d t h w e r e u s e d and t h e r e s u l t s a r e g i v e n i n T a b l e s I a n d 11.
where P c ( z ) i s t h e computed d e n s i t y f u n c t i o n and P ( z )
i s
t h e s t a n d a r d i z e d G a u s s i a n d e n s i t y f u n c t i o n . The t a s k of d e c i d i n g wgat d e v i a t i o n i sa c c e p t a b l e w i l l be p r e s e n t e d i n t h e f o l l o w i n g p a r a g r a p h s .
A s h a s been shown i n t h e p r e v i o u s s e c t i o n , knowing t h e c o r r e l a t i o n c o e f f i c i e n t and t h e r a t i o of t h e mean-square v a l u e s of t h e two components, i t i s p o s s i b l e t o p r e d i c t t h e v a l u e s of t h e s t a n d a r d d e v i a t i o n s [ a ( 1 ) and a ( y ) ] f o r b o t h t h e n o r m a l i z e d mean-square p r e s s u r e and t h e sound p r e s s u r e l e v e l s i f t h e components a r e t r u l y Gaussian d i s t r i b u t e d . Thus a t o l e r a n c e v a l u e f o r t h e mean-square d e v i a t i o n ( D ) may be chosen by comparing t h e p r e d i c t e d and t h e computed v a l u e s of o ( I ) and o ( y ) . Comparison u s i n g a ( y ) t u r n e d o u t t o be b e t t e r , b e c a u s e a few v e r y l a r g e v a l u e s of t h e sampled C1 and C2 c a n a f f e c t t h e c a l c u l a t e d o ( 1 ) s i g n i f i c a n t l y b u t t h e i r i n f l u e n c e on t h e computed p r o b a b i l i t y d e n s i t y f u n c t i o n s of C 1 and C2 w i l l be less s e v e r e , e x c e p t a t t h e t a i l end i n t e r v a l s . T h i s e f f e c t i s somewhat s u p p r e s s e d by t h e l o g a r i t h m i c o p e r a t i o n when sound p r e s s u r e l e v e l s a r e used.
F i g u r e s 1 t o 4 show t h e comparisons between t h e computed and t h e p r e d i c t e d s t a n d a r d d e v i a t i o n s of sound p r e s s u r e l e v e l s f o r t h e f o u r d i f f e r e n t s o u r c e p o s i t i o n s u s i n g a l l t h e sampled d a t a of d i f f e r e n t
f r e q u e n c i e s and d i f f e r e n t v a l u e s of b e t a . Data p o i n t s i n F i g u r e s 1 and 4 which show s i g n i f i c a n t d e v i a t i o n from t h e p r e d i c t e d v a l u e s a r e shown w i t h a
f l a g . These d a t a p o i n t s had v a l u e s of D l a r g e r t h a n 0.003 ( * i n
-
T a b l e s I and 1 1 ) . I f one c o n c l u d e s t h a t t h e two components t h a t make up p2 a r e non-Gaussian f o r t h e s e p a r t i c u l a r c a s e s , a t o l e r a n c e l i m i t of 0.003 c a n be chosen f o r t h e mean-square d e v i a t i o n (D) a s a c r i t e r i o n f o r a c c e p t i n g o r r e j e c t i n g t h e assumption t h a t t h e components a r e G a u s s i a n d i s t r i b u t e d . These r e s u l t s show t h a t t h e S c h r o e d e r c r i t i c a l f r e q u e n c y l i m i t d o e s n o t e n s u r e G a u s s i a n d i s t r i b u t i o n . V i o l a t i o n seems t o be r e s t r i c t e d o n l y t o c e r t a i n f r e q u e n c i e s depending o n t h e s o u r c e l o c a t i o n . For t h e p a r t i c u l a r room s t u d i e d , t h e s e were 250 and 315 Hz.Zero-Mean Value Assumption
R e s u l t s p r e s e n t e d i n T a b l e s I11 and I V i n d i c a t e t h a t , i r r e s p e c t i v e of S c h r o e d e r ' s c r i t i c a l f r e q u e n c y l i m i t , t h e f i r s t assumption of
cC1> = <C2> = 0 i s g e n e r a l l y m e t , e x c e p t f o r t h e 630 Hz c a s e s . No
improvement t o t h e s e i r r e g u l a r r e s u l t s was a c h i e v e d when a d i f f e r e n t set of randomly sampled p o i n t s was used. S i m i l a r c a l c u l a t i o n s a t 620, 635 and 640 Hz showed v e r y s m a l l v a l u e s of <C1> and <C2>. Thus, t h e 630 Hz c a s e s must b e t r e a t e d a s anomalous f o r t h e p a r t i c u l a r room s t u d i e d because e x c e p t i o n a l l y l a r g e v a l u e s of <C1> and cC2> a p p e a r e d a t a l l t h e chosen s o u r c e p o s i t i o n s . R e s u l t s of t h e s e p e c u l i a r c a s e s p r o v i d e a means of c h e c k i n g t h e c o n c l u s i o n t h a t t h e e f f e c t of non-zero mean
i s
n o t i m p o r t a n t , i r r e s p e c t i v e of t h e o t h e r a s s u m p t i o n s . I n T a b l e X, s t a n d a r d d e v i a t i o n s computed from t h e sampled d a t a w i t h and w i t h o u t c o r r e c t i o n f o r t h e non-zero v a l u e s of t h e components, a r e p r e s e n t e d f o r comparfson. Except f o r t h ec a s e s where 'C~>/<C:>' i s l a r g e r t h a n 0.5, t h e d i f f e r e n c e s a r e
1
Equal Mean-Square Values AssumptionA s r e s u l t s p r e s e n t e d i n T a b l e V i n d i c a t e , t h e Schroeder c r i t i c a l f r e q u e n c y l i m i t does p r o v i d e some form of boundary f o r t h e a c c e p t a b i l i t y of t h i s assumption. I n g e n e r a l , c a s e s below t h e * Schroeder l i m i t have v a l u e s
L L
much l a r g e r t h a n one f o r t h e r a t i o < C 1 > / < C 2 > . A v a l u e of t h r e e might b e chosen as t h e a l l o w a b l e t o l e r a n c e f o r t h i s r a t i o i f t h e Schroeder l i m i t i s
t o be a c c e p t e d a s t h e g u i d e l i n e f o r a p p l y i n g t h e s t a t i s t i c a l model. The 500 Hz c a s e s must b e viewed a s anomalous f o r t h e p a r t i c u l a r room chosen because e x c e p t i o n a l l y l a r g e v a l u e s of t h i s r a t i o a p p e a r e d a t a l l s o u r c e p o s i t i o n s a n d - s i m i i a r c o m p u t a t i o n s a t 495, 505 and 510 Hz show a c c e p t a b l e L L v a l u e s f o r cC1>/<C2> f o r t h e same s o u r c e p o s i t i o n s . O t h e r t h a n t h a t , t h e r e a r e o n l y f i v e v i o l a t i o n s of t h i s a s s u m p t i o n a t f r e q u e n c i e s above t h e S c h r o e d e r l i m i t f o r s o u r c e p o s i t i o n s n o t a t t h e c o r n e r of t h e room. Zero C o r r e l a t i o n Assumption
S c h r o e d e r ' s c r i t i c a l f r e q u e n c y l i m i t seems t o mark t h e boundary between c a s e s w i t h l a r g e c o r r e l a t i o n c o e f f i c i e n t s and t h o s e w i t h more a c c e p t a b l e v a l u e s ( s e e r e s u l t s i n T a b l e VI). Choosing 0.3 a s t h e a l l o w a b l e t o l e r a n c e f o r t h e c o r r e l a t i o n c o e f f i c i e n t , t h e z e r o - c o r r e l a t i o n assumption i s v a l i d f o r a l l c a s e s above t h e Schroeder l i m i t f o r t h e c o r n e r s o u r c e p o s i t i o n e x c e p t f o r t h e 500 Hz c a s e s . Again t h e s e i r r e g u l a r r e s u l t s , which a p p e a r e d a t a l l s o u r c e p o s i t i o n s , must be viewed a s anomalous, s i n c e s i m i l a r
computation a t 495, 505 and 510 Hz showed a c c e p t a b l e v a l u e s . However, t h e r e a r e q u i t e a number of v i o l a t i o n s of t h i s assumption e v e n a t f r e q u e n c i e s above t h e S c h r o e d e r l i m i t f o r s o u r c e p o s i t i o n s o t h e r t h a n t h e c o r n e r . Computed S t a n d a r d D e v i a t i o n s
The computed s t a n d a r d d e v i a t i o n s of t h e n o r m a l i z e d mean-square p r e s s u r e s and t h e sound p r e s s u r e s a r e shown i n T a b l e s V I I and V I I I , r e s p e c t i v e l y . Based on t h e t o l e r a n c e s of 3 and 0.3 a s s i g n e d f o r t h e
assumptions of e q u a l mean-square v a l u e s and z e r o c o r r e l a t i o n r e s p e c t i v e l y , c o r r e s p o n d i n g t o l e r a n c e s [15% f o r u ( I ) and 6% f o r u ( y ) l c a n b e s e t f o r t h e computed s t a n d a r d d e v i a t i o n s u s i n g r e s u l t s shown i n T a b l e I X . Values above t h e s e t o l e r a n c e s a r e shown w i t h
*
i n b o t h t a b l e s . These r e s u l t s b r i n g o u t c l e a r l y t h e anomalous s i t u a t i o n s a t 500 Hz f o r t h e p a r t i c u l a r room s t u d i e d . Unacceptably h i g h v a l u e s a p p e a r e d a t a l l s o u r c e p o s i t i o n s f o r t h i sf r e q u e n c y . I n g e n e r a l , one can r e l a t e t h e o t h e r u n a c c e p t a b l e v a l u e s w i t h t h e v i o l a t i o n s of t h e a s s u m p t i o n s p r e s e n t e d i n t h e o t h e r t a b l e s . However, one s h o u l d n o t e x p e c t complete c o r r e s p o n d e n c e because of t h e random e r r o r s i n v o l v e d .
When t h e s o u r c e was n o t l o c a t e d a t t h e c o r n e r , t h e computed s t a n d a r d d e v i a t i o n s exceeded t h e a l l o w a b l e t o l e r a n c e s i n numerous c a s e s even a t f r e q u e n c i e s above t h e Schroeder c r i t i c a l f r e q u e n c y l i m i t ( T a b l e s V I I and V I I I ) . Thus, b e s i d e s t h e Schroeder l i m i t r e q u i r e m e n t , t h e c o r n e r p o s i t i o n must be s p e c i f i e d f o r t h e s o u r c e b e f o r e t h e p r e d i c t i o n s from t h e s t a t i s t i c a l model can be f u l l y a c c e p t e d .
10
A computer e x p e r i m e n t u s i n g t h e normal mode t h e o r y of r e c t a n g u l a r rooms
h a s shown t h a t , w i t h i n c e r t a i n
-
a l l o w a b l e t o l e r a n c e s f o r random e r r o r s , t h e s p a t i a l sampling s t a t i s t i c s of p 2 p r e d i c t e d by t h e s t a t i s t i c a l model a r e g e n e r a l l y v a l i d o v e r a f a i r l y wide r a n g e of room boundary a b s o r p t i o n v a l u e s p r o v i d e d t h a t : (1) t h e S c h r o e d e r c r i t i c a l f r e q u e n c y l i m i t i s s a t i s f i e d and ( 2 ) t h e s o u r c e i s l o c a t e d a t a c o r n e r of t h e room. The e f f e c t of boundary a b s o r p t i o n seems t o be a d e q u a t e l y t a k e n c a r e of by the r e q u i r e m e n t of t h e S c h r o e d e r l i a i t f o r normal v a l u e s of a b s o r p t i o n .For t h e p a r t i c u l a r room s t u d i e d , t h e r e a r e anomalous r e s u l t s a t 500 Hz,
which a p p e a r a t a l l t h e f o u r p o s i t i o n s chosen. Such r e s u l t s would s u g g e s t t h a t f o r r e v e r b e r a t i o n room measurements i n v o l v i n g pure-tone o r v e r y
narrow-band s o u r c e s , a v e r a g i n g o v e r s o u r c e p o s i t i o n might n o t b e a d e q u a t e . Other t e c h n i q u e s s u c h a s r e s p o n s e a v e r a g i n g u s i n g r o t a t i n g d i f f u s e r may be a d v i s a b l e .
O t h e r non-valid r e s u l t s were caused mainly by t h e v i o l a t i o n s of two of t h e a s s u m p t i o n s i n v o l v e d i n t h e s t a t i s t i c a l model: t h e e q u a l mean-square v a l u e and t h e z e r o c o r r e l a t i o n a s s u m p t i o n s of t h e two components t h a t make
-
up p2. The o t h e r two a s s u m p t i o n s of z e r o l n e a n v a l u e and Gaussiand i s t r i b u t i o n a r e m o s t l y v a l i d even a t f r e q u e n c i e s below t h e Schroeder l i m i t . Again t h e r e a r e anomalous s i t u a t i o n s which might b e p e c u l i a r t o t h e
I
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11. D a v e n p o r t ,
W.B.,
Jr. and R o o t , W.L. An I n t r o d u c t i o n t o t h e Theory of Random S i g n a l s and Noise. McGraw-Hill, New York, 1958, p. 168.12. Bowers, H. a n d Lubman, D. D e c i b e l A v e r a g i n g i n R e v e r b e r a n t Rooms. LTVR Tech. R e p o r t 0-71200/8TR-130 ( 1 9 6 8 ) .
13. Morse,
P.M.
and I n g a r d , U. T h e o r e t i c a l A c o u s t i c s . McGraw-Hill, New York, 1968.14. S p r i n g e r , M.D. The A l g e b r a of Random V a r i a b l e s . J o h n Wiley & S o n s , New York, 1979, Ch. 3.
15. G r a d s h t e y n , I.S. and Ryzhik, I . M . T a b l e of I n t e g r a l s S e r i e s and P r o d u c t s . Academic P r e s s , New York, 1965-
16. McLachlan,
N.W.
B e s s e l F u n c t i o n s f o r E n g i n e e r s . Oxford Univ. P r e s s , London, 2nd E d i t i o n , 1961.17. C r a m e r , H. M a t h e m a t i c a l Methods of S t a t i s t i c s . P r i n c e t o n Univ. P r e s s , P r i n c t o n , N J , 1 9 4 6 , S e c t i o n 12.3.
TABLE
I.
Mean-square d e v i a t i o n of computed p r o b a b i l i t y d e n s i t y f u n c t i o n f r o m s t a n d a r d G a u s s i a n d e n s i t y f o r 1st component. S o l i d l i n e s mark b o u n d a r i e s of S c h r o e d e r ' s c r i t i c a l f r e q u e n c y l i m i t . BETA - - SOURCEAT
( 0 . 0 0 , 0.00, 0.00)SOURCE AT
( 2 . 3 4 , 1.57, 1.67) - - SOURCEAT
(4.50, 3.00, 2.30) - SOURCE AT (6.00, 4.20, 3 - 5 0 ]-. .-.----.- .. . . .. . .
14
TABLE 11. Mean-square d e v i a t i o n of computed p r o b a b i l i t y d e n s i t y f u n c t i o n from standard Gaussian d e n s i t y f o r 2nd component. S o l i d l i n e s mark boundaries of Schroederrs c r i t i c a l frequency limit.
BETA
F R E Q
0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100 SOURCE AT (0.00, 0.00, 0.00) 160 0.0008 0.0009 0.0010 0.0012 0.0011 0.0011 0.0009 200 0.0016 0.0013 0.0014 0.0010 0.0008 0.0016 r 0 . 0 0 1 1 250 0.0011 0.0015 0.0008 0.0011 0.0012]
0.0010 0.0012 315 0.0160* 0.0120* 0.0063*1
0.0026 0.0008 0.0009 0.0006 400 0.0023 0.00121
0.0014 0.0011 0.0009 0.0017 0.0012 500 0.0013 1 0 . 0 0 1 4 0.0009 0.0010 0.0011 0.0010 0.0009 630 0.0014 0.0013 0.0010 0.0012 0.0015 ' 0.0014 0.0012 800 0.0009 0.0010 0.0013 0.0010 0.0010 0.0006 0.0008 1000 0.0010 0.0009 0.0011 0.0007 0.0009 0.0010 0.0008 - SOURCE AT (2.34, 1.57, 1.67) SOURCE AT (4.50, 3.00, 2.30) 160 0.0008 0.0010 0.0011 0.0015 0.0013 0.0019 0.0020 200 0.0016 0.0016 0.0014 0.0016 0.0010 0.0011 1 0 . 0 0 1 0 250 0.0060* 0.0066* 0.0056* 0.0043* 0.00261
0.0014 0.0008 315 0.0190* 0.0170* 0.0140*[ 0.0100* 0.0064* 0.0033* 0.0017 400 0.0013 0.00161
0.0013 0.0008 0.0008 0.0009 0.0012 500 0.0009 [ 0 . 0 0 1 1 0.0010 0.0011 0.0011 0.0008 0.0008 630 0.0012 0.0012 0.0011 0.0010 0.0013 0.0010 0.0008 800 0.0011 0.0015 0.0009 0.0009 0.0011 0.0012 0.0016 1000 0.0010 0.0012 0.0016 0.0015 0.0022 0.0012 0.0013SOURCE
AT (6.00, 4.20, 3.50) 160 0.0011 0.0011 0.0010 0.0011 0.0014 0.0012 0.0015 200 0.0008 0.0009 0.0012 0.0017 0.0013 0.00161
0.0012 250 0.0016 0.0014 0.0007 0.0016 0.0013 [ 0 . 0 0 1 3 0.0012 315 0.0120* 0.0095* 0.0044*1
0.0010 0.0016 0.0015 0.0007 400 0.0013 0.00101
0.0010 0.0010 0.0012 0.0011 0.0013 500 0.00121
0.0011 0.0008 0.0012 0.0010 0.0012 0.0015 630 0.0019 0.0012 0.0015 0.0014 0.0011 0.0012 0.0008 800 0.0009 0.0010 0.0011 0.0009 0.0006 0.0011 0.0010 1000 0.0006 0.0010 0.0007 0.0018 0.0010 0.0012 0.000715
TABLE
111. Ratio of the 1 s t component's mean value and t h e square-root of i t s mean-square value, < c ~ > / < c ~ > + . S o l i d l i n e s markboundaries of Schroeder's c r i t i c a l frequency l i m i t .
BETA
FREQ 0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100SOURCE
AT (0.00, 0.00, 0.00) 160 0.010 200 0.006 250 0.021 315 0.008 400 0.005 500 0.004 630 0.015 800 0.047 1000 0.020SOURCE AT
(2.34, 1.57, 1.67) 160 0.020 200 0.001 250 0.027 315 0.018 400 0.035 500 0.007 630 0.006 800 0.036 1000 0.023SOURCE AT
(4.50, 3.00, 2.30) 200 0.009 250 0.015 315 U.013 400 0.018 500 0.041 630 0.005 800 0.002 1000 0.030SOURCE
AT (6.00, 4.20, 3.50) 160 0.011 200 0.005 250 0.003 315 0.001 400 0,000 630 0.030 800 0.021 1000 0.007 *Values >0.116
TABLE IV.
Ratio of the 2nd component's mean value and the square-root of its mean-square value,'c2>/<c2>*.
2 Solid lines markboundaries of Schroeder's critical frequency limit.
BETA FREQ 0.0010 0.0015 0.0025 0.0035 0 .OD50 0.0070 0.0100 SOURCE AT (0.00, 0.00, 0.00) 160 0.013 0.013 0.013 0.013 0.013 0.013 0.012 200 0.002 0.001 0.003 0.005 0.007 0.008
1
0.009 250 0.018 0.019 0.020 0.020 0.018I
0.015 0.010 315 0.012 0.010 0.007[
0.002 0.003 0.006 0.006 400 0.017 0.0191
0.021 0.02 1 0.019 0.018 0.016 500 0.006 1 0 . 0 0 6 0.006 0.007 0.007 0.007 0.004 630 0.313* 0.260* 0.191* 0.151* 0.115* 0.089 0.067 800 0.030 0.029 0.026 0.020 0.014 0.008 0.001 1000 0.037 0.033 0.024 0.018 0.013 0.008 0.003 SOURCE AT (2.34, 1.57, 1.67) 160 0.010 0.016 0.027 0.036 0.049 0.058 0.079 200 0.008 0.009 0.013 0.018 0.027 250 0.023 0.023 0.026 0.028 0.032 315 0.028 0.028 0.0301
0.030 0.030*lzr
0.027 0.02 1 400 0.001 0.001[
0.005 0.010 0.016 0.023 0.029 500 0.005 0.006 0.007 0.008 0.012 0.016 0.030 630 0.667* 0.609* 0.478* 0.392* 0.323* 0.273* 0.234* 800 0.023 0.024 0.025 0.026 0.026 0.024 0.019 1000 0.003 0 .OOO 0.006 0.009 0.011 0.013 0.019 SOURCE AT (4.50, 3.00, 2.30) 160 0.010 0.009 0.006 0.002 0.014 0.025 0.041 200 0.005 0.006 0.011 0.012 0.016 0.018]
0.016 250 0.006 0.005 0.002 0.001 0.0021
0.003 0 .OO 1 315 0.014 0.015 0.0151
0.015 0.015 0.013 0.004 400 0.002,
:
1
0.005 0.008 0.011 0.004 0.004 500 0.003 0.005 0.007 0.009 0.0 12 0.018 630 0.144* 0.095 0.045 0.022 0.004 0.004 0.012 800 0.007 0.005 0.000 0.004 0.008 0.012 0.013 1000 0.036 0.035 0.032 0 -030 0.028 0.029 0.028 SOURCE AT (6.00, 4.20, 3.50) 160 0.013 0.019 0.024 0.028 0.039 0.056 0.07 1 200 0.007 0.010 0.014 0.020 0.023 0.030]
0.030 250 0.003 0.002 0.001 0 .OO 1 0.0021
0.002 0.004 315 0.028 0.027 0.0231
0.018 0.012 0.011 0.017 400 0.015 0.0151
0.016 0.016 0.014 0.010 0.009 500 0.0011
0,000 0.000 0.001 0.003 0.005 0.001 630 0.589* 0.511* 0.412* 0.354* 0.297* 0.259* 0.210* 800 0.005 0.009 0.015 0.018 0.020 0.021 0.024 1000 0.023 0.021 0.020 0.019 0.015 0.013 0.011 *Values '0.117
TABLE
V. Ratio of the two components' mean-square v a l u e s ,<c:>/<c~
>.Solid l i n e s mark boundaries of Schroeder's c r i t i c a l frequency
l i m i t
.
BETA FREQ 0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100SOURCE AT
(0.00, 0.00, 0.00) -SOURCE AT
(2.34, 1.57, 1.67) - --SOURCE
AT
(4.50, 3.00, 2.30) 160 181.04* 83.62* 33.53* 19.71" 11.97" 7.79* 4.86* 200 2.58 4.18* 4.63" 4.42* 4.35" 4.41*[
3.94* 2 50 18.51* 8.79* 3.87" 2.56 1.93I
1.64 1.40 315 17.54* 7.94* 2.951
1.59 0.88 0.57 0.45 400 2.47 1.641
1.33 1.22 1 .OO 0.79 0.65 500 22.16* ] 1 1 . 1 9 * 5.54* 3.95* 3.04* 2.45 1.93 630 1.77 1.73 1.37 1.16 1.03 0.97 0.94 800 1.99 1.36 1.05 1.03 1.08 1.11 1.14 1000 0.87 0.84 0.90 0.94 0.99 1.05 1.13SOURCE AT
(6.00, 4.20, 3.50)TABLE
VI.
C o r r e l a t i o n c o e f f i c i e n t,
<c1c2>/[<c:><c$>]*,
of t h e two components. S o l i d l i n e s mark b o u n d a r i e s of S c h r o e d e r ' s c r i t i c a li
f r e q u e n c y l i m i t .BETA
FREQ
0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100 - - - - SOURCEAT
(0.00, 0.00, 0.00) 160 0.72* 0.71* 0.67* 0.63* 0.54" 0.4 3* 0.30 200 -0.53* -0.19 0.23 0.41* 0.47* 0.40*]
0.25 250 -0.75* -0.70* -0.57" -0.42* -0.261
-0.14 -0.08 3 15 0.00 -0.02 -0.07J
-0.11 -0.13 -0.14 -0.16 400 -0.45* -0.33*1
-0.16 -0.08 -0.02 0.02 0.02 500 -0.68*1
-0.64* -0.54* -0.43* -0.30 -0.19 -0.08 630 -0.11 -0.15 -0.16 -0.17 -0.17 -0.15 -0.13 800 0.07 0.07 0.00 -0.04 -0.06 -0.05 -0.02 1000 -0.03 -0 -04 -0.02 0.01 0.05 0.07 0 -08 SOURCEAT
(2.34, 1.57, 1.67) 800 0.05 0.04 -0.08 -0.16 -0.19 -0.17 -0.14 1000 -0.33* -0.25 -0.17 -0.13 -0.10 -0.06 0.00i
SOURCEAT
(4.50, 3.00, 2.30)SOURCE AT
(6.00, 4.20, 3.50)i 1
TABLE
VII.
S t a n d a r d d e v i a t i o n of normalized mean-square p r e s s u r e s ,- -
~ ( p ~ / < ~ ~ > ) . S o l i d l i n e s mark b o u n d a r i e s of Schroeder ' s c r i t i c a l f r e q u e n c y l i m i t .-
BETA
-~ SOURCE AT (0.00, 0.00, 0.00) - SOURCEAT
(2.34, 1.57, 1.67) SOURCEAT
(4.50, 3.00, 2.30) p-- - SOURCEAT
(6.00, 4.20, 3.50)TABLE VIII. Standard d e v i a t i o n of t h e sound p r e s s u r e l e v e l s ,
o(SPL).
S o l i d l i n e s mark boundaries o f Schroeder's c r i t i c a l f r e q u e n c y
l i m i t . BETA FREQ 0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100 SOURCE AT ( 0 . 0 0 , 0.00, 0 . 0 0 ) 800 5.69 5.77 5.82 5.8 3 5.57 5.64 5.62 1000 5.73 5.61 5.55 5.55 5.55 5.55 5.69 SOURCE AT ( 2 . 3 4 , 1.57, 1 . 6 7 ) 160 8.60* 8.29* 7.79* 7.47* 7.14* 7.01* 6.91* 200 7.73* 7.65* 7.25* 6.97* 6.71* 6.50*
1
6.38* 250 8.88* 8.37* 7.81* 7.39" 6.90*1
6.62* 6.23* 3 15 5.58 5.20* 4.87*1
4.83* 4.88* 4.99* 5.42 400 6.28* 5.93*1
5.57 5.47 5.39 5.40 5.52 500 7.33*1
6.94* 6.48* 6.18* 5.95* 5.65 5.51 630 4.87* 5.31 5.53 5.74 5.6 1 5.60 5.68 800 5.38 5.42 5.56 5.68 5.72 5.77 5.57 1000 5.89 5.87 5.64 5.69 5.47 5.75 5.4 1 SOURCE AT ( 4 . 5 0 , 3.00, 2 . 3 0 ) - - -- - - - - - SOURCE AT ( 6 . 0 0 , 4.20, 3 . 5 0 ) 160 7.64* 7.18* 6.57* 6.18* 5.84 5.69 5.69 200 6.22* 5.95* 6.07* 6.04* 6.09* 6.12*1
5.99* 2 50 7.64* 7.15* 6.6 l* 6.33* 5.861
5.62 5.43 3 15 4.64* 4.93* 5.46]
6.01* 6.27* 6.04* 5.85 400 7.37* 6.85*r
6.27* 5.99* 5.96* 5.76 5.92* 500 7.82*1
7.34* 6.67* 6.41* 6.11* 6. OO* 6.00* 630 5.26 5.52 5.53 5.62 5.61 5.74 5.69 800 5.76 5.58 5.89 5.921 5.77 5.49 5.42 1000 6.07* 6.07* 5.66 5.56 5.53 5.44 5.47TABLE
IX.
Predicted standard deviations for d i f f e r e n t values of the r a t i o of the components' mean-square values(RM)
and t h e i r correlation c o e f f i c i e n t(CC).
...
.:.>
.
...., ( A ) f o r normalized mean-square pressure
... ...
(B)
f o r sound pressure l e v e l 0.00 5.57 5.67 5.81 0.10 5.58 5.68 5.82 0.20 5.60 5.70 5.85 0.30 5.65 5.75 5.89 0.40 5.72 5.82 5.96 0.50 5.81 5.91 6.05 0.60 5.94 6.04 6.18 0.70 6.13 6.23 6.36 0.80 6.41 6.51 6 . 6 4 0.90 6.90 6.99 7.11 0.98 7.91 7.97 8.0622
TABLE X.
Comparison of s t a n d a r d d e v i a t i o n s computed w i t h (*) and w i t h o u t c o r r e c t i o n f o rthe
non-zero mean v a l u e s of t h e components-
630
Hz.
BETA Source p o s i t i o n 0.0010 0.0015 0.0025 0.0035 0.0050 0.0070 0.0100 (0.00,0.00,0.00) 1.04 1.03 1.03 1.03 1.04 1.05 1.06 1.04* 1.03* 1.02* 1.03* 1.04* 1 . 0 1.06* (2.34,1.57,1.67) 0.88 1.02 1.10 1.09 1.07 1.06 1.04 1.12* 1.19* 1.18* 1.12* 1.07* 1.05* 1.02*....
-
*. . . -...--.-,.--- . . .~.~~....,....-.--.--..-..- . . . . . .-. . . - .2
3
APPENDIX A
Listing of Program used in the Computer Experiment
IBM FORTRAN IV PROGRAM FOR COMPUTER SIMULATION OF REVERBERATION
SOUND FIELD
-
PURE-TONE EXCITATION
SAMPLING STATISTICS CALCULATION
EXCLUDING POINTS ON BOUNDARIES HALF LAMDA AWAY
11 BETA CASES, TRULY RANDOMLY SPACED POINTS
USE IBM SUBROUTINE TO GENERATE RANDOM NUMBERS
USEFUL DATA
ARE
STORED ON FILE DATA.SAMOOO$M
NL=LOWER MODE NO.
;NU-UPPER MODE NO.
NBI=INITIAL BETA NO. AND NBF=FINAL BETA NO.
XR,YR,ZR
=ROOM DIMENSIONS
XS ,YS,ZS
=SOURCE LOCATION
NX,NY,NZ
=NO. OF GRID LINES IN THE THIZEE CO-ORDINATES
INTEGER*4 IY/23
1/
REAL*4 YFL
DIMENSION
XF(3000),P1(3000),P2(3000)DIMENSION DISDIF(3000) ,PlU(3000),P2U(3000)
DIMENSION
PS(3000),TC(60),PD1(60),PD2(60),TPD(60)DIMENSION TS(60),TD(60),APD1(6O),TAPD1(60),APD2(60),TAPD2(60)
DIMENSION CS1(100),CS2(100),CS3(100),SPL(3000)
DIMENSION BETA(11), CF1(3000),CF12(3000)
DIMENSION CF2(100,3000),CF3(100,3000)
DIMENSION FKl(100) ,FK2(100),FK3(100)
DOUBLE PRECISION Pl,P2,P3
DATA NL,NU,NBI,NBF/0,40,3,9/,XR,YR,ZR/8.0,6.5,4.9/
DATA FREQ/505.0/
DATA XS,YS,ZS/O.O,O.O,O.O/
DATA NX,NY,NZ/15,12,9/
DATA
BETA/0.025,0.015,0.01,0.007,0.005,0.0035,- 10.0025,0.0015,0.0U10,0.0005,0.0001/FORMAT
( 'FREQUENCY=',F7.1,10X,' SOURCE AT ',F5.2,',',F5.2,',',-
1F5.2,5X,' BETA=',F8.6//)
FORMAT (20X,' MEAN VALUE OF M-S=',E14.6/20X,' SIGMA FOR M-S=',-
lF7.3/20X,' SIGMA FOR SPL(DB)z1,F7.3)
FORMAT (20X,' X-CORR. COEFF. BETWEEN COMPONENTS=',F7.3/-
120X,
'
VARIANCE FROM NORMAL OF 1 ST COMP.=',E14.6/-
220X,' VARIANCE FROM NORMAL OF 2 ND COMP.=',E14.6/-
320X,' VARIANCE FROM THEORY FOR APD(M-S)s1,E14.6/-
420X,' VARIANCE FROM THEORY FOR APD(SPL)=',E14.6//)
FORMAT (7X,3HM-S ,5X,
10H APD(M-S) ,1OH TAPD(M-S) ,3X,-
13HSPL,7X,lOH APD(SPL),lOH
TAPD(SPL),4X,3HMAG,6X,lOH AP~l(c-l),-
210H TAPD(COM),lOH APD2(C-2)//)
FORMAT
(3X,3F10.4,3X,3F10.4,3X,4F10.4)
FORMAT
( / / / I
FORMAT(20X,' LAMDA*S/V= ',F6.3/20X,' BOUNDARY COR FACTOR= ',F6.3)
FORMAT(20X,' TOTAL NUMBER OF MODES sUMMED;',EL~.~)
FORMAT(20X,' COMPONENTS MEANS
&RATIOS TO M-S z1,4E14.6)
FORMAT(20X,' COMPONENTS MEAN-SQUARES
&~~~1@',3E14.6)
FORMAT(' ROOM
DIMENSIONS=',F5.2,',',F5.2,',',F5.2)
FORMAT(20X,' NO.OF SAMPLED POINTS=',I4)
FORMAT(' SECOND CASE
:NON ZERO MEAN OF COMPONENTS
AKE-
2
4
14
F O R M A T ( ~ F ~ ~ . ~ )
15
FORMAT(7X,' FREQ=',F7.1,10X,' SOURCE AT ',F4.1,',',F4.1,',',-
lF4*1///)
16
FORMAT(7X,4HBATA,2X,12HRATIO OF M-S,~X,~~HX-CORR.
COEFF,2X,lOH-
lSIGMA(M-S) ,~X,~OHSIGMA(SPL)
,5X,llHvA~-TH(M-S)
,3X,-
~~~HVAR-TH(SPL)//)
17
FORMAT(3X,F8.5,F14.3,3Fl2.3,4X,ZE15.6)
18
FORMAT(8E15.6)
19
FORMAT(I4,E12.6,14,E15.6)
PI~3.1415926
AREA=2.*(XR*YR+YR*ZR+ZR*XR)
VOLU=XR*YR*ZR
SPEED OF SOUND
=343 M/S
FK=2
.*PI*FREQ/343.
SLOV=(343./FREQ)*AREA/VOLU
WAVLEN=34
3.
/FREQGENERATE CHARACTERISTIC FUNCTION FOK SOURCE
DO
80
L7=NL,NU
A=L7 *P I
L=L7+1
FKl(L)=A/XR
FK2(L)=A/YR
FK3(L)=A/ZR
CS~(L)=COS(FK~(L)*XS)
CS~(L)=COS(FKZ(L)*YS)
80
CS~(L)=COS(FK~(L)*ZS)
C
GENERATE RANDOM SAMPLING POINTS
LAMDA/2 FROM BOUNDARIES
N4=0
DO 20 L1-l,NX
DO 24 Ml=l,NY
DO 28 N1-l,NZ
CALL UDRAN(1Y ,YFL)
RNl=YFL
RNlT=O
.4*COS(PI*KNl)
IF(L1.EQ.l)GOTO
22
IF(L~.EQ.NX)GOTO
21
XFl=(WAVLEN/Z, )+(XR-WAVLEN)*(L~-1 .+RN~T)/
(NX-1)
GOT023
21
XF~=XR-(WAVLEN/~.)*(~.O+O.~*COS(RN~))
GOT0 23
22
XF~=(WAVLEN/~.)*(~.O+O.~*COS(RN~))
23
CALL UDRAN(IY,YFL)
RN2=YFL
RN~T=O.~*COS(PI*O.~*(RN~+RN~))
IF(M1 .EQ. 1
)GOTO 26
IF(M1.EQ.NY)GOTO
25
YF~=(wAVLEN/~.)+(YR-wAVLEN)*(M~-~.+RN~T)/(~-~)
GOT0 27
25YF~=YR-(WAVLEN/~.)*(~.O+O.~*COS(RNZ))
GOT0 27
26
YFl=(WAVLEN/2.)*(1.0+0.4*COS(RNZ))
27
CALL UDRAN(1Y ,YFL)
RN3=YFL
. - - - - .. . . . . . . . . - . . .
2
5
IF(N1 .EQ. 1
)GOTO 30
IF(N1 .EQ.NZ)GOTO 29
EN/2.)+(ZR-wAVLEN)*(N~-~.O+RN~T)/(NZ-1)
29ZFl=ZR-(WAVLEN/~.)*(~.O+O.~*COS(RN~))
EN/~.)*(~.O+O.~*COS(RN~))
R1 S=(XFl-XS)**2
R2S=(YFl-YS)**2
R3S=(ZFl-ZS)**2
DISDIF(N4)=SQRT(RlS+R2S+R3S)
XF(N4)=XF1
DO 40 I=NL,NU
I7=1+1
CF2(17,N4)=COS(FK2(17)*YFl)
CF3(17,N4)=CoS(FK3(17)*ZE'l)
40
CONTINUE
28
CONTINUE
24
CONTINUE
20CONTINUE
DO 500 KB=NBI,NBF
BATA-BETA
(KB)
P3-0.0
SMODE-0
.
DO 70
1-1,N4
Pl(I)=O.O
70
P2(1)=0.0
DO 280 L=NL,NU
E112.0
IF (L.EQ.O)
~ 1 ~ 1 . 0
L7
=L+1
Bl=El
/XR
FKll=FKl(L7)
FKlS=FKll*FKll
CSll=CSl
(L7)
DO 56 L1=1,N4
56
CFl(Ll)=COS(XF(Ll)*FKll)
DO 270 M=NL,NU
E2~2.0
IF (M.EQ.O)E2=1.0
M7
=M+1
B12=Bl+E2/YR
E12=El*E2
FK22=FK2
(M7
)FK2
S=FK2
2
*FK2
2
FK12SxFKl
S+FK2
S
CS12=CSl
l*CS2(M7)
DO 66 Ml=l,N4
66
C F ~ ~ ( M ~ ) I C F I ( M ~ ) * C F ~ ( M ~ , M ~ )
DO 260 N=IqL,NU
L4=L+M+N
IF (L4.EQ.O)GOTO
260
E3-2.0
IF (N.EQ.O)E3=1.0
N7=N+1
FK33=FK3(N7)
FK4=SQRT(FK12S+FK33*FK33)
FK~s(FK-FK~)*(FK+FK~)
B123=B12+E3/ZR
E123=E12*E3
SMODE=SMODE+l.
CS123=CS12*CS3(N7
)css=cs123*cs123
B9=2.*FK*BATA*B123
Dg=(B9**2+FK6**2)/El23
P3=P3+CSS*FK*FK/D9
DO 200 N1=1,N4
C123=CFl2(Nl)*CF3(N7,Nl)
C=CS123*C123/D9
Pl(Nl)=Pl(Nl)+C*FK6*FK
P2(Nl)=P2(Nl)+C*Eg*FK
CONTINUE
CONTINUE
CONTINUE
CONTINUE
STORE USEFUL
DATA
ON
FILE
DATA.SAIIOOOSM
WKITE(4,14) XR,YR,ZR,XS
,YS,ZS,FREQ,BATA
WRITE(4,lg)
NU
,SMODE,N4
,P3
WRITE(4,lS) (~1(1),~2(1),I=l,N4)
DATA FOR POINTS
CMIN DISTANCE FROM SOURCE
ARE
NOT TO BE USE0
DMIN=O.5636*SORT(BATA*AREA)
L9=0
DO 300 Ll=l,N4
DIST=DISDIF(Ll)-DMIN
IF(DIST.LE.O.O)GOTO
300
L9=L9+1
PlU(L9)=Pl(Ll)
P2U(L9)=P2(Ll)
CONTINUE
N3
=L9
CASE=
1.0
NCAS=O
COV=O.
PlS=O.
P2S=O.
P3S=O.
PlM=O.
P2M=O.
DO 282 I=l,N3
P~M=P~M+P~U(I)
P2M=P2M+P2U(I)
P ~ M = P ~ M / N ~
P2M=P2M/N3
PlM,P2M ARE
MEAN
VALUES OF
THE
COMPONENTS
THE
NON ZERO VALUE OF THE MEANS ARE SUBTRACTED OUT
IN THE SECOND CASE
PlMl=PlM*( 1 .O-CASE)
P2Ml=P2M*(l.O-CASE)
P1
l=PlU(J)-P1M1
P22=P2U(J)-P2Ml
PlS=PlS+Pl l*Pl1
P2S=P2S+P22*P22
COv=cov+Pl l*P22
PS(J)=Pll*Pll+P22*P22
P3S=P3S+PS(J)
CROSC=COV/SQRT(PlS*P2S)
SM=P3
S
/N3
CORECT=P3/SM
PlS=PlS/N3
P2S=P2S/N3
PSR=PlS/P2S
PlMR=PlM/SQRT(PlS)
P2MR=P2M/SQRT(P2S)
PlS,P2S
ARE
THE MEAN-SQUARE VALUES OF THE COMPONENTS
sv=o.
SPLM=O.
SPLV=O
.
DO 292 J=l,N3
PS(J)=PS(J) /SM
SPL(J)=lO.*ALOGlO(PS(J))
SV=SV+PS(J)**2
SPLM=SPLM+SPL(J)
SPLV=SPLV+SPL(J)**2
CONTINUE
VARMS=SV/N3-1.
VARM=
SQRT
(VARMS
)VARSPL=(SPLV/N3)- (SPLM/N3)**2
VARS=SQRT(VARSPL)
CALCULATION OF CUMULATIVE DENSITY FUNCTION OF COMPONENTS
AND COMPARE WITH STANDARD NORMAL DISTRIBUTION
PlSS,P2SS
ARE
THE STANDARD DEVIATION OF COMPONENTS
IF(CASE.EQ.O.0)GOTO
191
plM2=(PlM*CASE)**2
P2M2=
(P2M*CASE)**2
GOT0 192
P 1M2=0
.O
P2M2=0.0
P~ss=sQRT(P~S-PIM~)
P~SS=SQRT
(P2S-P2M2
)DO 196 I=1,60
TC(I)=I*O.~-3.0
L8
1=0
DO 193 J=l,N3
P~N=(PIu(J)-P~M>/P~SS
IF(P~N.GT.TC(I))GOTO
193
L81=L81+1
CONTINUE
ES=l.*L81
L8250
DO 194 J=l,N3
PZN=(P~U(J>-P~M)
/P2SS
IF(P2N.GT.TC(I))GOTO
194
L82=L82+1
CONTINUE
ES=
1. *L82
PD2(I)=ES/N3
CONTINUE
CALCULATION OF CUMULATIVE DISTRIBUTIONS
OF BOTH THE MEAN-SQUARE PRESSURE AND THE SPL
DO 296 I=1,60
TS(I)=(4.*1)/60.
L8
1=0
DO 293 J=1,N3
IF (PS(J).GT.TS(I))GOTO
293
L8
1
=L8
1+1
CONTINUE
ES=1
.*La1
APD1 (I)=ES/N3
TAPDl IS THE THEORETICAL DISTRIBUTION OF THE MEAN-SQUARE PRESSUKE
TAPD1
(I)=l .-EXP(-TS(1))
TD(I)=(40.*1)/60.-30.
L82=0
DO 294 J=l,N3
IF (SpL(J).GT.TD(I))GOTO
294
L82=L82+1
CONTINUE
E W 1
.*L82
APD2(I)=ED/N3
TAPD~(I)=~.-EXP(-(EXP(O.~~O~~~*TD(I))))
TAF'D2 IS THE THEORETICAL DISTRIBUTION OF THE SPL
CONTINUE
VCl=O.
VC2-0.
Vl=O.
v2=0.
DO
298 I=1,60
VCl=vCl+(TPD(I)-PDl(I))**2
VC2=vC2+
(TPD(1)-PD2 (I) )**2
Vl=Vl+(APDl(I)-TAPDl(I))**2
VZ=V2+(APD2(1)-TAPD2 (I) )**2
VCll-l=VC1/60.
VC2WVC2/60.
VlM=Vl/
60.
V2M=V2/60.
WRITE(3,ll) XR,YR,ZR
WRITE(3,l) FREQ,XS ,YS
,ZS
,BATA
WRITE (3,12) N3
WRITE
( 3 , 8 )SMODE
WRITE (3,9) PlM,P2M,PlMR,P2MR
WRITE (3,lO) PlS,P2S
,PSR
WRITE (3,2) SM,VARM,VARS
WRITE
(3,7
)SLO
V,CORECT
WRITE (3,3) CROSC,VClM,VC2M,VlM,V2M
WRITE (3,4)
WRITE (3,5) (TS(1) ,APDl(I) ,TAPD~(I),TD(I) ,APD2(I) ,TAPD~(I)
,-
1TC(I),PD1(I),TPD(I),PD2(I),I=l,60)
WRITE (3,6)
CASE=O .O
NCAS=NCAS+l
IF
(NCAS
.
EQ .2 )GOTO 4 5
0WRITE (3,13)
GOT0 283
WRITE(3,6)
CONTINUE
STOP
END
. . . .- --. . . . ... - --.- - - - --.---..
30
APPENDIX B
S c h r o e d e r ' s C r i t i c a l Frequency L i m i t Computation
I n g e n e r a l , t h e s t a t i s t i c a l model works a t f r e q u e n c i e s above t h e l i m i t g i v e n by S c h r o e d e r :
*
f c
'
2000 (T60/V) (B1)which c o r r e s p o n d s t o a modal o v e r l a p i n d e x of t h r e e . V i s t h e room volume i n c u b i c m e t r e s and TsO i s i t s r e v e r b e r a t i o n t i m e . A c c o r d i n g t o Morse and I n g a r d , l 3 t h e e n e r g y i n e a c h mode d e c a y s e x p o n e n t i a l l y a f t e r t h e e x c i t a t i o n
i s s w i t c h e d o f f ; i . e .
where bN i s t h e damping f a c t o r of Eq. ( 5 ) and c i s t h e s p e e d of sound. I f i t i s assumed t h a t t h e o b l i q u e modes d o m i n a t e (which i s t r u e a t h i g h
f r e q u e n c i e s ) and t h a t t h e s p e c i f i c normal a d m i t t a n c e ( 8 ) i s i n d e p e n d e n t o f f r e q u e n c y , t h e t o t a l e n e r g y a l s o d e c a y s e x p o n e n t i a l l y : where: 6 = BS/V, S = t o t a l s u r f a c e a r e a of t h e room. From Eq. ( B 3 ) , i t f o l l o w s t h a t :
-
-10 l o g ( p 2 ) = 20 (0.43429) 6 c t which l e a d s t o :Using Eqs. ( B l ) and (B4), t h e S c h r o e d e r l i m i t c a n b e computed f o r d i f f e r e n t v a l u e s o f 8.
. . .. . 3 1
APPENDIX
C D e t a i l e d D e r i v a t i o n of Eqs. ( 9 , 10, 1 1 ) A c c o r d i n g t o E q . ( 1 ) :-
P 2 = (f
A ~ c o s ~ ~ ) ~+
( ~ ~ s i n 9 ~ ) ~ i = l i= 1 2 2 = C 1+
C2 The f o l l o w i n g d e r i v a t i o n s a r e b a s e d on t h e a s s u m p t i o n s t h a t : ( 3 ) C 1 and C 2 a r e G a u s s i a n d i s t r i b u t e d .Assiimptions ( 2 ) and ( 3 ) e n s u r e t h a t C 1 and C 2 a r e i n d e p e n d e n t random v a r i a b l e s . The p r o b a b i l i t y d e n s i t y f u n c t i o n s of C 1 and C 2 a r e :
2 2
I f one d e f i n e s u = C 1 and v = C 2 , t h e p r o b a b i l i t y d e n s i t y f u n c t i o n s of u and v a r e :
S i n c e C 1 a n d C 2 a r e s t a t i s t i c a l l y i n d e p e n d e n t , u and v a r e a l s o i n d e p e n d e n t random v a r i a b l e s . I f one d e f i n e s
o n e c a n w r i t e , u s i n g s t a t i s t i c a l m e t h o d s , 1 4 f o r t h e p r o b a b i l i t y d e n s i t y o f (S1+S2)I :
![TABLE VI. C o r r e l a t i o n c o e f f i c i e n t , <c1c2>/ [<c:><c$>]*, of t h e two components](https://thumb-eu.123doks.com/thumbv2/123doknet/14298976.493724/20.933.67.905.115.1151/table-vi-c-lt-gt-lt-gt-components.webp)
