The influence of wind and
temperature gradients
on
outdoor soundpropagation.
WIND AND TEMPERATURE GRADIENTS
ON OUTDOOR SOUND PROPAGATION
BIBLIOTHEEK T U Delft P 1736 5354
T H E INFLUENCE O F
WIND A N D T E M P E R A T U R E GRADIENTS
ON OUTDOOR SOUND PROPAGATION
Development and evaluation of a calculation method
based on wavefield extrapolation
PROEFSCHRIFT ter verkrijging van
de graad van doctor in de
technische wetenschappen
aan de Technische Hogeschool Delft,
op gezag van rector magnificus,
prof.ir. B.P.Th. Veltman
voor en commissie aangewezen
door het college van decanen
te verdedigen op
dinsdag 3 mei 1983
te 14.00 uur door
BALTHASAR ADRIANUS DE J O N G
natuurkundig ingenieur
geboren te Gouda
Dit p r o e f s c h r i f t is g o e d g e k e u r d d o o r de p r o m o t o r prof.dr.ir. A . J . B e r k h o u t
PREFACE
The s t u d y d e s c r i b e d i n t h i s t h e s i s was made p o s s i b l e by t h e c o o p e r a t i o n o f t h r e e i n s t i t u t e s . The f i n a n c i a l s u p p o r t was g i v e n by t h e department o f N o i s e Abatement o f t h e f o r m e r D u t c h M i n i s t e r y o f P u b l i c H e a l t h and E n v i r o n m e n t a l P r o t e c t i o n (now b e i n g p a r t o f t h e M i n i s t e r y o f H o u s i n g , P h y s i c a l P l a n n i n g and E n v i r o n m e n t ) .
The a c t u a l work was c a r r i e d o u t a t t h e group o f a c o u s t i c s o f t h e d e p a r t m e n t o f A p p l i e d P h y s i c s , D e l f t U n i v e r s i t y o f T e c h n o l o g y . D u r i n g t h e s t u d y c l o s e c o n t a c t was k e p t w i t h t h e I n s t i t u e o f A p p l i e d P h y s i c s (TPD-TNO/TH), p a r t i c u l a r l y w i t h r e s p e c t t o t h e p r a c t i c a l a s p e c t s o f t h e p r o b l e m .
Thanks a r e due t o many p e o p l e w o r k i n g a t t h e above m e n t i o n e d i n s t i t u t e s , p e o p l e who have h e l p e d me w i t h good s u g g e s t i o n s and w i t h c l a r i f y i n g d i s c u s s i o n s . I am g r e a t l y i n d e b t e d t o P r o f . i r . D.W. v a n W u l f f t e n P a l t h e . H i s t h e o r e t i c a l knowledge was i n d i s p e n s a b l e i n f i n d i n g t h e c o r r e c t s o l u t i o n t o t h e p r o b l e m . H i s d e a t h i n A u g u s t 1981 meant a g r e a t l o s s t o me. O n l y a f t e r h i s d e a t h t h e f u l l i m p o r t a n c e o f t h i s s u g g e s t i o n s c o n c e r n i n g t h e t h e o r e t i c a l p a r t s o f t h i s s t u d y became o b v i o u s t o me.
F i n a l l y , I would l i k e t o thank P e t e r K o e r s , who has made a p a r t o f t h e p l o t s , P e t e r Mesdag, who h e l p e d me w i t h my E n g l i s h and G e r d a Boone, who has done t h e p e r f e c t t y p e w r i t i n g .
VI
CONTENTS
SUMMARY IX
1 INTRODUCTION 1 1,1 THE PURPOSE OF THIS WORK 1
1 .2 THE OUTLINE OF THIS THESIS 2 1.3 THE MODELLING OF SOUND PROPAGATION OUTDOORS 2
1.3.1 The d i s t a n c e a t t e n u a t i o n 3 1.3.2 A homogeneous ground and a homogeneous atmosphere 4
1.3.3 D e c o m p o s i t i o n by a i d o f t h e d i s c r e t e F o u r i e r t r a n s f o r m . . 6
1.3.4 I n t r o d u c t i o n o f an inhomogeneous ground s u r f a c e 8
1.3.5 The m e t e o r o l o g i c a l f a c t o r s 9 1.3.6 The ground e f f e c t w i t h w i n d and t e m p e r a t u r e g r a d i e n t s . . . 13
1.4 A BRIEF DESCRIPTION OF THE EXTRAPOLATION MODEL 14
2 THE WAVE EQUATION IN AN INHOMOGENEOUS MOVING MEDIUM 19
2.1 THE BASIC EQUATIONS 19 2.2 THE HELMHOLTZ EQUATION FOR A LAYERED STEADY WIND PROFILE . . 20
3 THE EXTRAPOLATION METHOD 25 3.1 INTRODUCTION TO FINITE DIFFERENCE TECHNIQUES 25
3.2 DERIVATIVES I N TERMS OF SPATIAL CONVOLUTION 30 3.3 EXTRAPOLATION I N THE SPATIAL FREQUENCY DOMAIN 32
3.4 FLOATING TIME REFERENCE 37 3.5 EXTRAPOLATION I N SPACE DOMAIN 41
4 EXTRAPOLATION METHOD IN THE Z-DIRECTION 47
4.1 INTRODUCTION 47 4.2 A UNIFORM WIND AND TEMPERATURE PROFILE 49
4.3 THE W.K.B.J.-SOLUTION 50 4.4 EXTRAPOLATION THROUGH A STRATIFIED MEDIUM 55
5 THE COMPUTATION MODEL 59
5.1 INTRODUCTION 59 5.2 CALCULATION OF THE SOUND FIELD ABOVE A HARD GROUND 60
5.3 INTRODUCTION OF AN ABSORBING GROUND 65 5.4 THE COMPUTATION OF THE W.K.B.J. METHOD 71
6 RESULTS AND DISCUSSION 73
6.1 INTRODUCTION 73 6.2 RESULTS AS ISOBAR PLOTS 75
6.3 SOUND LEVELS AS A FUNCTION OF FREQUENCY 82
6.3.1 G e n e r a l remarks 82 6.3.2 S e n s i t i v i t y t o t h e wind p r o f i l e 84 6.3.3 R e s u l t s w i t h a h a r d ground 89 6.3.4 R e s u l t s w i t h an a b s o r b i n g ground 89 6.3.5 A c o m p a r i s o n w i t h measurements 92 6.4 FINAL CONCLUSIONS 98 6.4.1 G e n e r a l remarks on t h e w i n d e f f e c t 98 6.4.2 E v a l u a t i o n o f t h e e x t r a p o l a t i o n model 98 APPENDIX 101 REFERENCES 105 SAMENVATTING 107
SUMMARY
I n t h i s t h e s i s w a v e f i e l d e x t r a p o l a t i o n t e c h n i q u e s a r e a p p l i e d t o t h e p r o b l e m o f o u t d o o r sound p r o p a g a t i o n . P a r t i c u l a r l y , t h e i n f l u e n c e o f w i n d and t e m p e r a t u r e g r a d i e n t s on t h e sound p r o p a g a t i o n a r e c o n s i d e r e d . An e x t r a p o l a t i o n model i s p r o p o s e d t o c o n s t r u c t t h e wave f i e l d o f a sound s o u r c e i n a moving, inhomogeneous medium and above an a b s o r b i n g ground s u r f a c e .
I n t h e d e r i v a t i o n o f t h e model some a p p r o x i m a t i o n s have been made. The inhomogeneous p a r a m e t e r s o f t h e medium (wind and t e m p e r a t u r e ) a r e c o n s i d e r e d t o be i n d e p e n d e n t o f t i m e , o r i n o t h e r w o r d s , t u r b u l e n c e i s n o t t a k e n i n t o a c c o u n t . S e c o n d l y we c o n f i n e o u r s e l v e s t o s i t u a t i o n s where w i n d and t e m p e r a t u r e depend on one c o o r d i n a t e o n l y ( s t r a t i f i e d medium).
I n c a s e s w i t h o u t w i n d and t e m p e r a t u r e g r a d i e n t s ( a homogeneous medium), t h e new model has been e v a l u a t e d w i t h w e l l e s t a b l i s h e d a n a l y t i c a l s o l u t i o n s . The i n f l u e n c e o f w i n d on t h e sound p r o p a g a t i o n has been c a l c u l a t e d f o r a number o f i n t e r e s t i n g s i t u a t i o n s . The r e s u l t s a r e compared w i t h some o f t h e few measurements, w h i c h c o u l d be found i n l i t e r a t u r e . I n s p i t e o f t h e e f f e c t o f t u r b u l e n c e s , w h i c h were l i k e l y t o be p r e s e n t d u r i n g t h e measurements, a good q u a l i t a t i v e agreement has been f o u n d . Some e x t e n s i o n s o f t h e new model a r e shown, such as t h e i n t r o d u c t i o n o f an inhomogeneous ground s u r f a c e ( h a r d - a b s o r b i n g t r a n s i t i o n ) and t h e e f f e c t o f a n o i s e s c r e e n . B o t h s i t u a t i o n s a r e c o n s i d e r e d w i t h and w i t h o u t a w i n d g r a d i e n t .
CHAPTER 1
INTRODUCTION
1.1 THE PURPOSE OF THIS WORK
I n a r e a s w i t h a h i g h p o p u l a t i o n d e n s i t y , n o i s e c o n t r o l i s an i m p o r t a n t p a r t o f e n v i r o n m e n t a l p r o t e c t i o n . I n 1981 p a r t s o f t h e D u t c h N o i s e Abatement A c t became e f f e c t i v e . To a b a t e n o i s e p o l l u t i o n t h i s l a w s e t s l i m i t s f o r " l o n g term L - l e v e l s " and p r e s c r i b e s s t a n d a r d methods f o r measurements and
eq c a l c u l a t i o n s . W i t h t h e i n t r o d u c t i o n o f t h i s l a w i t became n e c e s s a r y t o d e s i g n c a l c u l a t i o n methods, t o p r e d i c t n o i s e l e v e l s f o r f u t u r e s i t u a t i o n s and t o p r e d i c t t h e e f f e c t i v i t y o f n o i s e c o n t r o l measures i n e x i s t i n g s i t u a t i o n s . I n t h e c a l c u l a t i o n schemes t o p r e d i c t o u t d o o r sound p r o p a g a t i o n t h e i n f l u e n c e o f m e t e o r o l o g i c a l c o n d i t i o n s ( e . g . w i n d and t e m p e r a t u r e g r a d i e n t s ) i s v e r y i m p o r t a n t . I n most common c a l c u l a t i o n methods r a y - t r a c i n g t e c h n i q u e s a r e u s e d . These t e c h n i q u e s however, l e a d t o e r r o r s f o r g r e a t e r s o u r c e - r e c e i v e r d i s t a n c e s [ 1 ] . A l s o t h e c a l c u l a t i o n o f t h e i n f l u e n c e o f an a b s o r b i n g ground g i v e r i s e t o d i f f i c u l t i e s i n c o m b i n a t i o n w i t h r a y - t r a c i n g t e c h n i q u e s . These d i f f i c u l t i e s have n o t y e t been s o l v e d i n a p h y s i c a l l y p r o p e r way. O t h e r p r e d i c t i o n schemes c a l c u l a t e t h e i n f l u e n c e o f w i n d and t e m p e r a t u r e by a i d o f r u l e s d e r i v e d from measurements. The s t a n d a r d method, as p r o p o s e d i n t h e D u t c h a c t , makes u s e o f a c o m b i n a t i o n o f b o t h r a y - t r a c i n g t e c h n i q u e s and r u l e s f r o m measurements.
The work, d e s c r i b e d i n t h i s t h e s i s , was s u p p o r t e d by a g r a n t from t h e f o r m e r D u t c h M i n i s t e r y o f P u b l i c H e a l t h and E n v i r o n m e n t a l P r o t e c t i o n . The a i m was t o
2
d e v e l o p a p h y s i c a l l y a c c e p t a b l e method f o r c a l c u l a t i o n of the i n f l u e n c e o f w i n d and t e m p e r a t u r e g r a d i e n t s i n t h e p r e s e n c e o f an a b s o r b i n g g r o u n d . From
t h e r e s u l t s c o n c l u s i o n s c a n be drawn c o n c e r n i n g the a c c u r a c y o f e x i s t i n g s t a n d a r d methods. I n p a r t s the e x i s t i n g methods can be extended o r c o r r e c t e d , but t h a t l i e s beyond the scope o f t h i s s t u d y .
1.2 THE OUTLINE OF THIS THESIS
A f t e r the i n t r o d u c t i o n g i v e n i n t h i s c h a p t e r , the wave e q u a t i o n w h i c h i n c l u d e s w i n d and t e m p e r a t u r e e f f e c t s i s d e r i v e d i n c h a p t e r 2 . The t h e o r e t i c a l a s p e c t s , w h i c h form the b a s i s o f the e x t r a p o l a t i o n model, a r e d i s c u s s e d i n the c h a p t e r s 3 and 4. C h a p t e r 5 g i v e s the c o m p u t a t i o n a l a s p e c t s o f the model. I n c h a p t e r 6 t h e r e s u l t s a r e g i v e n and d i s c u s s e d , f o l l o w e d by the f i n a l c o n c l u s i o n s .
The e x t r a p o l a t i o n model i s t h e o r e t i c a l l y based on g e o p h y s i c a l e x t r a p o l a t i o n t e c h n i q u e s and on m a t h e m a t i c a l t e c h n i q u e s such as s p a t i a l F o u r i e r t r a n s f o r m s and t h e W.K.B.J.-method. F o r p e o p l e w o r k i n g on the p r a c t i c a l a s p e c t s o f n o i s e abatement, t h i s t h e o r e t i c a l p a r t may be hard t o d i g e s t . To h e l p the r e a d e r , who i s m a i n l y i n t e r e s t e d i n the r e s u l t s , a b r i e f d e s c r i p t i o n o f the model i s g i v e n i n t h i s c h a p t e r . T h i s knowledge w i l l be s u f f i c i e n t t o i n t e r p r e t the r e s u l t s i n c h a p t e r 6.
1.3 THE MODELLING OF SOUND PROPAGATION OUTDOORS
To p r e d i c t the p r o p a g a t i o n o f t r a f f i c and i n d u s t r i a l n o i s e i t i s n e c e s s a r y t o s i m p l i f y complex r e a l i s t i c s i t u a t i o n s i n t o b a s i c s i t u a t i o n s w h i c h c a n be m o d e l l e d . T h i s s i m p l i f i c a t i o n must be done i n such a way t h a t sound l e v e l s c a n be c a l c u l a t e d w i t h s u f f i c i e n t a c c u r a c y . I n t h i s t h e s i s a m o d e l l e d s i t u a t i o n f o r sound p r o p a g a t i o n o u t d o o r s , s c h e m a t i c a l l y shown i n f i g u r e 1, w i l l be used a s the b a s i c s i t u a t i o n i n w h i c h p r e d i c t i o n o f sound l e v e l s i s r e q u i r e d .
WIND P R O F I L E
S O U R C E
z GROUND S U R F A C E t
Here we w i l l assume one p o i n t s o u r c e above a ground s u r f a c e . E x t e n d e d s o u r c e s c a n a l w a y s be b u i l t up o f a number o f p o i n t s o u r c e s . The ground s u r f a c e i s n o t n e c e s s a r i l y homogeneous, but w i l l be assumed t o c o n s i s t o f a few homogeneous p a r t s . The open a i r i s a p p r o x i m a t e d by a medium, where wind and t e m p e r a t u r e o n l y depend on the a l t i t u d e ( a s t r a t i f i e d m o d e l ) . The sound l e v e l a t t h e microphone p o s i t i o n depends on t h e d i s t a n c e between s o u r c e and r e c e i v e r , t h e p r o p e r t i e s o f t h e ground s u r f a c e and t h e m e t e o r o l o g i c a l c o n d i t i o n s . The c a l c u l a t i o n model i n t r o d u c e d i n t h i s t h e s i s i s d e v e l o p e d t o d e r i v e a good p r e d i c t i o n o f t h e i n f l u e n c e w h i c h w i n d and t e m p e r a t u r e g r a d i e n t s have on t h e p r o p a g a t i o n o f sound. However, t h e i n t e r a c t i o n o f t h e a t t e n u a t i o n f a c t o r s makes i t n e c e s s a r y t o s t a r t w i t h a b r i e f i n t r o d u c t i o n o f t h e e n t i r e p r o p a g a t i o n problem.
1.3.1 The distance attenuation
The sound p r e s s u r e i s a t e m p o r a l l y harmbnic v a r i a b l e , w h i c h can be w r i t t e n as
p ( x , y , z , t ) = R e { | p | e J( U t +^ , ( 1 . 1 ) where a) = t h e c i r c u l a r f r e q u e n c y . I n t h e f o l l o w i n g we w i l l use t h e complex a m p l i t u d e n o t a t i o n , i n w h i c h t h e t i m e dependent term e x p ( j t o t ) i s o m i t t e d P ( x , y , z , o j ) = | P | eJ* . ( 1 . 2 )
Note t h a t t h i s n o t a t i o n i s i n t h e f r e q u e n c y domain and t h a t w i t h t h e o m i s s i o n o f exp(+jo)t) we have c h o s e n f o r t h e t e c h n i c a l n o t a t i o n where j = \ / - l .
I n a s t a t i o n a r y homogeneous medium t h e sound p r o p a g a t i o n has t o s a t i s f y t h e wave e q u a t i o n + f-L + k2P = 0, (1.3) 32P A 92P j . 92P A v2 where and 9 x2 9 y2 9 z2
k = o)/c = the wave number
c = t h e sound v e l o c i t y i n t h e medium.
W i t h a p o i n t s o u r c e i n ( x ,y ,z ) t h i s e q u a t i o n l e a d s t o t h e s o l u t i o n
r o o o
4 P(x,y,z,oj) = (1.4) w i t h r2 = ( x - xo)2 + ( y - yQ)2 + ( z - % )2. and = t h e p r e s s u r e a m p l i t u d e f o r k r = l . T h i s r e s u l t i s known a s t h e 1 / r - l a w .
1.3.2 A homogeneous ground and a homogeneous atmosphere
I n o r d e r t o s t u d y o u t d o o r sound p r o p a g a t i o n t h e ground e f f e c t must be i n t r o d u c e d . The l a s t t e n y e a r s a l o t o f r e s e a r c h has been done on t h i s s u b j e c t . The a n a l y t i c a l s o l u t i o n f o r a homogeneous g r o u n d , w h i c h i s used v e r y o f t e n , was i n t r o d u c e d by I n g a r d i n 1956 [ 2 ] . H i s d e r i v a t i o n s t a r t s w i t h t h e w e l l known e q u a t i o n f o r t h e p l a n e wave r e f l e c t i o n c o e f f i c i e n t , d e f i n e d as Pr( 9 j . ) = Rp( 61) P1( Q1) , ( 1 * 5 ) where P, = t h e i n c i d e n t p l a n e wave, P = t h e r e f l e c t e d p l a n e wave, r and R ( 8 , ) = t h e p l a n e wave r e f l e c t i o n c o e f f i c i e n t . P 1 I f we assume t h e ground s u r f a c e a t z=0, t h e n f o r z>0 we w i l l d e f i n e the a n g l e o f t h e i n c i d e n t p l a n e wave w i t h t h e z - a x i s , the d e n s i t y o f medium 1,
the sound v e l o c i t y i n medium 1.
F o r z<0 t h e s e v a r i a b l e s a r e d e f i n e d by
the a n g l e o f t h e t r a n s m i t t e d p l a n e wave w i t h t h e z - a x i s , the d e n s i t y o f medium 2,
a complex number, w h i c h c o n t a i n s t h e sound speed and t h e i n e l a s t i c a b s o r p t i o n i n medium 2.
W i t h t h i s d e f i n i t i o n the p l a n e wave r e f l e c t i o n c o e f f i c i e n t can be r e w r i t t e n as
p c cos6 -p c cos6
R = — J L ' ' *2. . ( 1 . 6 )
F o r most p r a c t i c a l ground s u r f a c e s the p r o p a g a t i o n i n the x and the y d i r e c t i o n can be n e g l e c t e d f o r z<0. Or i n o t h e r words t h e ground s u r f a c e i s c a l l e d " l o c a l l y r e a c t i n g " f o r cos92 1» ( 1 - 7 ) W i t h t h i s c o n d i t i o n the p l a n e wave r e f l e c t i o n c o e f f i c i e n t i s g i v e n by c o s e -6 w i t h g = p^c-^/p2C2 • t h e n o r m a l i z e d ground a d m i t t a n c e . To c a l c u l a t e the r e f l e c t i o n o f s p h e r i c a l waves t h e i n c i d e n t f i e l d i s decomposed i n t o i t s p l a v e wave components. A f t e r a p p l y i n g e q u a t i o n ( 1 . 5 ) , t h e r e s u l t has t o be i n t e g r a t e d t o d e r i v e the r e f l e c t e d f i e l d . I n g a r d has shown t h a t the sound p r e s s u r e can be w r i t t e n i n the f o r m ( u s i n g h i s n o t a t i o n i =-j )
i k r ^ i k r ^ P = P ^ — + Q ( 6 , r2, el) ^ — , (1.9) rl 2 w i t h where and Q = R + {1 - R } F ( B , r2, 91) , (1.10) r^ = t h e d i s t a n c e f r o m s o u r c e t o r e c e i v e r , r2 = t h e d i s t a n c e f r o m " m i r r o r - s o u r c e " t o r e c e i v e r F = a f u n c t i o n w h i c h i n t e g r a t e s o v e r a l l a n g l e s o f i n c i d e n c e .
I n e q u a t i o n (1.9) the f i r s t p a r t i s t h e i n c i d e n t sound f i e l d and t h e second p a r t i s the r e f l e c t e d sound f i e l d . The s o l u t i o n i s w r i t t e n i n s u c h a way t h a t t h e second p a r t d e f i n e s a m i r r o r s o u r c e , i n w h i c h r ^ e q u a l s t h e d i s t a n c e f r o m " s o u r c e " t o r e c e i v e r . Q can be seen as a complex s p h e r i c a l r e f l e c t i o n
6
c o e f f i c i e n t . F may be d e r i v e d by i n t e g r a t i o n o v e r a l l a n g l e s 9^ o f t h e i n c i d e n t p l a n e waves. I n g a r d makes use o f a " W e i l - t r a n s f o r m a t i o n " w h i c h i s l a t e r c o r r e c t e d by Thomasson [ 3 ] . More r e c e n t l y A t t e n b o r o u g h [4] has r e w r i t t e n t h e s o l u t i o n i n a c o m p u t a t i o n a l l y more e f f i c i e n t f o r m r - a2 F ( a ) = 1 + ii/ïïa e e r f c ( - i a ) , (1.11) where a2 = i k r2{ l + BcosOj^ - -J\ - B2 s i n B ^ , Re(a)>0, (1-12) and e r f c i s the e r r o r f u n c t i o n g i v e n by 2 C - t2 e r f c ( z ) = — — J e d t . vAt J z
F o r a homogeneous ground s u r f a c e and a homogeneous atmosphere, the e x t r a -p o l a t i o n model, -p r e s e n t e d i n t h i s t h e s i s , w i l l be checked w i t h above g i v e n s o l u t i o n .
1.3.3 Decomposition by a i d of the d i s c r e t e Fourier transform
I n the e x t r a p o l a t i o n model w h i c h i s i n t r o d u c e d i n t h i s t h e s i s we use the same a p p r o a c h as i n t r o d u c e d by I n g a r d . O n l y now we decompose the i n c i d e n t sound f i e l d i n t o i t s p l a n e wave components by a i d of the d i s c r e t e F o u r i e r t r a n s f o r m ( s e e s e c t i o n 3.2 and the a p p e n d i x ) . A d o u b l e F o u r i e r t r a n s f o r m o f x and y i s d e f i n e d as - j ( k nAx+k mAy) P ( kx, k ,z,u)) = X 2 > ( x , y , z , w ) e y (1.13) n m w h i c h w i l l be w r i t t e n as P(x,y,z,a)) „ P ^ . k y . z . u i ) , (1.14)
where the s i g n » d e n o t e s a F o u r i e r t r a n s f o r m a t i o n . T h i s t r a n s f o r m decomposes t h e sound f i e l d i n t o p l a n e wave components t r a v e l l i n g t h r o u g h the p l a n e z=0. The complex numbers o f t h e kx, k ^ - p l a n e d e f i n e phase and a m p l i t u d e o f a p l a n e
wave component. The d i r e c t i o n o f t h e p l a n e waves a r e r e l a t e d t o the v e c t o r s o f t h e s e complex numbers. The ground a b s o r p t i o n i s now d e f i n e d by t h e wave e q u a t i o n f o r the homogeneous c a s e and the boundary c o n d i t i o n s on t h e s u r f a c e , where P and v a r e c o n t i n u o u s . W i t h g i v e n d e n s i t i e s and wave numbers o f t h e
— z
f i e l d Pr and a t r a n s m i t t e d f i e l d Pf c » t h e s e boundary c o n d i t i o n s l e a d t o t h e r e l a t i o n s P 1+P r = P tr (1.15) and , D P . , 3P , 3P P j 3z p 3z P2 3z U s i n g e q u a t i o n ( 1 . 3 ) t h e d e r i v a t i v e o f P w i t h r e s p e c t t o z c a n be e x p r e s s e d as a f u n c t i o n o f t h e d e r i v a t i v e s o f P w i t h r e s p e c t t o x and y. S y m b o l i c a l l y , dz * 1 3 x2 3 y2 A f t e r F o u r i e r t r a n s f o r m a t i o n o f x t o k and o f y t o k , and s u b s t i t u t i o n o f x y' p r o p e r t i e s ( s e e s e c t i o n 3.2) « - k2r and l l F - o - l ^ P , ( 1 . 1 8 ) 3 x2 X 3 y2 Y we f i n d f o r z=0 Pt + Pr = Pt r ( 1 - 1 9 ) and V k2- k2- k2 A r k 2 -k 2 A ^ - k ' - k2 - — y 1 1 y ? . - 2 " y ^r. d . 2 0 ) P , 1 P , r p2 E l i m i n a t i o n o f P from e q u a t i o n s (1.19) and (1.20) g i v e s Pr = Rp Pj_ f o r z=0, (1.21) with P, V k2- k2- k2 - PV k2- k2- k2 Rp = 2 ; ' X 7 ' , 2 X 7 . (1.22) p„ V k2- k2- k2 + p , V k2- k2- k2 2 1 x y 1 2 x y E v e r y p l a n e wave, w h i c h i s i n c i d e n t on t h e p l a n e z=0 ( a t an a n g l e 6, t o z - a x i s ) , i s d e f i n e d by a s e t o f kx, k ^ - v a l u e s . The a n g l e o f i n c i d e n c e ( f o r z>0) and t h e a n g l e o f t r a n s m i s s i o n ( f o r z<0) a r e g i v e n by t h e V k2- k2- k2 V k2- k2- k2 c o s e , = x y and c o s e , = x y ( 1 - 2 3 ) kl k2
8
w i t h
k^ = o)/c^ and k2 = u>/c2. (1-24)
S u b s t i t u t i o n o f r e l a t i o n s (1.23) and (1.24) i n t o e q u a t i o n (1.22) shows t h a t t h e f o r m u l a t i o n s i n e q u a t i o n s (1.21) and (1.22) a r e e q u i v a l e n t t o t h a t i n e q u a t i o n s (1.5) and ( 1 . 6 ) . F o r a l o c a l l y r e a c t i n g ground s u r f a c e the u n e q u a l i t y kx+ky K < k2 (1,25) h o l d s . F o r t h i s c a s e the p l a n e wave r e f l e c t i o n c o e f f i c i e n t i s g i v e n by e q u a t i o n ( 1 . 8 ) .
1.3.4 Introduction of an inhomogeneous ground surface
The e x t r a p o l a t i o n model can e a s i l y be e x t e n d e d f o r an inhomogeneous g r o u n d , p r o v i d e d t h a t t h i s ground can be d e v i d e d i n t o p a r t s ( x ,x . ) , w i t h a
n n+1
c o n s t a n t a d m i t t a n c e Bn- T h i s e x t e n s i o n i s of g r e a t i m p o r t a n c e f o r t h e
i n d u s t r i a l and t r a f f i c n o i s e p r e d i c t i o n , because v e r y o f t e n a s p h a l t - g r a s s t r a n s i t i o n s ( h a r d - s o f t t r a n s i t i o n s ) o c c u r between s o u r c e and r e c e i v e r .
F o r the two d i m e n s i o n a l c a s e the p l a n e wave r e f l e c t i o n c o e f f i c i e n t f o r a homogeneous grond s u r f a c e i s d e f i n e d i n the s p a t i a l f r e q u e n c y domain by
Pr( kx, z , t o ) = Rp( kx) P ^ k ^ z . w ) , (1.26) where ( s e e e q u a t i o n s ( 1 . 8 ) , (1.23) and ( 1 . 2 4 ) ) VI " (k /k )2 - B R „ a O X , (1-27) w i t h P X y/l - ( kx/ k , )2 + ^ 1C1 ^ 2C2 = t*l e aorma^ze^ gr°>in<i a d m i t t a n c e .
We suppose t h a t t h e inhomogeneous ground s u r f a c e can be s p l i t i n t o a number o f "homogeneous" p a r t s ( x ^ x ^ , ( x2 >x3) , e t c . , w i t h a d m i t t a n c e s ¡3^, 32 >
e t c . Because t h e a d m i t t a n c e i s a f u n c t i o n o f x, we s t a r t out t h e d e r i v a t i o n i n t h e space domain. I n t h i s domain e q u a t i o n (1.26) i s g i v e n by a c o n v o l u t i o n
Pr( x , z , u ) ) = rp( x ) * P1( x , z ,t 0) , (1-28)
w i t h
r ( x ) « R (k ). P P x
We make the a s s u m p t i o n t h a t we can use t h e p l a n e wave r e f l e c t i o n c o e f f i c i e n t R (k , R )> a s d e f i n e d i n the homogeneous c a s e , f o r e v e r y p a r t ( x ,x ).
p x n n n+1 Here 8 i-s t n e a d m i t t a n c e of t h e c o n s i d e r e d p a r t . Thus i n the space domain,
n
c o r r e s p o n d i n g w i t h the above m e n t i o n e d d i v i s i o n , we decompose e q u a t i o n (1.28) i n t o Pr = r ( x , 61) * { P1f ( x1, x2) } + r ( x , 82) * { Pif ( x2, x3) } + ... (1.29) i n w h i c h f ( x , x . . ) i s the b o x c a r - f u n c t i o n d e f i n e d by n n+1 f ( a , b ) = 0 f o r x<a and x>b, f ( a , b ) = 1 f o r a<x<b, and f ( a , b ) = 1/2 f o r x=a and x=b.
E q u a t i o n (1.29) can be w r i t t e n i n the s p a t i a l f r e q u e n c y domain
?r = Rp(kx,Bl){?i * F ( x1, x2) } +
+ Rp( kx, B2) { P i * F ( x2, x3) } + ... (1.30)
So, e v e r y t i m e we m u l t i p l y the p r e s s u r e w i t h the p l a n e wave r e f l e c t i o n c o e f f i c i e n t R (k ) i n t
P x i n t h e inhomogeneous c a s e .
c o e f f i c i e n t R (k ) i n the homogeneous c a s e , we have t o a p p l y e q u a t i o n (1.30) P x 1.3.5 The meteorological f a c t o r s The m e t e o r o l o g i c a l c o n d i t i o n s a r e o f g r e a t i m p o r t a n c e f o r t h e o u t d o o r sound p r o p a g a t i o n . The most i m p o r t a n t f a c t o r s a r e : - h u m i d i t y - t e m p e r a t u r e g r a d i e n t - w i n d d i r e c t i o n and g r a d i e n t - t u r b u l e n c e . Now we w i l l c o n s i d e r t h o s e f a c t o r s i n more d e t a i l .
10
H u m i d i t y o n l y has i n f l u e n c e on the sound p r o p a g a t i o n as f a r as a t m o s p h e r i c a b s o r p t i o n i s c o n c e r n e d . I n the l i t e r a t u r e [5] the mechanism of sound a b s o r p t i o n by a i r i s f u l l y d e s c r i b e d as a f u n c t i o n o f the h u m i d i t y and the t e m p e r a t u r e . F o r e v e r y f r e q u e n c y the a t t e n u a t i o n due t o a t m o s p h e r i c a l a b s o r p t i o n can e a s i l y be t a k e n i n t o a c c o u n t as a c o r r e c t i o n on the d i s t a n c e a t t e n u a t i o n . I n t h i s s t u d y , however, h u m i d i t y i s n o t c o n s i d e r e d .
The sound v e l o c i t y v a r i e s w i t h t h e t e m p e r a t u r e a c c o r d i n g t o the e q u a t i o n
I n o u t d o o r a p p l i c a t i o n s the n u m e r i c a l v a l u e s a r e K=1.4 and PQ/pQ=287 T, so
t h e sound v e l o c i t y i s o n l y a f u n c t i o n o f the a b s o l u t e t e m p e r a t u r e
A u n i f o r m change i n t e m p e r a t u r e l e a d s t o a change i n v e l o c i t y and thus t o a change i n the s p a t i a l phase of the sound f i e l d .
S i m i l a r l y , a u n i f o r m w i n d p r o f i l e a l s o g i v e s r i s e t o a change i n s p a t i a l p h a s e . T h i s e f f e c t can be s e e n as an a p p a r e n t change o f t h e sound v e l o c i t y . So, w i t h a g i v e n u n i f o r m w i n d and t e m p e r a t u r e p r o f i l e the a p p a r e n t sound v e l o c i t y i s g i v e n by ( s e e s e c t i o n 3.3) c = K V V P o -(1.31) (1.32) w i t h (1.33) w = the u n i f o r m w i n d v e c t o r ,
a = the a n g l e between w i n d d i r e c t i o n and sound p r o p a g a t i o n , and
c = v e l o c i t y of sound f o r jw|cosa=0 and T=273°K.
A u n i f o r m wind and t e m p e r a t u r e p r o f i l e have n e g l i g i b l e i n f l u e n c e on t h e p r o p a g a t i o n l o s s , p r o v i d e d t h e d i s t a n c e a t t e n u a t i o n ( 1 / r - l a w ) c a n be a p p r o x i m a t e d by
I n r e a l i s t i c s i t u a t i o n s v a l u e s o f w and T a r e f u n c t i o n s o f t h e s p a t i a l c o o r d i n a t e s ( i n a s t r a t i f i e d medium o n l y o f t h e z - c o o r d i n a t e ) . These w i n d and t e m p e r a t u r e g r a d i e n t s g i v e r i s e t o i n h o m o g e n e i t i e s w h i c h have an i n f l u e n c e on t h e p r o p a g a t i o n l o s s . I n terms o f r a y s i n h o m o g e n e i t i e s g i v e c u r v e d sound p a t h s and i f a sound wave h i t s t h e ground s u r f a c e t h i s c u r v a t u r e a l s o l e a d s t o a change i n t h e phase d i f f e r e n c e s between t h e i n c i d e n t and t h e r e f l e c t e d f i e l d . T h i s i s why t h e t o t a l sound p r e s s u r e c a n change s i g n i f i c a n t l y and i n p r a c t i c e v e r y b i g d i f f e r e n c e s between w i n d y and n e u t r a l c o n d i t i o n s c a n be found.
I n o u r model t h e atmosphere i s s i m p l i f i e d by a s t r a t i f i e d medium. T h i s i s n e c e s s a r y t o d e r i v e a manageable wave e q u a t i o n ( s e e c h a p t e r 2 ) . As i n p u t t h e sampled v a l u e s o f t h e w i n d and t e m p e r a t u r e p r o f i l e s have t o be known as a f u n c t i o n o f a l t i t u d e . I n many c a s e s t h e s e g r a d i e n t s c a n n o t be measured d i r e c t l y a t t h e p l a c e where t h e sound t r a n s m i s s i o n must be c a l c u l a t e d . I n t h a t c a s e t h e w i n d and t e m p e r a t u r e p r o f i l e s have t o be p r e d i c t e d by a i d o f s t a t i s t i c a l v a l u e s known from comparable w e a t h e r c o n d i t i o n s . Here we w i l l c o n f i n e o u r s e l v e s t o a few r e p r e s e n t a t i v e examples g i v e n by M o e r k e r k e n [6] ( s e e f i g u r e 2) and a b r i e f q u a l i t a t i v e d e s c r i p t i o n .
The t e m p e r a t u r e p r o f i l e :
The m e t e o r o l o g i c a l c o n d i t o n s w h i c h d e f i n e t h e t e m p e r a t u r e g r a d i e n t a r e t h e w i n d v e l o c i t y , t h e t u r b u l e n c e , t h e amount o f c l o u d s and t h e time o f t h e day. The l a t t e r f a c t o r i s i l l u s t r a t e d i n f i g u r e 2. I n c o m p l e t e l y n e u t r a l w e a t h e r c o n d i t i o n ( i . e . no h e a t t r a n s p o r t t h r o u g h t h e a t m o s p h e r e ) t h e t e m p e r a t u r e d e c r e a s e s w i t h h e i g h t a t t h e r a t e o f 0.7°C/100 m.
The wind p r o f i l e :
The w i n d v e l o c i t y and t h e r o u g h n e s s o f t h e ground s u r f a c e a r e t h e main v a r i a b l e s w h i c h d e f i n e t h e w i n d p r o f i l e . I n t h e l o w e r 100 m o f t h e atmosphere
t h e w i n d p r o f i l e c a n be m o d e l l e d a p p r o x i m a t e l y by a power p r o f i l e
( 1 . 3 5 )
12
m/s
F i g u r e 2: Some examples o f w i n d and t e m p e r a t u r e g r a d i e n t s .
a. The t e m p e r a t u r e p r o f i l e s d u r i n g a c l e a r summer day measured i n Rye.
b. Three d i f f e r e n t w i n d p r o f i l e s measured i n V l a a r d i n g e n (mean v a l u e s o v e r one h o u r ) w i t h u n s t a b l e weather c o n d i t i o n s ( 1 ) , n e u t r a l w e a t h e r c o n d i t o n s ( 2 ) and s t a b l e weather c o n d i t i o n s ( 3 ) . A l s o a m o d e l l e d w i n d p r o f i l e i s g i v e n ( 4 ) , c a l c u l a t e d w i t h w(z)=3.0(z/10)°*2m/s.
W ^ Q = the mean wind v e l o c i t y a t 10 m h e i g h t ,
and
p = t h e exponent w h i c h depends on t h e roughness o f t h e s u r f a c e .
I n t h e l i t e r a t u r e [6] a l s o a l o g a r i t h m i c f u n c t i o n i s s u g g e s t e d t o model t h e w i n d p r o f i l e f o r t h e l o w e r a l t i t u d e s e s p e c i a l l y . F o r c o m p u t a t i o n a l r e a s o n we
have c h o s e n f o r a power p r o f i l e e v e r y w h e r e . I n o u r examples ( s e e c h a p t e r 6 ) we m a i n l y use a wind p r o f i l e where p=0.2 and a c o n s t a n t t e m p e r a t u r e .
The t u r b u l e n c e i s a f a c t o r w h i c h i s v e r y d i f f i c u l t t o implement i n a c a l c u l a t i o n model a t t h e moment. The p r e s e n t e d e x t r a p o l a t i o n model i s based on a wave e q u a t i o n , w h i c h o n l y h o l d s f o r weak t u r b u l e n c e ( s e e e q u a t i o n ( 2 . 1 1 ) ) . I n r e a l i s t i c s i t u a t i o n s t h i s c o n d i t i o n i s o f t e n n o t met. F l u c t u a t i o n s o f t h e p r o p a g a t i o n l o s s due t o t u r b u l e n c e a r e i n t h e range o f +5 dB f o r s t a b l e w e a t h e r c o n d i t i o n s ( c l o u d y , l i t t l e w i n d ) and +15 dB f o r u n s t a b l e w e a t h e r c o n d i t i o n s (sunny, s t r o n g w i n d ) . I n t r a f f i c and i n d u s t r i a l n o i s e c o n t r o l , however, n o i s e l e v e l s a r e d e f i n e d by L l e v e l s a l m o s t e v e r y w h e r e . These L v a l u e s a r e d e r i v e d by time eq eq
a v e r a g i n g o f t h e sound p r e s s u r e . The v a l u e s o f w1 Q and p i n e q u a t i o n (1.35)
a r e a l s o mean v a l u e s o v e r a s u i t a b l e p e r i o d . I t i s o u r h y p o t h e s i s t h a t t h e a v e r a g i n g o f w^ Q > P a n <* tne sound l e v e l a r e s t r o n g l y i n t e r r e l a t e d , b u t a
c o r r e c t i o n o f W ^ Q or p may be necessary t o f i t outdoor measurements. These p r o b l e m s , however, l i e beyond t h e scope o f t h i s s t u d y .
1.3.6 The ground e f f e c t with wind and temperature gradients
I n c a s e o f an inhomogeneous medium t h e wave f i e l d i s n o t s p h e r i c a l anymore and t h u s t h e s o l u t i o n g i v e n by I n g a r d c a n n o t be u s e d . I n t h e e x t r a p o l a t i o n model we a r e s t i l l a b l e t o decompose t h e sound f i e l d a t z=0 i n t o i t s p l a n e wave components by a i d o f d i s c r e t e F o u r i e r t r a n s f o r m a t i o n . A r e l a t i o n s h i p between t h e u p g o i n g sound f i e l d P , t h e downgoing f i e l d P , and t h e
up down t r a n s m i t t e d f i e l d Pt r> can De found from t h e boundary c o n d i t i o n s ( s e e a l s o
e q u a t i o n s (1.15) and ( 1 . 1 6 ) )
Pu p( x , y , z , a j ) + Pd o w n( x , y , z , a j ) - Pt r( x , y , z , u ) ) ( 1 . 3 6 )
and
Pj 3 z p 3 z p2 3 z
N o t e t h a t f o r z>0 we s p l i t up t h e sound f i e l d i n t o an u p g o i n g and a downgoing p a r t , i n s t e a d o f an i n c i d e n t p a r t P^, w h i c h i s d e f i n e d a s t h e f i e l d w i t h o u t a ground s u r f a c e and a p a r t o f t h e sound f i e l d due t o t h e ground s u r f a c e P . I n a homogeneous medium t h e r e i s no d i f f e r e n c e between t h o s e d e f i n i t i o n s , b u t when t h e medium i s inhomogeneous, r e f l e c t i o n o f t h e sound f i e l d i n t h e medium i t s e l f c a n o c c u r ( s e e c h a p t e r 4 ) . I n t h i s c a s e Pr c o n s i s t s o f b o t h an up- and
a downgoing f i e l d and t h u s t h e g i v e n d e f i n i t i o n s a r e n o t e q u a l anymore. A n a l o g o u s t o t h e homogeneous c a s e t h e d e r i v a t i v e o f P and P, w i t h
up down r e s p e c t t o z c a n be e x p r e s s e d i n t h e d e r i v a t i v e s t o x and y by a i d o f t h e wave e q u a t i o n . F o r a l a y e r e d medium we pose t h a t i n t h e f i r s t l a y e r t h e wind w i l l be a l m o s t z e r o (m(O)RO). F o r s m a l l z we w i l l a p p r o x i m a t e t h e wave e q u a t i o n by t h e H e l m h o l t z e q u a t i o n f o r a homogeneous medium ( s e e e q u a t i o n ( 1 . 1 7 ) ) w i t h
k^kXO)
14 By a i d o f t h e F o u r i e r t r a n s f o r m w i t h r e s p e c t t o x and y, t h e boundary c o n d i t i o n s a r e g i v e n by P = R' P, f o r z=0, (1-39) up p down ' R ; = ^ — ' 2 x y • (i.4o) p2v / k2( 0 ) - k2- k2 + P . v / k ^ - k2 N o t e t h a t t h i s e x p r e s s i o n o f R' i s e q u i v a l e n t t o R i n e q u a t i o n ( 1 . 2 2 ) . P P
1.4 A BRIEF DESCRIPTION OF THE EXTRAPOLATION MODEL
The model p r e s e n t e d i n t h i s t h e s i s i s based on a s t e p by s t e p e x t r a p o l a t i o n o f t h e sound f i e l d . I f e f f e c t s o f w i n d and temperature g r a d i e n t s a r e i n t r o d u c e d i n s m a l l s t e p s , t h i s method i s n e a r l y e x a c t . I n o t h e r words, w i t h a c o r r e c t c h o i c e o f t h e s t e p s i z e t h e e r r o r s c a n be k e p t below e v e r y d e s i r e d l e v e l . O t h e r advantages o f t h i s model a r e t h a t t h e a b s o r b i n g boundary c o n d i t i o n s a t
t h e ground s u r f a c e c a n be h a n d l e d i n a p h y s i c a l l y c o r r e c t manner and t h a t t h e model i s s t i l l v a l i d f o r g r e a t e r s o u r c e - r e c e i v e r d i s t a n c e s , w h i l e c a l c u l a t i o n models based on r a y t r a c i n g have r a y s f o r m i n g c a u s t i c s .
I n t h e t h e o r e t i c a l c h a p t e r s o f t h i s t h e s i s t h e e x t r a p o l a t i o n method i s d e s c r i b e d f o r t h e t h r e e d i m e n s i o n a l c a s e . F o r s i m p l i c i t y ' s sake t h e c o m p u t a t i o n a l work i s c o n f i n e d t o t h e c a s e o f two d i m e n s i o n s , v i z . an x d i r e c t i o n p a r a l l e l t o t h e ground s u r f a c e and a l t i t u d e i n z d i r e c t i o n . A f t e r -wards t h e r e s u l t s c a n be t r a n s f o r m e d t o a t h r e e d i m e n s i o n a l sound p r o p a g a t i o n ( s e e c h a p t e r 5 ) . B u t b a s i c a l l y i t r e m a i n s a model i n w h i c h i n h o m o g e n e i t i e s o f t h e medium o n l y c a n be i n t r o d u c e d i n a p l a n e w h i c h c o n t a i n s s o u r c e and r e c e i v e r . S i d e w i n d c o n d i t i o n s t h u s have t o be t r a n s l a t e d i n up o r downwind c o n d i t i o n s w i t h wx= | w | c o s a , a s i n t r o d u c e d i n e q u a t i o n ( 1 . 3 3 ) . We pose t h a t t h i s s i m p l i f i c a t i o n i s a l l o w e d , a l t h o u g h t h i s w i l l n o t be p r o v e n i n t h i s s t u d y . The e x t r a p o l a t i o n p r o c e d u r e s t a r t s w i t h t h e c a l c u l a t i o n o f t h e t o t a l sound p r e s s u r e above a h a r d g r o u n d . I n t h i s c a s e w i n d and t e m p e r a t u r e a s w e l l a s t h e s o u r c e c a n be m i r r o r e d i n t o t h i s h a r d s u r f a c e , so a f t e r e x t r a p o l a t i o n t h e t o t a l f i e l d i s found ( s e e f i g u r e 3 ) . I t i s f o r c o m p u t a t i o n a l r e a s o n s t h a t we s t a r t i n t h i s way ( s e e c h a p t e r 5 ) . The e x t r a p o l a t i o n i s c a r r i e d o u t r e c u r s i v e l y i n s t e p s Ax s t a r t i n g w i t h t h e f u n c t i o n P ( x . , z ) . So we assume t h a t
WIND P R O F I L E ; SOURCEo z=0-MIRROR S O U R C E
J
2 13 »EXTRAPOLATION Ax P ( x „ z ) f%2,Z) P( XJ #Z ) P(X„,Z)F i g u r e 3 : The e x t r a p o l a t i o n o f the t o t a l sound f i e l d above a hard ground s u r f a c e .
t h e ( h a r m o n i c ) sound p r e s s u r e P ( x , z ) i s known i n t h e p l a n e x=x^. The sound p r e s s u r e i n t h e p l a n e x=x1+Ax f o l l o w s from t h e T a y l o r s e r i e s :
P ( x + A x , z ) = P ( x . z ) + + M L l ! l + ... (1.41)
I.' 3x 21 3 x2
The d e r i v a t i v e s o f P w i t h r e s p e c t t o x a r e unknown. They may be d e r i v e d f r o m t h e d a t a P ( xn, z ) u s i n g t h e wave e q u a t i o n . We i n t r o d u c e two r e s t r i c t i o n s :
- t h e medium i s a p p r o x i m a t e l y s t a t i o n a r y ( l o w t u r b u l e n c e ) ; - t h e medium i s l a y e r e d (wind and t e m p e r a t u r e depend on z o n l y ) .
A f t e r some a p p r o x i m a t i o n s ( s e e c h a p t e r 2) t h e wave e q u a t i o n c a n now be r e w r i t t e n as
^-=- + — - 2 j k ( z ) m ( z ) | ^ + k2( z ) { l - m2( z ) }P = 0, ( 1 . 4 2 )
3 x2 3 y2 dx
where
k ( z ) = to/c(z) = t h e wave number,
m(z) = w ( z ) / c ( z ) = t h e Mach number, c ( z ) = t h e sound v e l o c i t y ( w h i c h depends on t h e t e m p e r a t u r e ) , and w(z) = t h e wind v e l o c i t y i n t h e x - d i r e c t i o n . One e x t r a p o l a t i o n s t e p c a n t h u s From t h e p r e s s u r e P ( xn >z ) i n t h e r e s p e c t t o z a r e c a l c u l a t e d . By a i d be c a r r i e d o u t by t h e f o l l o w i n g p r o c e d u r e : p l a n e x=x t h e d e r i v a t i v e s o f P ( x ,z) w i t h n n o f t h e wave e q u a t i o n , t h e d e r i v a t i v e s w i t h
16 r e s p e c t t o x a r e found n e x t . F i n a l l y t h e T a y l o r s e r i e s g i v e t h e r e l a t i o n , f r o m w h i c h t h e p r e s s u r e p(x n+ Ax) i n t h e p l a n e x=x +Ax i s c a l c u l a t e d . Or i n t h e f o r m o f a diagram: P ( x ,z)+ -s—j , e t c n dz 3 z2 3P 32P •5—, , e t c . wave e q u a t i o n T a y l o r s e r i e s * P ( x +Ax,z) I n g e n e r a l t h i s e x t r a p o l a t i o n p r o c e d u r e c a n be w i r t t e n w i t h t h e a i d o f an e x t r a p o l a t i o n m a t r i x W, d e f i n e d by P ( xn+ A x , z ) = ^ ( xn+ A x , xn) P ( xn, z ) . (1.43) I n c h a p t e r 3 t h i s m a t r i x i s d e r i v e d and d i s c u s s e d f o r s e v e r a l c o n d i t i o n s (no g r a d i e n t s , s m a l l g r a d i e n t s and l a r g e g r a d i e n t s ) .
The sound f i e l d above a h a r d ground b e i n g known, t h e n e x t s t e p i s t h e i n t r o d u c t i o n o f an a b s o r b i n g g r o u n d . To t h i s p u r p o s e we decompose t h e sound f i e l d i n t o i t s p l a n e wave components. We d e f i n e
i>^(k ,z,u>) = t h e p r e s s u r e o f t h e i n c i d e n t p l a n e wave ( e v e r y k v a l u e d e f i n e s a p l a n e wave component)
and
^ ( k^ j Z . o j ) = the p r e s s u r e of the r e f l e c t e d plane wave.
The t o t a l sound p r e s s u r e i s d e f i n e d a s Pt( kx, z , u ) - ^ ( k ^ z . U ) ) + Pr( kx, z , u ) . (1.44) The boundary c o n d i t i o n s f o r an a b s o r b i n g s u r f a c e a t z=0 a r e g i v e n by P = R P. ( 1 . 4 5 ) up p down w i t h R = t h e p l a n e wave r e f l e c t i o n c o e f f i c i e n t a s d e f i n e d i n e q u a t i o n P ( 1 . 4 0 ) .
W I N D P R O F I L E
plone f
w o v e G R O U N D S U R F A C E
///////////// / 7 / 7 //////;//; > ;
///////////// / 7 / / /; .
F i g u r e 4: The " r e f l e c t i o n " o f a p l a n e wave component due t o t h e i n h o m o g e n e i t y o f t h e medium.
Under z e r o o r up wind c o n d i t i o n s and w i t h z e r o o r n o r m a l t e m p e r a t u r e g r a d i e n t s , t h e downgoing sound f i e l d a t z=0 i s t h e i n c i d e n t f i e l d and t h e u p g o i n g sound f i e l d i s t h e r e f l e c t e d f i e l d . So f o r t h e i n t r o d u c t i o n o f t h e boundary c o n d i t i o n s we f i n d f o r zW
^down =? i and % * V t1-46)
F o r down w i n d and i n v e r s e t e m p e r a t u r e g r a d i e n t s some p l a n e wave components a r e " r e f l e c t e d " back t o t h e ground s u r f a c e by t h e medium. F o r s u c h a p l a n e wave component t h e s i t u a t i o n i s d e l i n e a t e d i n f i g u r e 4. The e x p l a n a t i o n o f t h i s e f f e c t i s d i s c u s s e d i n more d e t a i l i n c h a p t e r 4. F o r t h e boundary c o n d i t i o n s t h i s means t h a t t h e downgoing sound f i e l d has t o be r e p l a c e d by
^ d o w n ^ x '2'0^ = P i (k x>z. w ) + B ( kx, u ) ) Pr( kx, z , w ) . ( 1 . 4 7 )
The u p g o i n g sound f i e l d i s now g i v e n by
?u p( kx >z>u ) = A ( kx, u ) Pr( kx, z , a j ) . ( 1 . 4 8 ) F o r g i v e n w i n d and t e m p e r a t u r e g r a d i e n t s t h e ( c o m p l e x ) A and B v a l u e s c a n be c a l c u l a t e d f r o m t h e wave e q u a t i o n ( 1 . 4 2 ) by a i d o f t h e W.K.B.J.-method ( s e e c h a p t e r 4 ) . W i t h t h i s boundary c o n d i t i o n and e q u a t i o n ( 1 . 4 4 ) we a r e a b l e t o d e r i v e P.- V..,,J bv d e f i n i n g R =1 a t z=0 and P . by d e f i n i n g R + \ a t z=0 f r o m r , n a r a p r , s o r t p t h e t o t a l sound p r e s s u r e P.. , ,, w h i c h we had c a l c u l a t e d i n t h e f i r s t t , h a r d p l a c e . A t z=0 P , , and P , a r e known; f o r a l l z>0 t h e r e f l e c t e d r . h a r d r , s o f t f i e l d f o r b o t h c a s e s c a n be c a l c u l a t e d by e x t r a p o l a t i o n i n t h e z - d i r e c t i o n . The e x t r a p o l a t i o n m a t r i x i s d e f i n e d by
18
P ( x , zn+ A z , u ) = ^ ( zn+ A z , zn) P ( x , zn, o j ) (1.49)
and can be d e r i v e d by a i d o f t h e W.K.B.J.-method, o r , when no " r e f l e c t i o n s " o c c u r i n t h e medium, w i t h t h e a i d o f t h e same p r o c e d u r e as t h e e x t r a p o l a t i o n i n x - d i r e c t i o n ( s e e c h a p t e r 4 and 5 ) . The t o t a l p r e s s u r e f o r t h e s o f t c a s e , P , , can be c a l c u l a t e d from t h e t o t a l p r e s s u r e above a h a r d ground w i t h
t , s o f t t h e e q u a t i o n Pt , s o f t = Pt , h a r d ~ Pr , h a r d + Pr , s o f f (1-50) More t h e o r e t i c a l d e t a i l s o f t h e e x t r a p o l a t i o n model a r e g i v e n i n c h a p t e r s 3 and 4, and t h e c o m p u t a t i o n a l a s p e c t s a r e g i v e n i n c h a p t e r 5. R e s u l t s o f t h e model a r e g i v e n and d i s c u s s e d i n c h a p t e r 6. i
CHAPTER 2
THE WAVE EQUATION
IN AN INHOMOGENEOUS MOVING MEDIUM
2.1 THE BASIC EQUATIONS
F o r a good d e s c r i p t i o n o f t h e p r o p a g a t i o n o f sound i n an atmosphere w i t h w i n d and t e m p e r a t u r e g r a d i e n t s , we have t o c o n s i d e r t h e b a s i c r e l a t i o n s between t h e p a r t i c l e v e l o c i t y (_v), t h e p r e s s u r e ( Pt) , t h e d e n s i t y } and t h e t e m p e r a t u r e ( T ) . These r e l a t i o n s a r e g i v e n e.g. by G.K. B a t c h e l o r [ 7 ] , v i z . t h e e q u a t i o n o f m o t i o n , t h e c o n t i n u i t y o f mass, t h e e q u a t i o n o f s t a t e and t h e c o n t i n u i t y o f h e a t . I n o u r c a s e i t i s p o s s i b l e t o make some s i m p l i f i c a t i o n s . We do n o t need t h e e q u a t i o n s t o d e s c r i b e t h e w i n d i t s e l f , so e x t e r n a l f o r c e s w h i c h a r e r e s p o n s i b l e f o r t h e w i n d a r e n o t t a k e n i n t o a c c o u n t . The atmosphere c a n be seen as an i d e a l gas w i t h no v i s c o s i t y w h i l e t h e c o m p r e s s i o n due t o t h e sound f i e l d c a n be c o n s i d e r e d as an a d i a b a t i c p r o c e s s . I n t h i s c a s e t h e e q u a t i o n o f s t a t e i s g i v e n by ~K PtPt = c o n s t a n t , ( 2 . 1 ) where K = t h e r a t i o between t h e p r i n c i p a l s p e c i f i c h e a t s : c and c . p v
The r e l a t i v e p e r t u r b a t i o n o f t h e i n i t i a l s t a t e i s v e r y s m a l l when t h e sound p r o p a g a t i o n i s l i n e a r ( t h e sound p r e s s u r e l e v e l i s under c a . 100 d B ) . We i n t r o d u c e :
pt( t o t a l -) = pQ( r e f e r e n c e -) + p ( a c o u s t i c d e n s i t y )
20
and we impose t h a t
dp
_ ° « 4 a . ( 2 . 2 ) d t d t
Here we assume t h a t the change o f t h e t e m p e r a t u r e g r a d i e n t i n t i m e does n o t p l a y a dominant r o l e . I n a l l p r a c t i c a l s i t u a t i o n s t h e s t a b i l i t y o f t h e atmosphere i s such t h a t u n e q u a l i t y (2.2) h o l d s . F o r l i n e a r a c o u s t i c s we f i n d
|p| « Po a n d
IPI
k< PO » ( 2- 3)so a d i f f e r e n t i a l r e l a t i o n s h i p between p and p c a n be d e r i v e d from e q u a t i o n (2.1)
P
dp = K — — dp = c2d p ( 2 . 4 )
Po
w i t h c=c(T) i s t h e speed o f sound i n t h e atmosphere ( i n a i r K=1,4 and P /n =RT/M= 287 T ) . I t i s a l s o a c c e p t a b l e t o assume t h a t t h e sound f i e l d h a s
o Ko
no i n f l u e n c e on the w i n d . Because the energy per u n i t volume o f a sound f i e l d (= p /P ss3.10~6J/m f o r a sound p r e s s u r e l e v e l o f 100 dB) i s v e r y s m a l l
e f f . o
compared t o the energy o f the w i n d f i e l d (= ^p [w|« 5 J/m f o r 3 m/s).
W i t h t h e p r e v i o u s c o n s i d e r a t i o n s we a r r i v e a t t h e f o l l o w i n g two b a s i c e q u a t i o n s t o d e t e r m i n a t e t h e sound f i e l d 1 dp + V.v = 0 ( c o n s e r v a t i o n o f mass) ( 2 . 5 ) and w i t h 2 dt dv ~VP ° pn (equation of motion), ( 2 . 6 ) o dt J L = 3 + v.V • dt 3 t
-2.2 THE HELMHOLTZ EQUATION FOR A LAYERED STEADY WIND PROFILE
The p a r t i c l e v e l o c i t y c a n be d i v i d e d i n two p a r t s : The w i n d v e l o c i t y and t h e a c o u s t i c a l v e l o c i t y vi, w i t h _v = ^ + u .
Assuming l i n e a r a c o u s t i c s , u n e q u a l i t y (2.3) h o l d s and a l s o
2
C o n s e q u e n t l y we a r e a b l e t o n e g l e c t t h e terms w i t h u2, ( u . V)u and (_u.V)p. Now e q u a t i o n s ( 2 . 5 ) and ( 2 . 6 ) c a n be r e w r i t t e n as and _ L | E + J _ (w.v)p + p0V.(u+w) = 0 c2 8t c2 )(u+w) (2.8) VP + p0 ~ ~ + p0{(u+w>V}(u+w) = 0. ( 2 . 9 ) To s i m p l i f y t h e s e e q u a t i o n s we impose t h e f o l l o w i n g r e s t r i c t i o n s on t h e wind: V-w « y . u , (2.10) w h i c h i s q u i t e a c c e p t a b l e as shear f o r c e s a r e predominant i n t h e w i n d f i e l d and t h e c o m p r e s s i b i l i t y o f t h e medium p l a y s b u t a m i n o r r o l e . And f u r t h e r m o r e
3»
(2.11)
w h i c h i s t h e same r e s t r i c t i o n as was imposed on t h e s t a t i c d e n s i t y ( s e e e q u a t i o n ( 2 . 2 ) ) . T h i s r e s t r i c t i o n i s n e c e s s a r y i f o u r p r o b l e m i s n o t t o become unmanagable, t h u s s t r o n g t u r b u l e n c e s cannot be t a k e n i n t o a c c o u n t . E q u a t i o n s ( 2 . 8 ) and ( 2 . 9 ) now reduce t o
J _ | £ + J _ ( w .V) p + P oV- u = 0 2 d t 2 C C and VP + Po - j ^ T+ P0 (- 'v )- + P o ^ ' v ) " . + P0(*'V)w = °-We w i l l c a l c u l a t e t h e d i v e r g e n c e o f e q u a t i o n (2.13) and impose (2.12) (2.13)
I
V p0I
« j Vp ] = —I
VPI
c2 (2.14) T h i s l a t t e r r e s t r i c t i o n means t h a t t h e s p a t i a l v a r i a t i o n s o f p , w h i c h i n o u r c a s e a r e due t o t h e t e m p e r a t u r e g r a d i e n t , a r e much s m a l l e r t h a n t h e s p a t i a l d e n s i t y v a r i a t i o n s due t o t h e sound f i e l d . Note t h a t v a r i a t i o n s o f t h e l o c a l v a l u e s o f c and pQ a r e i m p o r t a n t f o r t h e p r o p a g a t i o n o f t h e sound f i e l d and22 A f t e r s u b s t i t u t i o n o f ^ . u , g i v e n i n e q u a t i o n ( 2 . 1 2 ) , and w i t h V.{(w.V)u}= ( w . v H V - u ) + V.{(u.V)w}, (2.15) we f i n d 1 32p _ 2 p - — Z-Z- - — (w.V) 4 £ - — (w.V)Vp + 2 ^ 2 2 — 3 t „ 2 — 2P oV- - f (Ji-V)w} + pQV.{(w.V)w} = 0. (2.16) W i t h o u t wind (w=0) e q u a t i o n (2.16) r e d u c e s t o the w e l l - k n o w n H e l m h o l t z e q u a t i o n v^p - - L Ü P = o. (2.17)
W i t h wind we have n o t y e t succeeded i n e l i m i n a t i n g ii and t h e r e f o r e t h e e q u a t i o n i s s t i l l unmanagably complex. An improvement i s a c h i e v e d i f we assume a p r a c t i c a l l y s t r a t i f i e d medium. I n such a medium t h e i n h o m o g e n e i t i e s o n l y depend on one c o o r d i n a t e , i n our c a s e the a l t i t u d e . W i t h t h e a l t i t u d e i n z - d i r e c t i o n we a p p r o x i m a t e t h e wind v e c t o r by dw dw dw dw << << dx dz dy dz w « w (2.18) A f t e r F o u r i e r t r a n s f o r m a t i o n o f t t o k, e q u a t i o n (2.16) r e d u c e s t o fa \ wA i \ du dw k2P + V2P - 2 j k ( — . V l P - — (—.VIVP + 2p - r — = 0, J \c / c\c / o dz dz ' (2.19) where
k = M / C = the wave number.
I n g e n e r a l o m i s s i o n o f t h e l a s t term i n e q u a t i o n (2.19) i s u n j u s t i f i e d , e s p e c i a l l y n e a r t h e g r o u n d , where |dw/dz| i s n o t n e c e s s a r i l y s m a l l w i t h r e s p e c t t o k|w_| . However, by c o n f i n i n g o u r s e l v e s t o waves t r a v e l l i n g a p p r o x i m a t e l y p a r a l l e l t o t h e x-y p l a n e , n e g l e c t i o n o f t h e l a s t term may be a c c e p t a b l e as f o r
du dz
P r o v i d e d the Mach numbers o f the wind v e l o c i t y a r e low enough ' 2 « 1, (2.21) and by a i d o f t h e z e r o t h o r d e r a p p r o x i m a t i o n o f ( e q u a t i o n ( 2 . 1 7 ) ) , we may r e w r i t e t h e f o u r t h term o f e q u a t i o n (2.19) w / w \ W2 ~2 - - . v Vp « — 1 1 . (2.22) c^c / •c" a t2 So t h e f o l l o w i n g wave e q u a t i o n w i l l be a p p l i e d i n t h i s t h e s i s k2^ l - — - ^ P + V2P - 2 j k ^ . V ^ P = 0. (2.23) I n l i t e r a t u r e [8] t h e f o u r t h term i n e q u a t i o n (2.19) u s u a l l y i s r e w r i t t e n i n a n o t h e r way and then t h e wave e q u a t i o n i s g i v e n as
k2P + V2p - 2 j | ( w . V ) P - -i(w.V)(w.VP) = 0, (2.24)
w h i c h i s t h e form u s e d t o i n t r o d u c e t h e e i k o n a l f u n c t i o n . I n f a c t , w i t h r e l a t i o n (2.21) i t i s a l l o w e d t o n e g l e c t t h e f o u r t h term i n e q u a t i o n (2.24) c o m p l e t e l y , b u t i n s e c t i o n 3.4 i t w i l l become c l e a r t h a t i t i s p r e f e r a b l e t o use t h e wave e q u a t i o n i n t h e form a s g i v e n i n e q u a t i o n ( 2 . 2 3 ) .
CHAPTER 3
THE EXTRAPOLATION METHOD
3.1 INTRODUCTION TO FINITE DIFFERENCE TECHNIQUES
The w a v e f i e l d e x t r a p o l a t i o n method i s a t e c h n i q u e t o s i m u l a t e sound p r o p a g a t i o n by p r e d i c t i n g t h e e x t e n s i o n o f t h e p r o p a g a t i n g sound, s t a r t i n g f r o m a known d i s t r i b u t i o n i n a g i v e n s o u r c e p l a n e . Assuming t h e sound p r o p a g a t i o n i s i n t h e x - d l r e c t i o n , t h e p r e s s u r e d i s t r i b u t i o n Pn i n a p l a n e x=x c a n be c a l c u l a t e d ( f o r a f i n i t e a r e a ) from t h e g i v e n p r e s s u r e n d i s t r i b u t i o n P^ i n t h e p l a n e x=x^ ( s e e f i g u r e 5 ) . The r e l a t i o n between P^ and P c a n be w r i t t e n a s a m a t r i x o p e r a t i o n P ( xn, y , z , a j ) = tV ( xn, x1) P ( x1, y , z , o j ) . ( 3 . 1 ) Throughout t h i s c h a p t e r we w i l l o m i t t h e c i r c u l a r f r e q u e n c y v a r i a b l e co. x = xn propogotion and extrapolation —>> G e o m e t r i c s i t u a t i o n : The s o u r c e p l a n e i n x=x^, t h e F i g u r e 5: e x t r a p o l a t i o n i n x - d i r e c t i o n and t h e sound p r o p a g a t i o n b e i n g m a i n l y i n t h e x - d i r e c t i o n .
26
One element o f W r e p r e s e n t s t h e t r a n s f e r f u n c t i o n f o r one " s o u r c e " p o i n t ( x ^ , y , z ) t o one " r e c e i v e r " p o i n t ( x , y , z ) . The p r o p a g a t i o n m a t r i x W i s d e f i n e d by t h e wave e q u a t i o n and t h e a c o u s t i c p r o p e r t i e s o f t h e medium. F o r t h e s i m u l a t i o n p r o c e s s two d i f f e r e n t methods can be used: a non r e c u r s i v e o r a r e c u r s i v e method.
S t a r t i n g from t h e g i v e n p r e s s u r e d i s t r i b u t i o n P^, a non r e c u r s i v e method c a l c u l a t e s t h e p r e s s u r e d i s t r i b u t i o n Pn f o r any n i n one e x t r a p o l a t i o n s t e p .
A r e c u r s i v e method i s a s t e p by s t e p e x t r a p o l a t i o n , where a f t e r e v e r y s t e p t h e c a l c u l a t e d p r e s s u r e d i s t r i b u t i o n i s t h e new i n p u t d i s t r i b u t i o n f o r t h e n e x t e x t r a p o l a t i o n s t e p . The r e c u r s i v e method needs more s t e p s t o c a l c u l a t e t h e p r e s s u r e on t h e p l a n e x=x , b u t t h e advantage i s t h a t the used p r o p a g a t i o n
n
m a t r i c e s f o r s m a l l e r s t e p s c a n be c a l c u l a t e d e a s i e r .
I f the p r o p a g a t i o n p r o p e r t i e s o f the medium a r e c o n s t a n t i n t h e x - d i r e c t i o n , t h e p r o p a g a t i o n m a t r i x w i l l be independent o f x. I n t h i s s i t u a t i o n e v e r y e x t r a p o l a t i o n s t e p , o v e r t h e same d i s t a n c e Ax, c a n be done w i t h t h e same m a t r i x o p e r a t i o n V(xn 'xl ) = ^ (xn 'xn - l) , ,'( xn - l >xn - 2) ' " ^x2 >xl > (3'2> o r ( / ( xn, x1) = j /n 1( A x ) , where A x = x1+1 - x± f o r ±»1,2, — ,n.
So, i n a r e c u r s i v e scheme t h e same o p e r a t o r (/(Ax) c a n be used t o p r e d i c t t h e sound f i e l d a f t e r e v e r y e x t r a p o l a t i o n s t e p Ax. F o r a s t a t i o n a r y homogeneous medium an a n a l y t i c e x p r e s s i o n f o r an e x t r a p o l a t i o n m a t r i x W c a n be d e r i v e d from t h e wave e q u a t i o n i! Z + H Z + 3 ^ + k2p = o. 3 x2 3 y 2 (3.3)
W i t h t h i s e q u a t i o n t h e sound f i e l d o f any s o u r c e c a n be computed i n t h e s o u r c e p l a n e x=x^ and w i t h t h e Huygens p r i n c i p l e t h e p r e s s u r e i n a p o i n t ^x2, y2, Z2 ^ Can ^e c alc ula t e^ from the p r e s s u r e d i s t r i b u t i o n i n t h e s o u r c e
P (X 2, y2, z2) - | 2 ƒ P ( x1 (y ,Z) e ^ d S ^ ( 3 . 4 )
" S, w i t h
r2 = A x2 + ( y2- y )2 + ( z2- z )2 ( 3 . 5 )
and where i s t h e s o u r c e p l a n e x=x^ . E q u a t i o n ( 3 . 4 ) i s known a s t h e R a y l e i g h I I i n t e g r a l [ 9 ] . I f we do t h e same f o r an a r b i t r a r y p o i n t ( x2 >y , z ) on t h e p l a n e x = x2 i n t h e a r e a o f i n t e r e s t , e q u a t i o n ( 3 . 4 ) c a n be w r i t t e n a s a c o n v o l u t i o n P ( x2, y , z ) = f P ( x1, y ' , z ' ) W ( x1, y - y ' , z - z,) d y,d z, J ( 3 . 6 ) o r s y m b o l i c a l l y P ( x2, y , z ) = Wx * P ( x1 > y,Z) . ( 3 . 7 ) A c o n v o l u t i o n p r o d u c t can a l s o be w r i t t e n i n m a t r i x n o t a t i o n : P ( x2, y , z ) = ^ ( x2, x1) P ( x1, y , z ) . ( 3 . 8 ) F o r a s t a t i o n a r y homogeneous medium e q u a t i o n ( 3 . 7 ) g i v e s a r e p r e s e n t a t i o n o f wave f i e l d p r o p a g a t i o n i n terms o f a c o n v o l u t i o n w i t h o p e r a t o r W. E q u a t i o n ( 3 . 8 ) g i v e s a r e p r e s e n t a t i o n i n terms o f m a t r i x m u l t i p l i c a t i o n s w i t h p r o p a g a t i o n m a t r i x W. The e x t r a p o l a t i o n w i t h o p e r a t o r W c a n be c a r r i e d o u t r e c u r s i v e l y o r non r e c u r s i v e l y , W b e i n g g i v e n by 1+ j k r - j k r W =W = . . . = WA= ^ 2 . L e ( 3 . 9 ) l " ^ X2 *X3 Ax 2TT 3 V ' o r r l , 1 + i k r , - j k r , O H ^ J j H , n-1 ( 3 a o ) V * n V X 2 V X 3 2V n-1 r e s p e c t i v e l y , w i t h r * = (nAx)2 + y2 + z2.
28
Ax """"
s o u r c e
F i g u r e 6: F o r a g i v e n a p e r t u r e a n g l e <t> t h e a r e a S o v e r w h i c h the sound p r e s s u r e must be c a l c u l a t e d , depends on the e x t r a p o l a t i o n s t e p Ax.
F o r a n o n s t a t i o n a r y and inhomogeneous medium i t i s not p o s s i b l e t o d e r i v e the sound f i e l d o f a p o i n t s o u r c e a n a l y t i c a l l y . I n t h i s c a s e a n o t h e r a p p r o a c h must be a p p l i e d t o f i n d W. The a r e a o v e r w h i c h t h e p r o p a g a t i o n of t h e sound f i e l d , due t o one p o i n t s o u r c e , has t o be c a l c u l a t e d t o f i n d the p r e s s u r e i n the n e x t p l a n e , depends on the s i z e o f the e x t r a p o l a t i o n s t e p ( s e e f i g u r e 6 ) . So, f o r s m a l l Ax v a l u e s u s e f u l a p p r o x i m a t i o n s a r e a l l o w e d t o s o l v e t h e p r o b l e m . Such a p p r o a c h e s a r e known as f i n i t e d i f f e r e n c e t e c h n i q u e s . We w i l l d i s c u s s t h e s e t e c h n i q u e s by s t a r t i n g w i t h t h e T a y l o r s e r i e s [10] P ( x 2 , y , z ) = P ( x1 ) y, z ) + ^ |?- + lf£ + ... (3.11) I! 2.' 8 x2 T h i s s e r i e s c o n v e r g e s f a s t f o r s m a l l Ax v a l u e s . I n t h i s r e l a t i o n between P^ and t n e unknown v a r i a b l e s a r e the d e r i v a t i v e s t o x. The wave e q u a t i o n
g i v e s a r e l a t i o n between 3P/ax and the p r e s s u r e d i s t r i b u t i o n a t the p l a n e x=x^ ( f o r e v e r y p o i n t ( x ^ , y , z ) ) . L e t us w r i t e t h i s r e l a t i o n i n terms of m a t r i c e s
(•^) P C x ^ y . z ) = ^ . x ^ P / x ^ y . z ) . (3.12) \ /x=Xj
One element o f m a t r i x A r e p r e s e n t s the c o n t r i b u t i o n o f one p o i n t o f t h e g i v e n p r e s s u r e d i s t r i b u t i o n to one p o i n t o f t h e d e r i v a t i v e p r e s s u r e d i s t r i b u t i o n . Second and h i g h e r d e r i v a t i v e s w i t h r e s p e c t t o x can be o b t a i n e d by a p p l y i n g t h i s m a t r i x A two o r more t i m e s (tV) ^ l . ^ = ^ P i + | U i (3.13) \dX /X=Xj [ — \ P ( x1 > y, z ) = AA£SX + A | | Pt + \3x3/x=x, x and
+ 2 | i ^ + — P1. (3.14)
dX 3 x2
I f we assume t h a t t h e medium p a r a m e t e r s a r e i n d e p e n d e n t o f x o v e r one e x t r a p o l a t i o n s t e p Ax, t h e d e r i v a t i v e s o f A w i t h r e s p e c t t o x become z e r o . I n t h i s a p p r o a c h we assume t h e medium t o be b u i l t up o f homogeneous l a y e r s i n t h e x - d i r e c t i o n . E r r o r s a r e i n t r o d u c e d a t t h e t r a n s i t i o n between t h e l a y e r s . These e r r o r s c a n be n e g l e c t e d f o r s m a l l changes i n t h e a c o u s t i c p r o p e r t i e s o f t h e medium. A more a c c u r a t e a p p r o a c h i s t o l i n e a r i z e t h e medium o v e r one s t e p ( s e e f i g u r e 7 ) . I n t h i s c a s e 3 A / 3 x i s i n c l u d e d and the second and h i g h e r d e r i v a t i v e s o f A t o x a r e n e g l e c t e d . MEDIUM P A R A M E T E R F i g u r e 7: Medium p r o p e r t i e s c a n be k e p t c o n s t a n t o r c a n be l i n e a r i z e d o v e r one e x t r a p o l a t i o n s t e p . S u b s t i t u t i o n o f e q u a t i o n (3.13) and (3.14) i n t o e q u a t i o n (3.11) g i v e s a s e r i e s e x p a n s i o n o f m a t r i x W. By n e g l e c t i n g t h e f i r s t and h i g h e r o r d e r d e r i v a t i v e s o f A we g e t ^(x+Ax.x) = ( J + — A + — AA + — AAA + . . . ) . ( 3 . 1 5 ) 1! 21 3! N o t e t h a t f o r x=0, W becomes t h e u n i t y m a t r i x . I n s e c t i o n 3.3 i t w i l l be shown t h a t f o r t h e homogeneous c a s e t h e r e s u l t g i v e n i n e q u a t i o n (3.15) i s t h e same as t h e o p e r a t o r i n e q u a t i o n ( 3 . 8 ) , w h i c h i s found by a i d o f t h e R a y l e i g h I I i n t e g r a l . I n t h i s c h a p t e r we w i l l d i s c u s s t h e f i n i t e - d i f f e r e n c e t e c h n i q u e i n more d e t a i l and we w i l l show how we c a n use t h i s t e c h n i q u e t o f i n d a s u i t a b l e o p e r a t o r f o r e x t r a p o l a t i o n o f t h e sound f i e l d i n t h e x - d i r e c t i o n o f a medium w i t h w i n d and t e m p e r a t u r e g r a d i e n t s . I m p o r t a n t t o o l s used i n t h i s t e c h n i q u e w i l l be
30
d i s c u s s e d f i r s t , i . e . " s p a t i a l c o n v o l u t i o n " i n s e c t i o n 3.2 and " f l o a t i n g t i m e r e f e r e n c e " i n s e c t i o n 3.4.
3.2 DERIVATIVES IN TERMS OF SPATIAL CONVOLUTION
The F o u r i e r t r a n s f o r m o f a f u n c t i o n f ( z ) i s d e f i n e d by ( s e e a p p e n d i x ) F ( kz) =
J
f ( z ) e 2 dz (3.16) o r , s y m b o l i c a l l y , f ( z ) « F ( k ). z From t h i s d e f i n i t i o n i t f o l l o w s t h a tƒ
3f J V H , jkzz •7— e dz = f ( z ) e dz j k z j k f ( z ) e d z . —CO —CO (3.17) I f f ( z ) = 0 a t t h e i n t e g r a l l i m i t s , t h e F o u r i e r t r a n s f o r m o f t h e d e r i v a t i v e o f f ( z ) w i t h r e s p e c t t o z i s g i v e n by ï ^ ï l « - j k F ( k ). dz z z (3.18) Because we a r e d e a l i n g w i t h p h y s i c a l phenomena, t h e f u n c t i o n F ( kz) w i l l be z e r o f o r h i g h e r s p a t i a l f r e q u e n c i e s (kz-*»). We i n t r o d u c e a band l i m i t e d f u n c t i o n D ( kz) d e f i n e d a s and w i t h D ( kz) = - j kz D(k ) = 0 z f o r k < k „„ z max f o r k co z ' (3.19a) (3.19b)k = t h e maximum wave number o f i n t e r e s t , max
So, we a r e a b l e t o w r i t e 3 f ( z ) / 3 z i n terms o f s p a t i a l c o n v o l u t i o n
F i g u r e 8: A band l i m i t e d f i l t e r f o r t h e second o r d e r d e r i v a t i v e i n s p a t i a l f r e q u e n c y domain ( a ) and i n space domain ( b ) .
w i t h d ( z ) » D ( kz) . (3.21) N o t e t h a t i n t h i s c a s e e q u a t i o n (3.20) i s o n l y e x a c t i f F ( kz) = 0 f o r k >k . H i g h e r o r d e r d e r i v a t i v e s c a n a l s o be o b t a i n e d by c o n v o l u t i o n z max 32f(z) = d ( z ) * d ( z ) * f ( z ) = d2( z ) * f ( z ) Jf ( z ) 3 z3 ! t C . = d ( z ) * d2( z ) * f ( z ) = d3( z ) * f ( z ) (3.22) An example o f s u c h a band l i m i t e d d e r i v a t i v e " f i l t e r " i s g i v e n i n f i g u r e 8. I n f i n i t e - d i f f e r e n c e t e c h n i q u e s a d e r i v a t i v e f i l t e r i s an i m p o r t a n t t o o l . W i t h t h e wave e q u a t i o n t h e d e r i v a t i v e o f t h e p r e s s u r e w i t h r e s p e c t t o one c o o r d i n a t e c a n be computed as a t w o - d i m e n s i o n a l c o n v o l u t i o n a l o n g t h e o t h e r two c o o r d i n a t e s .
F o r i n s t a n c e t h e wave e q u a t i o n f o r a s t a t i o n a r y homogeneous medium,
l ! l + i!*L + k2P = 0, (3.23)
3 x2 3 y2 8 z2
c a n be w r i t t e n i n terms o f s p a t i a l c o n v o l u t i o n as
i!l + { d2( y , z ) + k * 6 ( y , z ) } * P = 0 ( 3 . 2 4 )
