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IMPULSE RESPONSE OF THE ACOUSTIC DIFFRACTION : A THEORETICAL ESTIMATION
AND ITS EXPERIMENTAL VERIFICATION
T. Otani, S. Kubota
To cite this version:
T. Otani, S. Kubota. IMPULSE RESPONSE OF THE ACOUSTIC DIFFRACTION : A THEORET-
ICAL ESTIMATION AND ITS EXPERIMENTAL VERIFICATION. Journal de Physique Colloques,
1990, 51 (C2), pp.C2-423-C2-426. �10.1051/jphyscol:19902100�. �jpa-00230380�
COLLOQUE DE PHYSIQUE
Colloque C 2 , suppl6ment au n02, Tome 51, F6vrier 1990 l e r Congres Franqais dlAcoustique 1990
IMPULSE .RESPONSE OF THE ACOUSTIC DIFFRACTION : A THEORETICAL ESTIMATION AND ITS EXPERIMENTAL VERIFICATION
T. OTANI and S. KUBOTA
Department o f E l e c t r i c a l Engineering, Faculty o f Engineering, Doshisha U n i v e r s i t y , Kamigyo-ku, 602 Kyoto, Japan
RBsum6
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Une f o r m u l a t i o n t h g o r i q u e d e l a d i f f r a c t i o n e s t p r g s e n t 6 e p o u r o b t e n i r l a r 6 p o n s e i m p u l s i o n n e l l e d e l a d i f f r a c t i o n p a r un b o r d l i n 6 a i r e d l u n 6 c r a n m i n c e . L e s c a l c u l s t h g o r i q u e s s o n t v e r i f i g s e x p g r i m e n t a l e - ment p a r o b s e r v a t i o n d i r e c t e d e l a r g p o n s e i m p u l s i o n n e l l e B l ' a i d e d ' u n e m i c r o - 6 t i n c e l l e comme l a s o u r c e d ' i m p u l s i o n .A b s t r a c t
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T h e o r e t i c a l f o r m u l a t i o n o f t h e d i f f r a c t i o n i s p r e s e n t e d t o o b t a i n t h e i m p u l s e r e s p o n s e o f t h e d i f f r a c t i o n a t a s t r a i g h t e d g e . T h e o r e t i c a l c a l c u l a t i o n s a r e e x p e r i m e n t a l l y v e r i f i e d by d i r e c t o b s e r v a - t i o n o f t h e i m p u l s e r e s p o n s e u s i n g a m i c r o s p a r k .1
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INTRODUCTIONThe phenomenon o f d i f f r a c t i o n o r s c a t t e r i n g i s n o t o n l y a n i m p o r t a n t s u b j e c t a s a f u n d a m e n t a l c o n c e p t p e c u l i a r t o t h e wave p r o p a g a t i o n , b u t a l s o a n i m p o r t a n t knowledge f o r a n a p p l i c a t i o n o f t h e a c o u s t i c wave o r f o r t h e a c o u s t i c i m a g i n g . The s i g n a l p r o c e s s i n g o r t h e a n a l y s i s o f echo s i g n a l s i n t h e a c o u s - t i c imaging s u c h a s s o n a r , n o n d e s t r u c t i v e i n s p e c t i o n , m e d i c a l d i a g n o s i s , i s t o o b t a i n c l u e s o n a p o s i t i o n , a s h a p e a n d a s t r u c t u r e o f e a c h t a r g e t i n o r d e r t o r e c o n s t r u c t t h e t a r g e t image. E s s e n t i a l l y , t h e r e c o n s t r u c t i o n o f t a r g e t s i s c o n s i d e r e d a s a n i n v e r s e o p e r a t i o n o f wave p r o p a g a t i o n o r a n i n v e r s e p r o b l e m o f d i f f r a c t i o n o r s c a t t e r i n g . The t h e o r e t i c a l a n a l y s i s o f d i f f r a c t e d a c o u s - t i c f i e l d i s a p r i o r knowledge f o r t h e i n v e r s e e s t i m a t i o n o f t a r g e t s .
K i r c h h o f f - H e l m h o l t z i n t e g r a l i s g e n e r a l l y u s e d f o r t h e o r e t i c a l c a l c u l a t i o n o f t h e d i f f r a c t i o n . However, it i s n o t e v i t a b l e t o i n t r o d u c e more o r l e s s a p p r o x i m a t i o n t o a c h i e v e t h e c a l c u l a t i o n e x c e p t some s p e c i a l c a s e s . T h e r e f o r e ,
it i s d e s i r a b l e t o c o n f i r m t h e t h e o r e t i c a l r e s u l t s by e x p e r i m e n t . A sound s o u r c e t o v e r i f i y t h e t h e o r e t i c a l e s t i m a t i o n o f d i f f r a c t i o n i s r e q u i r e d t o h a v e a u n i f o r m s p h e r i c a l wave f r o n t o v e r a w i d e f r e q u e n c y r a n g e . We h a v e r e a l i z e d a p o i n t l i k e s o u r c e a s a p r a c t i c a l s p h e r i c a l wave s o u r c e by u s i n g a s m a l l s p a r k gap d i s c h a r g e ( a m i c r o s p a r k ) which e m i t s a v e r y b r i e f a c o u s t i c i m p u l s e . We r e p o r t h e r e a t h e o r e t i c a l e s t i m a t i o n i n t h e t i m e domain f u n c - t i o n and some r e s u l t s o f e x p e r i m e n t a l o b s e r v a t i o n o f t h e i m p u l s e r e s p o n s e o f t h e d i f f r a c t i o n .
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TIME DOMAIN ANALYSIS OF THE DIFFRACTION BY A STRAIGHT EDGED i f f r a c t e d f i e l d i s e x p r e s s e d by t h e K i r c h h o f f i n t e g r a l a t t h e s t a r t i n g p o i n t o f t h e a n a l y s i s a s t h e g e n e r a l b a s i c f o r m u l a t i o n . The a c o u s t i c p r e s s u r e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902100
C2-424 COLLOQUE DE PHYSIQUE
p(xp, yp, zp, t ) a t a o b s e r v a t i o n p o i n t P i s w r i t t e n i n terms of t h e i n t e g r a l o v e r a s u r f a c e S a s
,.
where [ p ] d e n o t e s p(x, y , z , t - % I o r a r e t a r d e d p r e s s u r e , a t t h e p o i n t ( x , y , z ) on t h e s u r f a c e S and
[El
=& Lp]. We c o n s i d e r a c a s e of a p l a n e wave i n c i - dence whose wave f r o n t i s p a r a l l e l t o t h e s t r a i g h t edge of a r i g i d h a l f p l a n e
i n t h e x - y p l a n e a s shown i n F i g . 1 . The a c o u s t i c p r e s s u r e of d i f f r a c t e d wave a t t h e p o i n t P i s g i v e n a s
where [ p ] i s t h e i n c i d e n t wave p r e s s u r e on t h e x-y p l a n e a t z = 0 i n t e r m s of r e t a r d e d v a r i a b l e t
- 5 .
S u r f a c e element d s i s e x p r e s s e d by a n a n n u l u s of r a d i u s R and w i d t h as a sdS = 2 R x d R = 2 r x dr rosin 0 - - ro sin 8
where x = -
J r 2- ( rOcos B )Z J I - ( O c o s e j2 r
Thus t h e d o u b l e i n t e g r a l o f Eq. (2) becomes a s t h e s i g l e i n t e g r a l . Then t h e d i f f r a c t e d wave i s e x p r e s s e d by
3
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THEORETICAL ESTIMATION AND EXPERIMENTAL VERIFICATIONWe have r e a l i z e d an a c o u s t i c impulse s o u r c e by u s i n g a s m a l l s p a r k d i s - c h a r g e gap t o o b t a i n e x p e r i m e n t a l l y t h e impulse r e s p o n s e of wave phenomenon and have v e r i f i e d by experiment t h a t a micro s p a r k e m i t s a s p h e r i c a l l y s p r e a d - i n g o m n i d i r e c t i o n a l wave w i t h a s i m p l e p u l s e waveform and a v e r y s h o r t p u l s e width.'") F i g u r e 2 shows t y p i c a l examples of e m i t t e d waveform by a micro s p a r k and t h e i r f r e q u e n c y s p e c t r a . I n t h i s e x p e r i m e n t a l s t u d y , a micro s p a r k was s e t a t a d i s t a n c e o f O.S(m) from t h e s t r a i g h t edge. T h i s d i s t a n c e i s f a r enough t o r e g a r d t h e wave f r o n t a s t h e p l a n e wave i n c i d e n c e . The ob- s e r v a t i o n p o i n t P was a t 0.1 (m) from t h e s t r a i g h t edge o r r, =O.l (m). Wave- forms have been o b s e r v e d by 1/4" condenser microphone of B & K 4 1 3 5 .
E x p e r i m e n t a l r e s u l t s of t h e d i f f r a c t i o n by t h e s t r a i g h t edge a r e shown i n F i g . 3 and r e s u l t s o f t h e o r e t i c a l s e t i m a t i o n a r e shown i n F i g . 4 . The r e p r e - s e n t a t i o n of d i f f r a c t e d wave i n Eq. (4) a p p l i e d t o t h e t h e o r e t i c a l e s t i m a t i o n i s v a l i d f o r t h e p l a n e wave i n c i d e n c e . Then some s l i g h t d i s c r e p a n c y must e x i s t i n t h e d e c a y i n g t a i l o f t h e e s t i m a t e d waveforms.
Fig. 1. Geometry of the diffraction by the straight edge of a rigid half plane.
100
Frequency [kHz]
a) Capacitor = 0.1 ,u F , Resistance = 2 Q Charging voltage = 1 B V
10 100
Frequency [kHz]
b) Capacitor = 4 ,u F , Resistance = 10 52 Charging voltage = 400 V
H 2 0 p sec
Fig. 2. Acoustic pressure waveforms emitted by a micro spark and their frequency spectrums.
COLLOQUE DE PHYSIQUE
H 20 p sec
( a ) I n c i d e n t wave.
H 20 p sec
( b ) D i f f r a c t e d wave.
Fig. 3 . Diffracted wave by the edge of a half plane.
(Experimentally observed waveforms)
t--l 20 p sec
Fig. 4 . Diffracted wave by the edge of a half plane.
(Theoretically estimated waveforms)
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CONCLUSIONT h e o r e t i c a l f o r m u l a t i o n o f t h e i m p u l s e r e s p o n s e o f d i f f r a c t i o n i s con- s i d e r e d a s a s o r t o f t h e c o n v o l u t i o n i n t e g r a l o v e r a s p a c e d i s t r i b u t i o n o f s e c o n d a r y s o u r c e s on t h e boundary. Then, t h e c o n c e p t o f t h e waveform d e f o r - m a t i o n o f i m p u l s i v e wave i s i l l u s t r a t e d by t h e i m p u l s e r e s p o n s e o f t h e d i f - f r a c t i n g boundary and t h e c o n v o l u t i o n i n t e g r a l . S i n c e t h e i m p u l s e r e s p o n s e i s d i r e c t l y r e l a t e d t o t h e boundary g e o m e t r y , a c o n t r a s t between t h e i m p u l s e r e s p o n s e and t h e boundary geometry make it p o s s i b l e t o g r a s p i n t u i t i v e l y t h e wave phenomenon o f d i f f r a c t i o n . Moreover, t h e t i m e domain a n a n l y s i s i s a d i r e c t method t o o b t a i n t h e t h e o r e t i c a l e s t i m a t i o n o f t h e i m p u l s e r e s p o n s e o f d i f f r a c t i o n . A c c o r d i n g l y , t h e t i m e domain a n a l y s i s , b o t h i n t h e o r e t i c a l e s t i m a t i o n and i n i t s e x p e r i m e n t a l v e r i f i c a t i o n i s r e g a r d e d a s a s i m p l e r and a c l e a r e r method a s compared w i t h t h e f r e q u e n c y domain a n a l y s i s o r t h e harmon- i c r e p r e s e n t a t i o n o f wave p r o p a g a t i o n which demands t h e i n v e r s e F o u r i e r t r a n s - form t o o b t a i n t h e i m p u l s e r e s p o n s e .
REFERENCES
1) O t a n i , T . and Takeda, M . , Revue d T A c o u s t i q u e , h o r s s & r i e l l e Congrgs I n t e r - n a t i o n a l d T A c o u s t i q u e , v o l . l ( 1 9 8 3 ) 341.
2 ) O t a n i , T . and F u j i o , K . , P r o c e e d i n g s o f t h e 1 3 t h I n t e r n a t i o n a l Congress on A c o u s t i c s , v o l . 4 (1989) 4 5 7 .