Chapter 13
Guided-Wave-Tube Technique for Materials Characterization R. Harrison Naval Surface Warfare Center, Silver Spring, MD 20903-5000
The acoustic properties of a layer of visocelastic material were determined from measurements made in a water f i l l e d guided wave tube. Such a tube consists of a pipe f i l l e d with water, several inches in diameter, and of a length corresponding to three or four wavelengths of the lowest frequency for which measurements are desired. Tone bursts, several cycles in duration and originating at one end of the tube, were reflected by the sample and their attenuation and phase shift measured. A method of obtaining the phase from the Fourier transform coefficients was employed with the advantage of working with a small set of data points rather than a large array. An extended frequency range was obtained by joining two pipes of unequal diameters. The propagation constants were calculated from the complex reflection coefficient. Results for silicone rubber are given. In the i n i t i a l selection of an acoustic absorbing material for an underwater application, the f i r s t considerations are often the density, and the complex dynamic shear modulus. These quantities can be measured in the laboratory,requiring only small sample sizes and hence are useful as a guide to material development. As the design matures, the direct measurement of the acoustic properties becomes necessary. These properties include the longitudinal wave speed, the coefficient of attenuation and the acoustic impedance, which can be obtained from measurements of the reflection and transmission of sound by the material. Two acoustic techniques are available for these measurements, the impedance tube and the panel test. In the panel test, a sample of the material is submerged in water and a sound projector is placed distant from the sample. A second hydrophone is placed close to the sample so that i t can receive both the transmitted and reflected tone burst. These tone This chapter not subject to U.S. copyright Published 1990 American Chemical Society
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249
b u r s t s a r e u s u a l l y two o r t h r e e c y c l e s i n time d u r a t i o n o f t h e l o w e s t f r e q u e n c y t o be measured. I n o r d e r t h a t t h e sample measurements approximate t h o s e made i n f r e e - f i e l d c o n d i t i o n s , t h e d i s t a n c e between the sample and t h e p r o j e c t o r s h o u l d be such t h a t t h e r e i s no c u r v a t u r e t o t h e sound wave and t h e sample w i d t h s h o u l d be a t l e a s t s e v e r a l w a v e l e n g t h s . However, measurements made i n a t e s t i n g tank a r e l i m i t e d by t h e s i z e o f t h e tank; a s a m p l e - p r o j e c t o r d i s t a n c e o f 170 cm and a sample s o u r c e d i s t a n c e o f 34 cm f o r a p a n e l m e a s u r i n g 76 cm by 76 cm a r e t y p i c a l o f t h e geometry c u r r e n t l y i n use. Such a geometry w i l l have an approximate lower f r e q u e n c y l i m i t o f 6 kHz, s e t by wave d i f f r a c t i o n e f f e c t s a t t h e sample edges r e a c h i n g t h e r e c e i v i n g hydrophone a t t h e same time as t h e r e f l e c t e d wave from t h e sample. M. P. H a g e l b e r g e r and R. D. C o r s a r o (1) d e s c r i b e t h e u s e o f an a v e r a g i n g l a r g e p l a n a r hydrophone w i t h a r i g i d l y f i x e d t e s t geometry. In t h i s c a s e , t e s t d a t a was o b t a i n e d a t f r e q u e n c i e s as low as 10 kHz w i t h o n l y a one f o o t square sample. H i g h e r f r e q u e n c i e s do n o t p r e s e n t a measurement p r o b l e m u n t i l t h e i r wavelengths become comparable t o g e o m e t r i c i r r e g u l a r i t i e s i n t h e sample and t e s t s t r u c t u r e a t w h i c h p o i n t t h e measurements a r e no l o n g e r c h a r a c t e r i s t i c o f t h e e n t i r e sample. While t h e p a n e l t e s t has become a c c e p t e d as a measurement s t a n d a r d , t h e a c o u s t i c impedance tube (2-5) o f f e r s a means o f i n e x p e n s i v e l y and r a p i d l y s u r v e y i n g a l a r g e v a r i e t y o f samples. These tubes a r e n o r m a l l y about two 5 cm i n d i a m e t e r and 6 meters l o n g w i t h the e l e c t r o n i c i n s t r u m e n t a t i o n s i m i l a r t o t h a t u s e d i n a p a n e l t e s t . The sample i s p l a c e d a t one end and two hydrophones a r e mounted a t the o t h e r end. S i g n a l s a r e g e n e r a t e d by one hydrophone and r e c e i v e d by t h e o t h e r . Indeed, a s i m p l e system c a n be c o n s t r u c t e d from a 6 f o o t l e n g t h o f two i n c h d i a m e t e r s t e e l p i p e , f i l l e d w i t h water and h a v i n g r u b b e r s t o p p e r s a t t h e ends. Leads from s m a l l hydrophones c a n be pushed t h r o u g h h o l e s d r i l l e d s l i g h t l y s m a l l e r t h a n t h e i r d i a m e t e r s i n t h e s t o p p e r s . Such a d e v i c e w i l l r e a d i l y i l l u s t r a t e t h e c h a r a c t e r i s t i c s and u t i l i t y o f a p u l s e tube. F o r p u l s e d o p e r a t i o n w i t h two m i l l i s e c o n d p u l s e s , e s p e c i a l l y e v i d e n t w i l l be t h e o n s e t o f n o n - p l a n a r wave p r o p a g a t i o n a t about 16 kHz as e v i d e n c e d b y t h e extreme s p r e a d i n g o f t h e echos due t o d i s p e r s i v e e f f e c t s o f t h e n o n - p l a n a r waves. METHOD Knowledge o f t h e p r o p a g a t i o n c o n s t a n t s o f a m a t e r i a l p r o v i d e s u s e f u l p h y s i c a l i n s i g h t s i n t o t h e m o l e c u l a r c o n f o r m a t i o n , s h e a r modulus and d e n s i t y . I f t h e a c o u s t i c a l impedance which i s a measure o f t h e r e s i s t a n c e o f t h e sample t o p e n e t r a t i o n b y an a c o u s t i c a l wave i s known, t h e n t h e p r o p a g a t i o n c o n s t a n t s c a n be o b t a i n e d t h r o u g h a known m a t h e m a t i c a l r e l a t i o n s h i p . One way o f o b t a i n i n g t h e impedance o f a sample i s t h r o u g h t h e use o f an impedance tube. Such a tube c o n s i s t s o f a p i p e , f i l l e d w i t h water, and o f s u f f i c i e n t l e n g t h t o a l l o w s e v e r a l wavelengths o f t h e sound wave t o p r o p a g a t e a t t h e same time i n t h e water. A sound p r o j e c t o r i s l o c a t e d a t one end o f t h e tube as i s a r e c e i v i n g hydrophone. The sample i s p l a c e d a t t h e o t h e r end. Sound waves a r e e m i t t e d from t h e p r o j e c t o r and t r a v e l t h r o u g h t h e water i n t h e tube t o t h e sample. Here t h e y a r e r e f l e c t e d b a c k t o r w a r d the r e c e i v i n g hydrophone where t h e i r r e l a t i v e a m p l i t u d e s and phase
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SOUND AND VIBRATION DAMPING WITH POLYMERS
are measured. The complex r e f l e c t i o n c o e f f i c i e n t , w h i c h i n c l u d e s b o t h the amount o f sound r e f l e c t e d from t h e sample and t h e phase s h i f t e x p e r i e n c e d b y t h e sound wave i s t h e n c a l c u l a t e d . The impedance c a n t h e n be found from t h i s c o e f f i c i e n t . We have d e v e l o p e d a water f i l l e d impedance tube system f o r m e a s u r i n g t h e p r o p a g a t i o n c o n s t a n t s as a f u n c t i o n o f f r e q u e n c y f o r p o l y m e r i c m a t e r i a l s i n c l u d i n g those which e x h i b i t h y s t e r e t i c o r o t h e r forms o f damping. F u l l y automated, t h e system uses a f a s t F o u r i e r t r a n s f o r m p r o c e d u r e t o e x t r a c t t h e phase a n g l e and c a l c u l a t e s t h e rms v a l u e o f t h e c o e f f i c i e n t o f r e f l e c t i o n b y a Simpson's r u l e i n t e g r a t i o n o f the d i g i t a l l y a c q u i r e d data. C a l c u l a t i o n o f the p r o p a g a t i o n c o n s t a n t s i s p e r f o r m e d b y a s m a l l computer code w h i c h a c c e p t s as i n p u t t h e complex c o e f f i c i e n t o f r e f l e c t i o n . The t o t a l time f o r a n a n a l y s i s o f a m a t e r i a l over two decades o f f r e q u e n c y i s l e s s t h a n an hour, w h i c h c o u l d be f u r t h e r d e c r e a s e d w i t h f a s t e r s i g n a l p r o c e s s i n g and i n s t r u m e n t a t i o n . THEORETICAL BACKGROUND We c o n s i d e r a p l a n e wave p r o p a g a t i n g a l o n g t h e z - a x i s i n a m a t e r i a l c h a r a c t e r i z e d by i t s speed o f sound c and d e n s i t y p. L e t t i n g p m
e q u a l t h e p r e s s u r e a t p o i n t z and time p = = P o
t , we c a n w r i t e :
e-(iKz-iwt)
where u> i s t h e a n g u l a r f r e q u e n c y i n r a d i a n s / s e c and K i s t h e p r o p a g a t i o n c o n s t a n t . F o r m a t e r i a l s which e x h i b i t damping, t h e p r o p a g a t i o n c o n s t a n t i s complex and i K z i s r e p l a c e d b y (a+ik)z yielding p=p e-az-ikz-iwt 0
T h i s i s t h e e q u a t i o n o f a s i n u s o i d a l p r e s s u r e wave which h a s a n i n i t i a l a m p l i t u d e o f p and decays e x p o n e n t i a l l y w i t h a decay c o n s t a n t o f a . a and k a r e known as t h e p r o p a g a t i o n c o n s t a n t s and k=o?/c i s a l s o c a l l e d t h e w a v e v e c t o r number. An a c o u s t i c wave which i s t r a v e l i n g i n a medium c h a r a c t e r i z e d b y an impedance e n c o u n t e r s a change o f impedance Z a t t h e f a c e o f t h e sample. T h i s change causes t h e wave t o be p a r t i a l l y r e f l e c t e d w i t h a complex r e f l e c t i o n c o e f f i c i e n t r , g i v e n b y Q
w
Zm"Zw
(1)
r +
^m Z
w
where Z i s t h e impedance o f t h e m a t e r i a l . F o r a m a t e r i a l w h i c h e x h i b i t s damping, t h e a c o u s t i c impedance c a n be shown t o be m
Z
0
top = i (a+ik)
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However, the l e n g t h o f the sample, i f i t i s comparable t o the l e n g t h o f the a c o u s t i c waves can cause wave i n t e r f e r e n c e e f f e c t s t o a r i s e a t the sample-water i n t e r f a c e and t h e s e must be t a k e n i n t o a c c o u n t i n a t h e o r e t i c a l d e s c r i p t i o n . The m a t h e m a t i c a l t r e a t m e n t o f t h e s e e f f e c t s can be found i n the l i t e r a t u r e o f t r a n s m i s s i o n l i n e t h e o r y which p r o v i d e s the f o l l o w i n g e x p r e s s i o n f o r the impedance o f a l e n g t h 1 o f material z
m
=
Z *coth[(a+ik)l] Q
o r n o r m a l i z i n g t o the impedance o f the water Z
m
— Z
w
iwp * coth[(a+ik)l]
(2)
(a+ik)
E q u a t i o n (2) c a n be combined w i t h e q u a t i o n (1) t o y i e l d an e x p r e s s i o n f o r r , the complex r e f l e c t i o n c o e f f i c i e n t . I n p r a c t i c e , we c a n use the impedance tube t o f i n d the complex c o e f f i c i e n t o f r e f l e c t i o n and t h e n v a r y the two p r o p a g a t i o n c o n s t a n t s i n the t h e o r y t o produce the same complex r e f l e c t i o n c o e f f i c i e n t . These v a r i a t i o n s a r e n o t easy t o p e r f o r m as the e q u a t i o n i s t r a n s c e n d e n t a l ; however, t h e r e a r e computer programs a v a i l a b l e t o do this (7). EXPERIMENTAL The d e s i g n o f a p u l s e tube i s u s u a l l y d i c t a t e d by c o n s i d e r a t i o n s o f the maximum w o r k i n g f r e q u e n c y and the sample d i a m e t e r . Where the sample c o n t a i n s v o i d s o r i n c l u s i o n s , the d i a m e t e r o f the tube must be l a r g e enough so t h a t the specimen d i a m e t e r w i l l c o n t a i n a r e p r e s e n t a t i v e number o f v o i d s o r i n l u s i o n s . On the o t h e r hand, the d i a m e t e r o f the tube d e t e r m i n e s an upper f r e q u e n c y o f o p e r a t i o n . When the tube d i a m e t e r i s s m a l l compared t o a wavelength o f the sound i n water i n the tube, o n l y p l a n e wave p r o p a g a t i o n i n the tube t a k e s p l a c e . However, when the wavelength becomes comparable t o the tube d i a m e t e r , r a d i a l modes o f p r o p a g a t i o n a r e p o s s i b l e making a c c u r a t e measurements d i f f i c u l t . I n t h i s s i t u a t i o n , the echos from the sample w i l l have a time d u r a t i o n e q u a l to l a r g e m u l t i p l e s o f the i n c i d e n t p u l s e l e n g t h . T h i s o c c u r s when the f r e q u e n c y o f o p e r a t i o n i s n e a r o r exceeds the c u t o f f f r e q u e n c y g i v e n by .581c f
cutoff = d
where c i s the speed o f sound i n the tube and d i s the d i a m e t e r o f the tube ( 8 - 9 ) . I n the c a s e where the speed o f sound i n the tube would h a v e a c u t o f f f r e q u e n c y o f 16 kHz, a 3.5 i n c h tube would have a c u t o f f f r e q u e n c y o f 9 kHz, and a 7 i n c h tube a c u t o f f f r e q u e n c y o f 4.6 kHz. These a r e t h e o r e t i c a l l i m i t s and the e x p e r i m e n t a l l y o b s e r v e d l i m i t s w i l l be d i f f e r e n t . F i g u r e 1 shows t h i s i n f o r m a t i o n graphically. The l e n g t h o f the tube i s d e t e r m i n e d by the r e q u i r e m e n t t h a t the i n t e r r o g a t i n g p u l s e c o m p l e t e l y i n s o n i f y the sample thus a l l o w i n g i n t e r f e r e n c e p a t t e r n s a r i s i n g w i t h i n the sample t o be g e n e r a t e d . F o r
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SOUND AND VIBRATION DAMPING WITH POLYMERS
FIGURE 1. Pulse tube cutofffrequencyvs. pulse tube diameter. Radial wave formation takes place atfrequencieshigher than the cutoff frequency.
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t h i s r e a s o n , use o f v e r y s h o r t p u l s e s s h o u l d be r e s t r i c t e d t o s i t u a t i o n s where t h i s i s n o t a problem. I n g e n e r a l , f o r t h e f r e q u e n c y range o f 4 kHz t o 60 kHz, t h r e e c y c l e s w i l l c o m p l e t e l y i n s o n i f y a sample s e v e r a l c e n t i m e t e r s t h i c k . Thus t h e tube l e n g t h must be a t l e a s t t h r e e w a v e l e n g t h s l o n g a t the l o w e s t f r e q u e n c y o f i n t e r e s t . O r i g i n a l l y , p u l s e tubes were c o n s t r u c t e d w i t h a w a l l t h i c k n e s s e q u a l t o t h e i r i n s i d e d i a m e t e r . T h i s p r e v e n t e d t h e tube from b u l g i n g as t h e p r e s s u r e wave t r a v e l e d from t h e t r a n s d u c e r t o t h e sample. R e c e n t l y , i n t h i s l a b o r a t o r y and a t o t h e r s , tubes w i t h t h i n w a l l s o f o n l y one o r two c e n t i m e t e r s have been u s e d . T h i s p r e v e n t s r i n g i n g i n t h i c k o u t e r w a l l s b u t a l l o w s b u l g i n g as t h e p r e s s u r e wave t r a v e l s t h e l e n g t h o f t h e tube. Any such b u l g i n g o f t h e tube would lower t h e p r o p a g a t i o n v e l o c i t y o f sound i n the tube and, as t h e impedance i s the p r o d u c t o f t h e speed o f sound m u l t i p l i e d by t h e d e n s i t y , t h e impedance o f t h e water t o sound. As a d i f f e r e n c e i n impedance between the sample and t h e water w i l l g i v e r i s e t o a r e f l e c t i o n , t h e v e l o c i t y o f sound i n t h e tube must be a c c u r a t e l y known and c e r t a i n l y u s e d i n any c a l c u l a t i o n s f o r sample impedance i n any p u l s e tube ( 2 - 3 ) . The impedance tube d e v e l o p e d c o n s i s t s o f a 7 f o o t l o n g s e c t i o n o f 2" b o r e s t a i n l e s s s t e e l t u b i n g c o u p l e d t o a 6 f o o t l o n g s e c t i o n o f 1/2" b o r e s t a i n l e s s s t e e l t u b i n g . C o n n e c t i n g t h e two tubes i s an 18" l o n g t r a n s i t i o n s e c t i o n so d e s i g n e d t h a t i t f l a r e s from 2" I.D. a t one end t o 1/2" I.D. a t t h e o t h e r end. W i t h t h e e x c e p t i o n o f t h e t r a n s i t i o n s e c t i o n which has an o u t s i d e d i a m e t e r o f 6 i n c h e s , t h e 1/2" s e c t i o n o f t u b i n g has a w a l l t h i c k n e s s o f 1/4" w h i l e t h e 2" s e c t i o n o f t u b i n g has a w a l l t h i c k n e s s o f 1/2". These two w a l l t h i c k n e s s e s mean t h a t t h e tube i s c h a r a c t e r i z e d as a t h i n w a l l tube w i t h a v e l o c i t y o f sound i n t h e tube s l i g h t l y lower t h a n i t would be f o r a c o r r e s p o n d i n g t h i c k w a l l tube. I n o u r case, t h e 2" tube p e r m i t t e d a r e a s o n a b l e sample s i z e w h i l e t h e 1/2" tube a l l o w e d us t o c o l l e c t d a t a up t o 60 kHz. The f i f t e e n f o o t l e n g t h o f t h r e e u n e q u a l tubes was chosen t o o b t a i n a low f r e q u e n c y l i m i t o f 2 kHz and t o l o c a t e t h e s i g n a l echo i n a time span away from o t h e r echos. F i g u r e 2 i s an i l l u s t r a t i o n o f t h e appearance o f t h e computer d i s p l a y t e r m i n a l a f t e r t h e a c q u i s i t i o n o f t h e d a t a from t h e waveform r e c o r d e r . Note i n p a r t i c u l a r how t h e echo from t h e sample i s l o c a t e d i n a q u i e t r e g i o n away from o t h e r echos which c o u l d cause measurement e r r o r s . The f i r s t s i g n a l on t h e r i g h t i s l e a k a g e t h r o u g h t h e T-R s w i t c h from t h e t r a n s m i t t e d p u l s e . The n e x t l a r g e s i g n a l i s a r e f l e c t i o n from t h e t r a n s i t i o n s e c t i o n due t o t h e impedance mis- match o f t h i s s e c t i o n . The t h i r d s i g n a l , l o c a t e d i n t h e gate, i s t h e d e s i r e d echo. O b s e r v a t i o n o f t h e waves a r r i v i n g a t t h e t r a n s d u c e r , w i l l show an i n i t i a l l y l a r g e wave due t o i n c o m p l e t e c a n c e l l a t i o n o f t h e o u t p u t p u l s e i n t h e t r a n s m i t - r e c e i v e s w i t c h , an echo from t h e t r a n s i t i o n s e c t i o n and o t h e r echo t r a v e l i n g i n t h e s t e e l w a l l s o f t h e tube. The echoes i n t h e w a l l s o f t h e tube w i l l be f o u n d t o be superimposed on the sample echo i f t h e l e n g t h o f the tube/tube s e c t i o n s i s n o t c h o s e n w i t h c a r e . The p a r t i a l l y c a n c e l l e d o u t p u t echo must be b l o c k e d o u t i n the p r e a m p l i f i e r as i t s a m p l i t u d e would h i n d e r a c c u r a t e measurement o f t h e sample echo. T h i s c a n be done by i n t r o d u c i n g a n e g a t i v e g a t e i n the p r e a m p l i f i e r t o b i a s o f f the input t r a n s i s t o r s . Any a i r i n t h e tube o r i n t h e water w i l l cause g r i e v o u s measurement problems as t h e sound p u l s e s w i l l be r e f l e c t e d from a b u b b l e o f a i r . ( T h i s c a n be checked by o b s e r v i n g t h e echo on an o s c i l l i s c o p e w h i l e t h e tube p r e s s u r e i s v a r i e d . ) Our tube i s
1 1/2' 1
EXPONENTIALLY SHAPED TRANSITION SECTION 1/2"WALL THICKNESS
-
ALL TUBING TYPE 303 STAINLESS STEEL
r
7' SAMPLE SECTION WITH 1/2" WALL THICKNESS
FIGURE 2. The j o i n i n g o f two tubes o f u n e q u a l s i z e a l l o w s o p e r a t i o n up t o 60 kHz a n d a two i n c h d i a m e t e r sample s i z e .
T LOW PASS FILTER SECTION WITH 1/4" WALL THICKNESS
ELECTRICAL CONNECTIONS FOR HYDROPHONE
2" THICK BACKING PLATE FOR MOUNTING SAMPLES
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e v a c u a t e d and t h e n f i l l e d w i t h d e - a e r a t e d water, p r o d u c e d b y e i t h e r b o i l i n g o r by d e g a s s i n g u s i n g a m e c h a n i c a l vacuum pump. A s m a l l s t a i n l e s s s t e e l hand pump from Enerpac i s u s e d f o r p r e s s u r i z a t i o n ; p i s t o n t y p e , a i r d r i v e n , s t a i n l e s s s t e e l pumps c o u l d a l s o be used. For t r a n s f e r r i n g water, a p o s i t i v e d i s p l a c e m e n t , screw type pump i s n e c e s s a r y : c e n t r i f u g a l pumps w i l l i n t r o d u c e c a v i t a t i o n when t h e y have e i t h e r a r e s t r i c t e d imput o r o u t p u t stream. Even w i t h a screw t y p e pump, i t i s a d v i s a b l e t o i n s e r t a b u b b l e t r a p between t h e pump and the tube. Such a t r a p c o n s i s t s o f a s t e e l c y l i n d e r , perhaps a f o o t h i g h , i n which t h e water i s pumped i n t o n e a r t h e t o p and e x t r a c t e d from t h e bottom. Bubbles w i l l r i s e t o t h e t o p where t h e a i r t h e y c a r r y c a n be removed by a v a l v e on t h e c y l i n d e r . I n a d d i t i o n , perhaps due t o t h e w a l l t h i c k n e s s o f 1/2", we have found i t n e c e s s a r y t o p r e s s u r i z e t h e tube t o 100 p s i t o o b t a i n a c c u r a t e d a t a . Bath c l e a r (VWR S c i e n t i f i c , I n c . ) o r i o d i n e c a n be added t o t h e water t o c o n t r o l b a c t e r i a l growth. Two m i l l i s e c o n d p u l s e s o f v a r y i n g f r e q u e n c y were u s e d t o i n t e r r o g a t e t h e sample. The l e n g t h o f time o f t h e p u l s e i s s e t b o t h by t h e need f o r complete i n s o n n i f i c a t i o n o f t h e sample and f o r s t e a d y s t a t e c o n d i t i o n s t o o c c u r . A B r u e l & K j a e r Model 8103 hydrophone i s l o c a t e d a t one end o f t h e tube and i s u s e d f o r b o t h s e n d i n g and r e c e i v i n g . A d i o d e a r r a y ( F i g u r e 3) i s u s e d as a t r a n s m i t - r e c e i v e s w i t c h and s e r v e s b o t h t o s h o r t o u t t h e preamp d u r i n g t h e t r a n s m i t time and t o d i s c o n n e c t t h e t r a n s m i t t e r d u r i n g t h e r e c e i v i n g t i m e . A H e w l e t t - P a c k a r d s y n t h e s i z e r under computer c o n t r o l g e n e r a t e s a c o n t i n u o u s s i n e wave which i s t h e n modulated by G e n e r a l Radio tone b u r s t g e n e r a t o r s . A K r o h n - H i t e 75 watt a m p l i f i e r p r o v i d e s u n u s a l l y c l e a n o u t p u t s i g n a l s t o t h e T r a n s m i t - R e c e i v e s w i t c h . The r e c e i v e d s i g n a l i s f i l t e r e d by an e l l i p t i c a l l y d e s i g n e d f i l t e r and t h e n c a p t u r e d by a B i o m a t i o n waveform r e c o r d e r where i t i s d i g i t i z e d and t r a n s m i t t e d t o t h e computer. Because o f t h e need f o r phase measurements,accurate t i m i n g i s c r u c i a l t o t h e e x p e r i m e n t . I t was f o u n d t h a t t h e b e s t system was t o use a d i f f e r e n t i a l s c a n n i n g analyzer developed f o r n u c l e a r p h y s i c s research to o b t a i n a t i m i n g p u l s e from t h e p o s i t i v e s l o p e o f t h e modulated s i g n a l . T h i s p u l s e was u s e d t o i n i t i a t e waveform r e c o r d i n g by t h e waveform r e c o r d e r . The a c q u i s i t i o n system a l l o w e d a gate t o be s e t i n t h e r e c o r d e d d a t a as shown i n F i g u r e 4. The rms v a l u e o f t h e s i g n a l was c a l c u l a t e d w i t h i n t h i s g a t e . However, t h e F o u r i e r t r a n s f o r m i s a l s o t a k e n o f t h e s i g n a l w i t h i n t h e gate so t h a t t h e gate l e n g t h must be an i n t e g r a l number o f h a l f - w a v e l e n g t h s t o a v o i d l e a k a g e . Furthermore, t h e f r e q u e n c y o f t h e s y n t h e s i z e r must be s e t a t an i n t e g e r m u l t i p l e o f the f r e q u e n c y g i v e n by 1.0 freq = time spanned by gate The phase i s f o u n d by c o n s i d e r i n g o n l y t h e t r a n s f o r m a t t h e s y n t h e s i z e r f r e q u e n c y and n o t i n g t h a t :
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Guided-Wave-Tube Technique
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