COMPARISON OF EMPIRICAL AND THEORETICAL LAWS OF PARAMETER VARIATION OF EXPLOSION WAVES IN THE SEA

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COMPARISON OF EMPIRICAL AND THEORETICAL LAWS OF PARAMETER

VARIATION OF EXPLOSION WAVES IN THE SEA

V. Fridman

To cite this version:

V. Fridman. COMPARISON OF EMPIRICAL AND THEORETICAL LAWS OF PARAMETER

VARIATION OF EXPLOSION WAVES IN THE SEA. Journal de Physique Colloques, 1979, 40 (C8),

pp.C8-62-C8-67. �10.1051/jphyscol:1979813�. �jpa-00219518�

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JOURNAL DE PHYSIQUE Colloque C8, supplementau n°ll, tome 40, novembre 1979, page C8- 62

COMPARISON OF EMPIRICAL AND THEORETICAL LAWS OF PARAMETER VARIATION OF EXPLOSION WAVES IN THE SEA,

V.E. Fridman

Radiophysioal Reseavah Institute, Lyadov Street, 25/14, Gorky, USSR.

Résumé.- On a comparé les relations théoriques obtenues par les méthodes géométriques de l ' a c o u s t i - que non linéaire aux l o i s empiriques donnant la variation spatiale des paramètres des signaux hy- droacoustiques résultant de sources explosives. Pour cette comparaison on a f i x é un point proche de la source, pour lequel le nombre de Mach est très inférieur à l ' u n i t é . Après avoir adapté les valeurs théoriques aux valeurs empiriques obtenues pour ce point, la minimalisation des différences entre ces valeurs est possible sur un intervalle d'espace assez grand. Une t e l l e minimalisation peut être obtenue en choisissant dans les formules théoriques un paramètre ajustable qui détermine l'amplitude et la durée caractéristique du signal tomme fonction de la masse de la charge. La com- paraison des résultats montre un bon accord entre les formules théoriques et empiriques donnant les variations en amplitude et en énergie d'une onde d'explosion. Quelques écarts apparaissent en ce qui concerne les variations de la constante de temps et de la quantité de mouvement, ce qui peut être dû au f a i t d'avoir négligé les pulsations des bulles gazeuses.

Abstract.- Theoretical relations obtained by means of nonlinear geometrical acoustic method are compared with empirical ones for spatial variation of parameters of hydroacoustic signals from explosion sources. At the comparison a fixed value of the coordinate near the source is chosen, where the Mach number is considerably less than unit. After matching of parameters values obtained by theoretical and empirical relations at this point the minimazation of the difference between these values is possible for rather a large spatial interval. Such minimization may be obtained by a

choice of an adjustable parameter in the theoretical formulae which determines the amplitude and the characteristic duration of the signal as a function of the charge weight. Comparison of results shows a good agreement of empirical and theoretical formulae for the amplitude and the energy variation of an explosive wave. Some disagreement takes place for the time constant and momentum variations which may be caused by neglecting the gas bubble pulsations.

1.- To calculate the explosive wave parameters in the ocean, empirical scaling laws are usually used (see, for example, /1,2/). Coefficients in these laws are defined by averaging over a large ensemble of experimental data obtained in different hydrological and athymetric conditions in different parts of oceans and in laboratory basin. These laws are confirmed by checking in the laboratory experiments with explosive waves in a homogeneous medium, where reflections from the bath walls are carrefully se- lected. It is clear that the averaging of results of different measurements leads to relations which do not take into account the inhomogeneity of the ocean medium, its surface, the bottom, etc. Thus, the accuracy of calculation of explosive waves un- der the conditions of the real ocean is not high.

One may speak only on the qualitative agreement of the measurement results with the empirical relations.

The consideration of the hydrological and other peculiarities when explosive waves propagate in the concrete part of the ocean is possible calculating by the method of the nonlinear geometrical acoustics (see, for example, /3-7/). The method given permits to predict the fundamental physical effects

observed at the propagation of explosive waves in the homogeneous ocean. As an example we may take the interpretation of the anomalous behaviour of a signal in the caustic region / 8 / which was observed at the natural measurements /9,10/. However, calculation formulas of the nonlinear geometrical acoustics (NGA) are not associated with the weigh and the type of the explosive charge. Using these formulas when calculating the explosive waves in the real ocean it is necessary :

1) to make sure of their quantitative coincidence when compared with the empirical relations, 2) to associate these formulas with parameters of

the explosive sources.

The necessity of comparison of the theoretical and empirical relations has attracted attention for a long time /11-13/. However, in these papers the comparison of asymptotic relations has been made only for the pressure disturbances in an explosive wave. The given paper presents the comparison of the empirical and theoretical relations for the amplitude of the pressure disturbance, the time constant, the impulse and the energy of the exponential form shock wave from the explosive source.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979813

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JOURNAL DE PHYSIOUE T h i s comparison gives a good coincidence o f r e l a -

t i o n s f o r t h e amplitude o f the pressure disturbance and the energy o f t h e explosive wave i n a l l t h e range o f distances and t h e charge weight t y p i c a l o f t h e l a r g e number o f experiments. Results o f t h e can- p a r i s o n made p e r m i t t o o b t a i n a r e l a t i o n o f parame- t e r s appearing i n f o r m u l a r o f t h e n o n l i n e a r geome- t r i c a l a c o u s t i c s w i t h t h e charge weight. When d e r i - v i n g t h e formulas i t i s assumed t h a t a t a small dis- tance from t h e e x p l o s i v e source t h e ocean medium may be considered homogeneous and e m p i r i c a l re1 a t i o n s may be used. Everywhere beside t h i s small area around t h e source t h e c a l c u l a t i o n i s c a r r i e d o u t by NGA method. A t boundaries o f t h i s r e g i o n "matching"

p f e m p i r i c a l and t h e o r e t i c a l s o l u t i o n s i s made. The r e l a t i o n o f NGA formulas w i t h parameters o f explo- s i v e sources p e r m i t s o t broaden t h e p o s s i b i l i t i e s o f c a l c u l a t i o n s , i . e . t o t a k e i n t o account t h e medium inhomogeneity, t h e depth o f t h e d e t o n a t i o n o f t h e e x p l o s i v e source, t h e presence o f t h e r a y s t r u c t u r e , e t c . It should be noted t h a t t h e r e l a t i o n o f t h e t h e o r e t i c a l conclusions w i t h the weight o f t h e ex- p l o s i v e charge may be found u s i n g t h e t h e o r y o f a s t r o n g explosion. However, t h e present r e l a t i o n i s e s s e n t i a l l y simple.

2.

-

E m p i r i c a l r e l a t i o n s describe t h e parameter changes o f t h e e x p l o s i v e impulse which has t h e po- n e n t i a l form :

? ( t ) = Pe exp (-t/Te).

F o r e x p l o s i v e charges THT e m p i r i c a l r e l a t i o n s f o r t h e amplitude o f t h e pressure d i s t u r b a n c e Pe

,

t h e t i m e constant Te

,

t h e impulse

I,=

j P

( t ) d t

0

and the energy

We = ( c u ) - l

1

^P' ( t ) d t

0

have t h e f o l l o w i n g form /11/ :

We (Pa .m) =

l o 5 Q1l3

( Q ~ / ~ / R ) ' , ~

Here c and p a r e t h e sound v e l o c i t y and t h e medium d e n s i t y . I n r e l a t i o n ( I ) t h e charge weight Q i s i n kilograms o f t h e mass, t h e d i s t a n c e from t h e source R i s i n meters. I n r e l a t i o n ( I ) t h e l a v e - rage value o f the c o e f f i c i e n t and exponents a r e g i v e n according t o t h e data o f d i f f e r e n t authors (see /11,12/). The spread of c o e f f i c i e n t v a l u e i s w i t h i n 20% i n t e r v a l /12,14/ though t h e maximum spread i s considerably higher. For example, expe- r i m e n t a l values o f t h e time constant have t h e maximum spread h i g h e r than 20% /1,14/. The spread o f t h e exponents i s comparatively small ( s m a l l e r than 3%) i n r e l a t i o n s f o r t h e amplitude and t h e energy, b u t considerably l a r g e r i n r e l a t i o n s f o r t h e im- p u l s e (13%) and t h e time constant (20%) /11/. Au- t h o r s t a k e t h e range o f distances a t which t h e f u n c t i o n o f (I) were measured d i f f e r e n t l y : t h e value R / Q ' / ~ v a r i e s i n t h e experiments from 0.7 t o 3.5

l o 4

m/kg1l3 /1,2,11,12,14,15/.

3.- The dependence o f parameters o f a c o u s t i c waves on t h e d i s t a n c e when propagating i n t h e inhomoge- neous ocean i s c a l c u l a t e d i n the frames o f t h e method o f the n o n l i n e a r geometrical acoustics /5,16/.

For t h e e x p l o s i v e pressure wave t h e p r o f i l e o f which i s described by the exgonential curve near t h e source

P = Pm exp (-t/Tm)

t h e dependence of t h e amplitude PT, t h e t i m e cons- t a n t TT

,

t h e impulse IT and t h e energy WT on t h e d i s t a n c e a r e given by t h e f o l l o w i n g expressions /16/ :

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V.E. Fridman

r e t i c a l and e m p i r i c a l laws a r e coincided. I n t h e designations given expressions ( I ) and (2) have t h e

de' ( 2 ) form :

pT = 2 / ( 1 + ( 1 + 2 j ~ n x f / ~ 1 ; pe = x -0,13 Here Lo i s t h e coordinate close t o t h e .source

which s e t t h e boundary c o n d i t i o n s ,

e

= ( 1 + 2 ~ & n x f / ~ - z ~ & x / L [ l + ( l + ~ j L n x ) ~ / ~ 1 ;

T

i s t h e c h a r a c t e r i s t i c scale o f t h e n o n l i n e a r i t y , P and c are t h e d e n s i t y and t h e sound v e l o c i t y , a i s t h e n o n l i n e a r parameter, x i s t h e s e c t i o n area o f t h e r a y tube d e f i n e d by t h e inhomogeneity o f t h e sound v e l o c i t y c ( L ) . The index "0" denoted values a t the d i s t a n c e Lo from t h e source o f t h e sound. R e l a t i o n s (2) take i n t o account t h e n o n l i - near d i s s i p a t i o n and t h e increase o f t h e s i g n a l d u r a t i o n a t t h e propagation.

4.- T h e o r e t i c a l dependences (2) a r e d e f i n e d by two parameters to and R*. The coordinate Lo i s chosen from t h e f o l l o w i n g c o n d i t i o n : t h e Mach num- b e r a t t h a t p o i n t must be e s s e n t i a l 1 y s m a l l e r than u n i t and t h e value Lo must be e s s e n t i a l l y smaller than t h e scale o f t h e medium inhomogeneity. To ob- t a i n c a l c u l a t i o n a l formulas i t i s necessary t o associate the amplitude Pm and t h e time constant Tm a t t h e p o i n t w i t h t h e charge weight. For t h i s purpose one may use e m p i r i c a l laws t a k i n g i n t o account t h a t a t t h e comparatively small d i s t a n c e Lo from t h e e x p l o s i v e source t h e inhomogeneity o f t h e ocean i s n o t s i g n i f i c a n t and t h e amplitude i s a l - ready s u f f i c i e n t l y small. Thus, one may use r e l a - t i o n s f o r weak waves. L e t us d e f i n e the r e l a t i o n o f t h e parameters Lo and R* w i t h t h e charge weight.

F o r t h i s we pass i n ( 2 ) t o t h e c o n s i d e r a t i o n o f a s p h e r i c a l wave i n a homogeneous medium ( l = R ) and compare t h e expressions o b t a i n e d w i t h t h e cor- responding e m p i r i c a l laws ( I ) . L e t us i n t r o d u c e dimensionless values

R e l a t i o n s (3) a r e comparable s i n c e e m p i r i c a l 1 aws i n t h i s form do n o t depend on t h e charge weight Q.

Since t h e o r e t i c a l laws a r e described by one-parameter f a m i l y o f curves d e f i n e d by t h e r e l a t i o n

then choosing t h e value j one may c o i n c i d e theo- r e t i c a l and e m p i r i c a l laws f o r t h e given medium.

The c r i t e r i o n o f closeness o f these 1 aws wi 11 be t h e ninimum o f the f o l l o w i n g f u n c t i o n s :

I f we r e q u i r e t h a t t h e f u n c t i o n #(j,x) a t t h e d e f i n i t e i n t e r v a l of distances does n o t exceed 10% Pe- ( t h a t corresponds t o t h e spread o f t h e experimental data) we o b t a i n t h e numerical value j.

For t h e i n t e r v a l o f d i s t a n c e s from Ro up t o 10 4 R i t h e given c r i t e r i o n i s f u l f i l l e a t

Consideration o f the value spread f o t h e exponent i n t h e e m p i r i c a l formula ( 3 ) f o r PL from 0.09 t o 0.13 (see, f o r example, I l l / ) shows t h a t t h e values PT are, mainly, i n s i d e the confidence i n t e r v a l o f t h e experimental data ( F i g . 1). The comparison o f t h e remaining expressions o f (3) w i t h t h i s value o f j shows a good agreement o f laws o f t h e e x p l o s i v e wave energy change. Laws o f v a r i a t i o n o f t h e time constant and t h e impulse have some disagreements; by choosing t h e value j i t i s d i f f i c u l t t o c o i n c i d e t h e laws a t l a r g e space i n t e r v a l s . Thus, t h e n o n l i n e a r t h e o r y gives t h e value o f t h e t i m e constant o f t h e e x p l o s i v e s i g n a l e s s e n t i a l l y s m a l l e r than experiments show.

N o r m a l i z a t i o n denotes t h a t on t h e sphere RO theo-

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JOURNAL DE PHYSIQUE

The mechanism o f t h i s phenomenon i s n o t q u i t e c l e a r . However, i t i s probable t h a t f o r o b t a i n i n g t h e accurate t h e o r i t i c a l laws o f v a r i a t i o n o f t h e time constant and the impulse i t i s necessary t o come o u t o f t h e frames o f t h e exponential approach o f t h e e x p l o s i v e wave form.

f i g . 1 : Comparison o f dimenless r e l a t i o n s (3) f o r t h e amplitude o f the e x p l o s i v e wave. A t j=1 the t h e o r e t i c a l curve (T) i s i n s i d e t h e confidence i n t e r v a l (marked by v e r t i c a l l i n e s ) . For t h e e m p i r i c a l curves ( e ) t h e i n t e r v a l i s g i v e n which corresponds t o t h e spread o f t h e exponent from 0.09 t o 0.13.

5-. Coincidence o f laws o f t h e shock wave parameter v a r i a t i o n i n a homogeneous medium p e r m i t s t o asso- c i a t e t h e parameters Ro, Pm, Tm o f t h e t h e o r i t i c a l laws w i t h t h e charge weight Q. For t h i s purpose one may use two independent laws from system (I), f o r example, t h e dependence of the amp1 i t u d e and t h e time c o n s t a n t on t h e d i s t a n c e and t h e charge weight.

These equations i s necessary t o complete w i t h t h e numerical values o f 'the parameter j when t h e theore- t i c a l and e m p i r i c a l lairs are c l o s e t o one anothkr.

From t h e d e f i n i t i o n o f t h e value j i t f o l l o w s t h a t R = j R*=R* ; t a k i n g account o f t h e dependence

0

o f Rx on Tm and Pm we o b t a i n t h e t h i r d equation.

F i n a l l y , t h e system o f equations f o r Pm$Tm and KO has t h e form

P (Pa) = 2,52 !O 8 53,23 m

5 l / ? <-0,63 Tm ( S ) = 6,54. 10- Q

Ro (m) = 0,24 E-z386

Here t h e f o l l o w i n g averaqe values o f t h e d e n s i t y , sound v e l o c i t y and t h e n o n l i n e a r parameter are

3 3

used :

3

= 10 kg/m3,

?

= 1,5 10 m/s, E = 3,6.

~2

_--^P,

The f u n c t i o n 5 = c h a r a c t e r i z e s

c3

,ao

t h e d i f f e r e n c e o f t h e medium parameters a t t h e p o i n t of t h e source d e t o n a t i o n from t h e g i v e n average value. Thus, dependences (2) f o r the amplitude, t i n e constant, impulse and energy o f the shock wave a r e associated w i t h t h e charae w e i g h t by expressions

( 5 ) which take i n t o account d i f f e r e n t c o n d i t i o n s o f t h e sound g e n e r a t i o n due t o t h e medium parameter v a r i a t i o n s near t h e source w i t h t h e v a r i a t i o n o f t h e depth o f i t s detonation.

6 . L e t us discuss now the comparison o f e m p i r i c a l and t h e o r e t i c a l formulas f o r t h e peak pressure and t h e time constant of the shock wave when i t propa- gates i n a homogeneous medium. From t h e diagramm f o r Pe and PT ( f i g . 2) i t i s seen t h a t a t a l a r g e s e c t i o n o f v a r i a t i o n o f t h e value R / Q ~ / ~ e m p i r i c a l and t h e o r e t i c a l laws a r e c l o s e as t h e f u n c t i o n

W =

20 l o p (PT/P!) (db) shows which changes i n the i n t e r v a l 2db. ccnpsriscn of e x ? r e s s i ~ n fo r the shock wave t i m e constant ( f i g . 3 ) gives t h c agreementof t h e t h e o r e t i c a l and experimental laws i n t h e i n t e r - v a l c f v a r i a t i o n o f l7/?lI3 f r o n 0.25 t o approxima- t e l y 10'. The f u n c t i o n @ =

/

(Te

-

TT)/TeI does n o t exceed 20 %. As i t i s seen from F i g . 3 IiGA t h e o r y gives a more slow growth o f t h e f u n c t i o n Ti/Q 1/3 than one may o b t a i n w i t h t h e h e l p o f e m p i r i c a l r e l a - t i o n s . However, o f i n t e r e s t i s t h e comparison w i t h t h e r e s u l t o f t h e paper /14/ on measurement OF t h e shock wave d u r a t i o n from t h e e x p l o s i v e ones o f v e r y l a r g e depths. I f we take t h a t t h e d u r a t i o n of t h e shock wsve TS, approximatejy 1.7 t i n e s 1arr;er than t h e t i n e c o c s t a n t Te ( t h i s corresponds t o o s c i l l o - grams given i n / 1 4 / ) than r e l a t i o n f o r TS w i t h t h e average values o f t h e ~ e d i u m parameters

( 5

= 1) i s c o r r e l a t e d w i t h t h e e x p e r i ~ e n t a l data (curve I , F i - gure 4 ) . A t t h e same time e m p i r i c a l r e l a t i o n ( I ) gives valuer T ~ / nary times enlsrged. Since a t ~ ~ / ~ t h e depth f r o m 1000 t o 6500m where t h e a e t o n a t i o n

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I f \ was made t h e sound v e l o c i t y and o t h e r parameters

smoothly increase so t h a t t h e parameter 5 changes

from 0.95 t o 1 , I t h e n t h e c a l c u l a t i o n according ¹

i 7 1 , ~ .

⁵^,

2

U

¹

t o formulas ( 2 ) and ( 5 ) r e g a r d i n g t h e changed parame

i ⁱ

t e r s o f t h e medium near t h e e x p l o s i v e source gives

F i g . @

-

D u r a t i o n o f t h e e x p l o s i v e wave a t l a r g e depth d e t o n a t i o n (curve 1) and t a k i n g i n t o account d i f f e r e n t d e t o n a t i o n depth ( c u r v e 2). Experimental date a r e marked by dots.

ACKNOWLEDGFIENT.

-

^Iwould l i k e t o thank Drs L.

O s t r o v s k i j and E. P e l i n o v s k i j f o r h e l p f u l discussion,

F i g . 3 Comparison o f t h e t h e o r i t i c a l ( T ) and e m p i r i - c a l (e) laws o f v a r i a t i o n o f the e x p l o s i v e wave time constant.

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JOURNAL DE PHYSIQUE

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-

/16/ P e l i n o v s k i j , E.N., Petukhov, Yu.V., Fridman, V.E., I z v . AS USSR, F i z i k a Atmosf.i.Oceana 15 (1979) 4, 436-444.