SOUND GAUSSIAN NOISE AND ITS CERTAIN TRANSFORMATION CAUSED BY MOVING BOUNDARY CONDITION EFFECT

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SOUND GAUSSIAN NOISE AND ITS CERTAIN TRANSFORMATION CAUSED BY MOVING

BOUNDARY CONDITION EFFECT

R. Dyba, B. Zoltogorski

To cite this version:

R. Dyba, B. Zoltogorski. SOUND GAUSSIAN NOISE AND ITS CERTAIN TRANSFORMATION

CAUSED BY MOVING BOUNDARY CONDITION EFFECT. Journal de Physique Colloques, 1979,

40 (C8), pp.C8-25-C8-28. �10.1051/jphyscol:1979805�. �jpa-00219510�

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JOURNAL DE PHYSIQUE Colloque C8, supplement au n° 11, tome 40, novembre 1979, page C8- 25

SOUND GAUSSIAN NOISE AND ITS CERTAIN TRANSFORMATION CAUSED BY MOVING BOUNDARY CONDITION EFFECT R. Dyba and B. Zoltogorski

Institute of Telecommunication and Acoustics - Wroclaw Technical University - 50-S70 Wroclaw, Poland

Résumé.- Le but de cet article est de préparer et de présenter une discussion de certains résultats sur la théorie statistique des ondes acoustiques concernant une certaine transformation non linéaire du spectre du mouvement des sources d'onde. Du point de vue physique cette transformation non linéaire est due au caractère cinétique d'excitation des ondes. Ce mouvement de source à amplitude finie crée les déformations du spectre primaire, qui dépendent des deux causes suivantes : effet de condition de frontières mouvementées et effet de la propagation non linéaire. Notre attention est concentrée sur l'analyse de l'aspect mathématique du premier effet. Cela consiste à changer les variables de Lagrange en variables de Euler (L.E conversion) en admettant que le mouvement des sources d'onde est décrit par le processus stationnaire de Gauss. Cette méthode d'analyse est en partie basée sur les travaux de Rudenko et de Tchêrkine, /6,7/, mais le résultat final des calculs théoriques est beaucoup plus complexe que Rudenko et Tchêrkine le présentent, même si les équations de base dans les deux cas - la leur et la notre - du point de vue formel sont semblables. Il est nécessaire d'ajouter, que nous décrivons un autre phénomène physique. Nous avons déjà étudié ce problème auparavant mais par des méthodes approximatives, / 2 /.

Abstract.- The purpose of the paper is to present and discuss a certain result of statistical wave theory which consists in determining a nonlinear spectrum transformation caused by the kinematic character of the wave excitation. Large-amplitude source motion results in excitation spectrum distortions which are caused by two reasons : the moving boundary condition effect and the medium nonlinearity effect ; here attention is focussed on the first one. The essence of the paper consists in analysis of mathematical aspects of the exchange of the Lagrangian description to the Eulerian form (the so-called L-E conversion), when the source motion is to be Gaussian. The method presented here is partially based upon papers of Rudenko and Chirkin, /6,7/, but the final result of theore- tical calculations is more complex. It should be noted that a similar problem as considered here, was researched earlier but using approximate methods, / 2 / .

1-INTRODUCTION- An intensive acoustical signal spreading out in a medium is submitted to nonlinear deformations (connected to nonlinearity of hydrodynamic equations). According to the way of wave exci tations in the medium and to the frequency range of a propagated signal, one may accept different mo- dels of the medium (e.g. : ideal medium, medium with losses, disperssive medium, etc.), and analysis of nonlinear effects may be carried out in different coordinate systems (Lagrangian variables, Eulerian variables). In the case of the kinematic way of wave excitations, apart from nonlinear effects corresponding to nonlinear propagation, an additional effect arises, which is closely connected to the movement of the source surface (the so-called moving boundary condition effect), / l , 2, 3, 4, 5/. It can be shown that in the low frequency range this effect results mainly in deformations of acoustical signal, and description of the effect is implied by conversion of Lagrangian coordinates into Eulerian coordinates. Most frequently this conversion is described by approximate methods.

A more complex and advanced situation is in

the case of stochastic excitations of finite-amplitude acoustic waves. The problem has been solved for plane, spherical and cylindrical waves in a perfect gas by a method based upon Taylorian approximations of solutions of hydrodynamic equations.(1,2).

Below an accurate method of analysis of acoustic Gaussian noise deformations is presented. This method is partially based upon the works of Rudenko and Chirkin /6,7/.

2 - THEORY - Let u (x,t) denote the particle velocity, and let 5(t) be a function describing the source surface motion. Assuming the medium to be lossless and taking into account one-dimensional right-running propagation, the dependence between the particle velocity and the source motion has the following form, (Earnshow's equations), see e.g./8/:

u (x,t) = § (6), (la)

t - e =

x

- * W , (i

b) c0 + ^ §(8)

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:1979805

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C8- 2 6 JOURNAL DE PHYSIQUE

m m

where c i s small-signal sound velocity and y i s

0 u (0.t) =

&

⁷¹

lj

^{( 0 )}

-

^c

c

^{( 9 )}

the r a t i o of s p e c i f i c heats. ^{m m}

The problem may be formulated as follows :

I

determine an analytical dependence between the par-

( O

-

c - l 5 ( 9 )

-

t ) d~ dw ;

t i c l e velocity power spectrum and t h e source velo-

I

^{( 4 )}

c i t y power spectrum, when the following assumptions are accepted :

( i ) the source motion i s described by the stationary Gaussian process with the mean equal t o zero ;

( i i ) t h e source velocity power spectrum i s concentrated a t the range of s u f f i c i e n t l y low f r e - quencies, ( c f . /1, 2, 3 / ) .

Assumption ( i i ) needs an extra commentary.

Namely i t can be shown t h a t the postulate about

by the way, i t i s worth noticing, t h a t as opposed t o eq. ( 2 ) , the l e s t equation has a d i r e c t form.

Multiplying u ( 0 , t l ) by u (0, t 2 ) , which a r e given by eq. ( 4 ) , and next averaging s t o c h a s t i c a l l y , we obtain t h e autocorrelation of the p a r t i c l e velocity in the form :

concentration of the "input" spectrum a t the low

frequency range i s equivalent, in a c e r t a i n sense, Do m

t o s i t u a t i o n when the nonlinear propagation e f f e c t ^J - i ( u l e l ⁺ )

e

?

do1 de2 dml du2

.

^{( 5 )}

i s neglected ( i . e . when the medium i s assumed t o -a -a)

be l i n e a r ) . Then i t i s s u f f i c i e n t t o r e s t r i c t our-

where selves t o a description of the conversion of the

moving boundary condition (given by 5 ( t ) ) into the

boundary condition u ( 0 . t ) and next to obtain the

?

⁼

?

( e l , el. u13 u2) dependence between the input spectrum and the out-

put spectrum ( i . e . the spectrum of u ( 0 , t ) ) . Since

under the above established assumptions ₌

{ _[

^{( e l )}

^-

u ( x , t ) = u ( 0 , t

- L ) ,

then the p a r t i c l e velocity Co

power spectrum i s not the function of the space variable x.

Hence the s t a r t i n g equations take the form :

u ( 0 , t ) = 5 ( e ) , (2a)

(2b) in which E

{.I

denotes the expected value operator.

Performing s u b s t i t u t i o n s under the integral Taking i n t o account the Dirac d e l t a function twice-over, and a f t e r i t u t i l i s i n g the f a c t t h a t

7

f i l t e r i n g property, u ( 0 , t ) can be rewritten in i s stationary with respect t o the time axis such a manner : t r a n s l a t i o n , ( i . e .

?

( e l , O,, w l , w2) u ( o , t ) =

-

2lr

J J

u ( 0 , s ) e ^{i w} ( s - t ) dw ds. ( 3 )

= ('1

-

O23 W 1 ' u2) = F ( 9 % ul, u 2 ) ) , we obtain :

-m -m

m cn

i w ~ -iwO

Using eqs. (2a). (Zb), ( 3 ) one obtains the integral B ( 0 = e

j

^e ^F(8,u.-u) de do 47)

dependence between 5 ( t ) and u (O,t), namely : cO rn

where T = t ₁

-

^t2'

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R. DYBA and B. ZOLTOGORSKI C8- 27

By virtue of the Wiener-Khinchin theorem, the w w w

C (v! .v2.vZ) = C?

(%. T, ^- -

,Vl.V2,V3'1

p a r t i c l e velocity power spectrum takes the form : o

m -iwe

s

U ( ~ ~ 0 ) = Su ( w , x ) =

1

^e ^F ^(0,^{W ,} ^-a)^de ⁽⁸⁾

-m

This equation has a d i r e c t form, b u t i n considera- tion of the complicated dependence between F and

J

the parameters of the input process ( i . e . S ( t ) ) ,

i t i s not convenient t o f u r t h e r calculations and D (v1 ,v2sv3) = C3 (-9 w

- TY

w ^TYVIw ^sv23v3) applications. Then we will pay some a t t e n t i o n t o Co

transforming eq. (8) t o a s l i g h t l y simpler form.

I t can be shown t h a t R (vl-v2)- R (v1-v3)

2 a2

% Co

F ( e l 7 e 2 , ~ , -w) = -2

W

ae, ae2

A ⁽ ⁵ ³ O2)

in which

!.'here Cn, n = 2,3,n, denotes the c h a r a c t e r i s t i c function of the n-dimentional probability density d i s t r i b u t i o n of t h e caussian type and R (.) i s t h e normalized autocorrelation of 5 (t l . Inserting eq. ( 9 ) i n t o eq. (8), we obtain :

and next expandina the Function exp in the power s e r i e s , a f t e r simple

we obtain t h e final formula f o r the particle velocity power spectrum :

where the d i f f e r e n t i a l operator L

CRI

has t h e form:

L. .. .. J

in wh~ch R = R (e!, i? = R(e) and 5 = R (0)

.

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JOURNAL DE PHYSIQUE

3-. CONCLUSIONS

( i ) Formula (4) i s v a l i d f o r any s t o c h a s t i c process d e s c r i b i n g t h e motion o f t h e plane wave source ;

( i i ) Formula (12) e s t a b l i s h e s t h e acurate s o l u t i o n o f the problem o f t h e L-E conversion when t h e source motion i s o f t h e Gaussian type.

The procedure o u t l i n e d above may be used f o r low-frequency a n a l y s i s o f t h e f i n i t e - a m p l i t u d e d i s t o r s i o n .

F i n a l l y , we would l i k e t o n o t e t h a t determi- n i s t i c v e r s i o n o f t h e s a i d method was worked o u t by Kessl er, /4/.

4-. REFERENCES

/1/ Dyba R., Contr. Papers o f t h e 9 t h I.C.A.

Madrid 1977, Vol. 11, N-38, 746.

/2/ Dyba R. and Z o l t o g o r s k i B., Spectral a n a l y s i s o f f i n i t e - a m p l i t u d e e f f e c t s f o r sound waves i n i d e a l medium, S c i e n t i f i c Papers o f t h e I n s t i t u t e o f Telecommunication and Acoustics o f Wroclaw Technical U n i v e r s i t y , Monographs, Wroclaw 1978

/3/ F r o s t P.A. and Harper E.Y., J. Acoust. SOC.

Am., 1975,

-

58, 318

/4/ Kessler H.C., J. Acoust. Soc. Am. 1962,

-

34 1958

/5/ Z o l t o g o r s k i B., Contr. Papers o f t h e 9 t h ICA Madrid 1977, Vol 11, N-35, 743

/6/ Rudenko O.V. and C h i r k i n A.S., Dokl AN USSR, 1974, 214, 1045.

/7/ Rudenko O.V. and C h i r k i n A.S., Dokl

,

AN USSR 1975, 225, 540.

/8/ Blackstock D.T., 3. Acoust. Soc. Am., 1962, 34, 9

-