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NONLINEAR SOUND GENERATION BY HADRON SHOWERS
Peter Westervelt
To cite this version:
Peter Westervelt. NONLINEAR SOUND GENERATION BY HADRON SHOWERS. Journal de
Physique Colloques, 1979, 40 (C8), pp.C8-95-C8-97. �10.1051/jphyscol:1979817�. �jpa-00219522�
JOURNAL DE PHYSIQUE CoVLoque C8, supplement au n°U, tome 40, novembre 1979, page c8- 95
NONLINEAR SOUND GENERATION BY HADRON SHOWERS Peter J. Westervelt
Department of Physios, Brown University, Providence, Rhode Island 02912, USA.
Résumé.- En relation avec le projet DUMAND qui consiste à utiliser l'océan comme un énorme détecteur acoustique de neutrinos, nous avons étudié l'application d'une théorie récente des émetteurs thermo- acoustique (Peter J. Westervelt and Richard S. Larson, J. Acoust. Soc. Am. 54 (1973) 121). Dans le cas statique ou aux très basses fréquences, on peut attribuer à peu près 1 0 T du coefficient de dila- tation thermique de l'eau à 20°C aux modes de Debye. Ces modes engendrent le son par un mécanisme non-linéaire responsable du fonctionnement du réseau paramétrique acoustique (Peter J. Westervelt, J. Acoust. Soc. Am. 35 (1963) 535). La contribution des modes de Debye au coefficient de dilatation thermique et ainsi qTT'à la pression acoustique est pratiquement indépendant de la température am- biante de l'eau. Par conséquent, si les modes de Debye ne sont pas complètement excités comme nous le supposons à haute fréquence, alors le coefficient de dilatation thermqiue sera inférieur à la valeur statique et disparaîtra aux environs de 6°C au lieu de 4CC, la température de densité maxima de l'eau. Cette théorie est en accord avec les mesures récentes de la variation thermique du son engen- dré par le bombardement de l'eau par les protons (L. Sulak et al., Proceedings of the La Jolla Work- shop on Acoustic Détection of Neutrinos, 25-29 July 1977, Scripps Institute of Oceanography,
U.C.L.A., San Diego, Hugh Bradner, Ed.). Une autre conséquence de notre travail impliquerait une mo- dification de la théorie thermo-acoustique conventionnelle par soustraction des termes de source pa- ramétrique â hautes fréquences.
Abstract.- In connection with Project DUMAND, the proposal to utilize the ocean as a giant acoustic detector of neutrinos, we have studied the applicability of a recent theory of thermoacoustic arrays (Peter J. Westervelt and Richard S. Larson, J. Acoust. Soc. Am. ^ 4 (1973) 121). In the static case or at very low frequencies, about 10 % of the coefficient of thermal expansion for water at 20°C can be attributed to Debye-like modes. Debye-like modes generate sound via the non-linear mechanism res- ponsible for the operation of the parametric acoustic array (Peter J. Westervelt, J.Acoust. Soc.Am.
35 (1963) 535). The contribution of the Debye-like modes to the thermal expansion coefficient, and TTius to the sound pressure, is essentially independent of the ambient water temperature. Hence if the Debye-like modes are not fully excited as we postulate to be the case at high frequencies, then the thermal expansion coefficient will be less than the static value by an amount that causes it to vanish at about 6°C instead of at 4°C, the temperature of maximum water density. This theory is in agreement with recent measurements of the temperature dependence of sound generated by proton depo- sition in water (L. Sulak et al., Proceedings of the La Jolla Workshop on Acoustic Detection of Neu- trinos, 25-29 July 1977, Scripps Institute of Oceanography, U.C.L.A., San Diego, Hugh Bradner, Ed.).
A further consequence of our work implies that the conventional thermoacoustic theory be modified by subtracting the parametric source terms at high frequencies.
When high energy neutrinos interact with nu- cleons in the ocean a jet of hadrons is produced which deposits thermal energy. This thermal energy
is expected to produce a sonic pulse which will, enable the energy and direction of the incident neutrino to be estimated. Initial laboratory experi- ments intent on investigating the feasibility of this detection scheme have been performed by L.Sulak et al., at Harvard University and this paper is de- voted to an analysis of an unexpected result of these experiments.
The theory of sound production by a line dis- tribution of thermal energy was first formulated in 1973 / l / and experimental verification of this theo- ry appeared in 1976 12/. After a brief review of this theory, a modification will be suggested which enables the theory to account for a previously
inexplicable experimental finding of L. Sulak et al. / 3 / , namely, that sound generated by depositing protons in water vanishes when the ambient water temperature is about 6°C rather than 4°C, the tem- perature of maximum water density.
The inhomogeneous acoustic wave equation for the sound pressure p generated by H, the thermal deposition of energy per unit volume and per unit time, is :
c^ 3tz c 3t
where c is the velocity of sound, 0 is the logarith- mic coefficient of thermal expansion, c is the specific heat par unit mass. In the example of Ref.
1 in which the heat deposition results from the energy lost from a laser beam modulated at
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:1979817
C8- 96 JOURNAL DE PHYSIQUE
frequency w
,
= a10 e-az-iwt
Y
where z i s the coordinate along which t h e l a s e r beam propagates, IO i s the o u t p u t i n t e n s i t y o f the l a s e r and a i s t h e a t t e n u a t i o n c o e f f i c i e n t f o r the 1 aser beam.
The asymptotic s o l u t i o n t o Eq. ( 1 ) i n t h e li- m i t r >> w/a c where r i s t h e d i s t a n c e from t h e 2 l a s e r i s :
iawBPO ei k r - i w t
=-rlncpr
a + i k core (3) where Po i s the power o u t p u t o f t h e l a s e r , k = w/c, ande
i s the angle between the l i n e connecting the f i e l d p o i n t and t h e l a s e r and the d i r e c t i o n o f pro- pagation o f the l a s e r beam. This s o l u t i o n e x h i b i t s t h e well-known behavior o f Cerenkov r a d i a t i o n i n t h e case where the e m i t t i n g p a r t i c l e i s progressing a t the speed o f l i g h t , f a r g r e a t e r than the speed o f sound. Thus t h e predominant a c o u s t i c emission occurs i n a d i s c perpendicular t o the l a s e r beam w i t h an i n t e n s i t y h a l f w i d t h g i v e n by := iT/2
+
a/k. ( 4 )This theory has r e c e n t l y received experimental confirmation /2/ w i t h respect t o t h e dependence o f t h e sound pressure on 0 (see F i g . I ) , and w i t h res- p e c t t o absolute magnitude (see F i g . 2 ) .
EXPERIMENT
Eq. 5
- 2 0
6 0 7 0 8 0 9 0 100 110 1 2 0 ANGLE
-
degF i g . 1 : F a r f i e l d beam p a t t e r n .
E i
0.8 1.2 2.0 4 . 0 6.0
L A S E R POWER INTO W A T E R - kW
F i g . 2 : Source l e v e l dependence on power.
The e x p e r i m e n t a l l y determined pressure ap- pears t o be about 1 decibel, t h a t i s , about 10 %, below t h e t h e o r e t i c a l p r e d i c t i o n , a f e a t u r e which i s i n accord w i t h t h e conclusions o f t h i s study.
These experiments were performed i n water and thus should be a p p l i c a b l e t o t h e r e l a t e d problem, t h a t o f sound generated by heat deposited when h i g h energy p a r t i c l e s are absorbed i n water. Such an experiment has r e c e n t l y been r e p o r t e d by L. Sulak e t a l . /3/. Sulak measured t h e amplitude dependetxe o f the a c o u s t i c o u t p u t on t h e ambient water tempe- r a t u r e f o r 160 MeV protons dumped i n t o water (see F i g . 3 ) .
F i g . 3 : Signal amplitude as a f u n c t i o n o f tempera- t u r e .
Peter J. Westervelt C8- 97
Note t h a t the sound vanishes a t 6OC i n s t e a d o f 4OC, the temperature a t which t h e s t a t i c c o e f f i - c i e n t o f thermal expansion f o r water vanishes.
Sound can be generated by sound, a mechanism which i n t h e s t a t i c l i m i t represents t h e c o n t r i b u - t i o n t o thermal expansion o f t h e Debye-like e x c i t a - t i o n s . The wave equation f o r t h e pressure ps o f t h e sound generated by primary waves whose pressure i s pi i s /4/ :
where :
and po and Co a r e t h e ambient d e n s i t y and speed o f sound, r e s p e c t i v e l y , f o r water. The q u a n t i t y i n t h e square brackets i n Eq. ( 6 ) has a value o f 3.5 f o r water.
The r a t i o o f t h e Debye-like source s t r e n g t h t o the t o t a l thermal source s t r e n g t h can be o b t a i - ned from Eqs. ( 1 ) - ( 6 ) . I n o b t a i n i n g t h i s r a t i o i t i s l e g i t i m a t e t o i g n o r e t h e time d e r i v a t i v e s s i n c e they appear t w i c e i n each source type. The r a t i o R i s :
i n which E i s t h e time dependent thermal energy dew s i t y o f which p f p i 1 ~ i 2 i s t h e Debye-like p a r t . I n - t r o d u c i n g c = 4.2 x
l o 7
erg/gm°C,B
= 3 x~ o - ~ / " c
P 5
a t 20°C and co = 1.5 x 10 cm/sec i n t o Eq. (7), t h e value o f R v a l i d a t 20°C i s :
I f R i s about 0.1 and a l l Debye-like modes a r e f u l l y e x c i t e d below some c r i t i c a l frequency, t h e 6OC zero c r o s s i n g i n F i g . 3 would be s h i f t e d down t o about 4°C.
An estimate o f t h e energy d e n s i t y D r e s i d i n g i n Debye-like modes i s g i v e n by :
47l kT O % t : 3 -
'mi n
i n which Xmin i s the Debye s h o r t wavelength l i m i t and k i s Boltzman's constant. An e s t i m a t e o f t h e t o t a l thermal energy d e n s i t y E i s g i v e n by : E = p c T
0 P (10)
The r a t i o D/E i s found by i n s e r t i n g k = 1.4 x 10-16 e r g l a c , c = 4.2 x
l o 7
erg/gm°C and po = 1 gm/cm 3 i n t o Eqs. ( 9 ) and (10) P :I f t h e r a t i o D/E
,
0.01 ,as i s c a l l e d f o r by t h e ex- perimental r e s u l t s o f L. Sulak e t a l . , t h i s leads t o a value o f A,,, 5 cm. A crude check on t h e i n - t e r n a l consistency o f t h i s r e s u l t can be made by e s t i m a t i n g d i r e c t l y t h e value o f Xmin from t h e f o r - mula :i n which n i s t h e number o f water molecules p e r u n i t volume. The value n = 3 x
l o z 2
leads t o Amin 5 10-
7cm, a s a t i s f a c t o r y r e s u l t .
This work has been supported i n p a r t by NORDA and by t h e U.S. Department o f Energy ; t h i s i s Brown U n i v e r s i t y r e p o r t Brown-HET-363, COO-3130TA- 341. A p r e l i m i n a r y r e p o r t o f t h i s work was presen- t e d a t t h e Neutrino 78 Conference, Purdue U n i v e r s i - t y , West Lafayette, Indiana.
References
/1/ Westervelt, Peter J. and Larson, Richard S., J. Acoust. Soc. Am.
54
(1973) 121./2/ Muir, T.G., Cul bertson, C.R. and Clynch, J .R., J. Acoust. Soc. Am.
2
(1976) 735,./3/ Sulak, L. e t a1
. ,
Proceedings o f t h e La J o l l a Workshop on Acoustic D e t e c t i o n o f Neutrinos, 25-29 J u l y 1977, Scripps I n s t i t u t e o f Oceano- graphy, U.C.L.A., San Diego, Hugh Bradner, Ed./4/ Westervelt, P.J., J. Acoust. Soc. Am.
