Group Properties Of The Acoustic Differential Equation

165 Pages·1995·7.57 MB·English

by  L V Poluyanov

Group Properties of the
Acoustic Differential
Equation

Group Properties of the
Acoustic Differential
Equation
Leonid V. Poluyanov, Antonio Aguilar
and Miguel Gonzilez
UK Taylor & Francis Ltd., 4 John Street, London WClN 2ET
USA Taylor & Francis Inc., 1900 Frost Road, Suite 101, Bristol, PA 19007
Copyright 0 Taylor & Francis Ltd 1995
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any form or by any means, elec-
tronic, electrostatic, magnetic tape, mechanical, photocopying, recording or
otherwise, without the prior permission qf the copyright owner.
British Library Cataloging in Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0 7484 0280 2
Library of Congress Cataloging in Publication Data are available
Cover design by Hybert Design and Type.
Typeset by Santype International Limited, Salisbury, Wilts.
Printed in Great Britain by Burgess Science Press, Basingstoke, on
paper which has a specified pH value on final paper manufacture of not
less than 7.5 and is therefore ‘acid free’.
Contents
PI-eface vii
S. mbols ix
. . .
III troduction XIII
1 Symmetry of the main equation 1
1.1 Symmetry operators of acoustic equations. Main operator equation 2
1.2 Central system of differential equations 2
1.3 Symmetry operators for homogeneous media 5
1.4 Non-homogeneous media with different symmetries 8
1.4.1 Media with translational symmetry 8
1.4.2 Media with dilatational symmetry 9
1.4.3 Media with rotational symmetry 10
1.4.4 Media with rotational and dilatational symmetry 11
1.4.5 Media with spiral symmetry 11
1.5 Lie symmetry of the acoustic equation 13
1.6 Lie algebra of infinitesimal operators for homogeneous media 21
1.7 Particular cases of non-homogeneous media: Lie symmetries 24
1.7.1 Media with translational symmetry 25
1.7.2 Media with dilatational symmetry 27
1.7.3 Media with rotational symmetry 27
1.7.4 Media with rotational and dilatational symmetry 28
1.7.5 Media with spiral symmetry 29
2 Separation of variables. Exact solutions 31
2.1 General principles of the separation of variables in linear differential equations 31
2.2 Separation of variables in the acoustic equation for homogeneous media 32
2.3 Review of different methods of separation of variables for non-homogeneous
media 38
2.4 Non-homogeneous media with translational symmetry 39
2.5 Non-homogeneous media with spherical symmetry 41
2.6 Non-homogeneous media with spherical and dilatational symmetry 44
2.7 Non-homogeneous media with cylindrical symmetry 47
2.8 Use of group properties of acoustic equations to produce new solutions for
homogeneous media 51
2.9 Use of group properties of acoustic equations to produce new solutions for
non-homogeneous media 62
3 Short wave approximation 71
3.1 Dimensionless form of the main equation 71
3.2 Structure of outer asymptotic expansion 72
3.3 Acoustic trajectories are characteristic of phase acoustic equations 76
3.4 Trajectory calculations of pre-exponential factors 78
3.5 Lie symmetry of the phase equation 79
3.6 Contact symmetry of the phase equation 91
V
Vl Contents
3.7 Separation of variables. Construction of short wave asymptotical solutions 99
3.7.1 Cartesian coordinates 99
3.72 Cylindrical coordinates 103
3.7.3 Spherical coordinates 107
3.8 Analysis of a particular case 111
4 Momentum representation in acoustics 117
4.1 Integral transformation of the main equation 117
4.2 Momentum representation of the acoustic equation for non-homogeneous
media of a particular type 118
4.2.1 Media of linear type 118
4.2.2 Media of quadratic type 120
4.3 Lie symmetry of the acoustic equation for linear media 121
4.4 Operator symmetry of the acoustic equation for linear media 124
4.5 Analysis of a particular case of linear media 125
4.6 Operator symmetry of the acoustic equation for quadratic media 132
4.7 Lie symmetry of the acoustic equation for quadratic media 137
4.8 Lie symmetry of the acoustic equation for homogeneous media 139
4.9 Discussion on the momentum representation 144
References 147
Index 149
Preface
This book shows that the Lie group and Lie algebra methods are very useful for the
analysis of differential acoustic equations. These methods allow us to carry out the
separation of variables and to construct exact solutions in the most complete form.
Moreover, group methods allow us to predict situations with desirable symmetry
properties and to classify them in accordance with these properties.
This book has a methodological character. The methods and ideas developed
here are of a general character and may therefore be applied to linear differential
equations in other scientific fields.
In Chapter 1 we have applied two methods (the Lie method (Ovsjannikov 1978)
and the operator method (Miller 1981)) for studying the symmetry of the main
acoustic equation. Special attention has been paid to the symmetry operators and to
the infinitesimal operators of homogeneous and non-homogeneous elastic media
(Brevhovskih and Godin 1989) with different symmetry properties. We have shown
that, essentially, both methods lead to identical results.
In Chapter 2 the general group principles of separation of variables in linear
partial differential equations (Miller 1981, Poluyanov and Voronin 1983, 1984) are
considered and applied to the separation of variables in equations for homogeneous
and non-homogeneous elastic media of different symmetry (Varshalovich et al.
1975). Then, algebraic and group properties of acoustic equations have been used to
determine a wide set of exact solutions, and a natural classification of exact solu-
tions has been performed.
In Chapter 3 the simplified acoustic equation in the short-wave approximation
(phase equation) is deduced and the general structure of the outer (direct) asymp-
totic solution is discussed (Osherov et al. 1985, 1989, Poluyanov and Voronin 1989).
We then consider the group properties of the phase equation, and the transform-
ations of point and contact symmetries are treated in accordance with Lie’s formal-
ism (Ibragimov 1983). In this way, we have described a special type of symmetry
that is similar to the Fock SO(4) symmetry group of the hydrogen atom. Different
types of the separation of variables in the phase equation are performed (Landau
and Lifshitz 1988) and a particular case is considered in detail.
In Chapter 4, it is shown that, for polynomial media, the Fourier integral trans-
formation of the main acoustic equation leads to a differential acoustic equation (of
another type). We then consider the Lie (Ovsjannikov 1978, Poluyanov and
Voronin 1984) and the operator (Barut and Ronchka 1980, Osherov et al. 1983)
symmetries of the resulting equation for linear and quadratic media. The results
obtained above were used for the discussion of the problems of separation of vari-
ables (Poluyanov and Voronin 1989) and production of exact solutions in momen-
tum representation. Special attention was paid to homogeneous media. We have
found that the Lie analysis in the momentum representation leads to a much wider
vii
VIII Preface
set of symmetries. The correlation between symmetries in the coordinate and
momentum representations is discussed.
Chapters 1 and 4 have been written by Leonid V. Poluyanov, Chapter 2 by
Antonio Aguilar and Chapter 3 by Miguel Gonzalez. Some of the topics considered
in Chapters 2 and 3 have been written in collaboration with L. V. Poluyanov.
We conclude this preface with thanks to Vicente Navarro, Director of the
‘Servei de Textos de la Divisio 111 (Ciencies Experimentals i Matematiques)’ of the
University of Barcelona for his support, and to Niria Millat and Milagros Lozano
for their efforts in typing the material of this book. Also, one of us (LVP) is grateful
to the ‘Direction General de Investigation Cientifice y Tecnico, Ministerio de Edu-
cation y Ciencia’ of Spain for a Research Grant.
L. V. Poluyanov, A. Aguilar and M. Gonzsilez
Department of Physical Chemistry, University of Barcelona
REFERENCES
BARUT, A., and RONCHKA, R., 1980, The Theory of Group Representations and Applications.
(Moscow: Mir) vol I, 455pp, vol II, 395~~.
BREHOVSKIH, L. M., and GODIN, 0. A., 1989, The Acoustics of Layer Media. (Moscow:
Nauka) 408~~.
GOLDSTEIN G., 1950, Classical Mechanics. (London: Addison-Wesley) 672~~.
IHRAGIMOV, N. H., 1983, The Transformation Groups in Mathematical Physics. (Moscow:
Nauka) 280~~.
LANDAU, L., and LIFSHITZ, E. M., 1988, Mechunics. (Moscow: Nauka) 215~~.
MILLER, W., 1981, Symmetry and Separation of Variables. (Moscow: Mir) 342~~.
OSHEROV, V. I., POLUYANOV, L. V., and VORONIN, A. I., 1983, Bound states of two identically
charged particles in a harmonic oscillator field. Journal of Physics E, 16, L3055L308;
1985, On the dynamics of a triatomic system in the vicinity of a conical intersection
between potential energy surfaces. Chemical Physics, 93, 13320; 1989, Quantum and
semiclassical dynamics of quasilinear triatomic systems near the adiabatic term intersec-
tion point of C+ = n = Z- type. Molecular Physics, 66, 1041-1055.
OVSJANNIKOV, L. V., 1978, Group Analysis ofozjferential Equations. (Moscow: Nauka) 399~~.
POLUYANOV, L. V., and VORONIN, A. I., 1983, On the dynamic symmetry of the stationary
Schrodinger equation. Journal of Physics A 16, 340993420; 1984, On the Lie-Bgcklund
symmetry of linear ordinary differential equations. Journal of Physics A, 17, 1787-1791;
1986, Weak symmetry of linear differential operators. Journal of Physics A, 19, 2019-
2031; 1989, Non-adiabatic quantum dynamics near the adiabatic term intersection of
II = A = 0 type. Journal ofPhysics B, 22, 1771-1784.
VARSHALOVICH, D. A., MOSKALYOV, A. N., and HERONSKYI, V. K., 1975, Quantum Theory cf
Angular Momentum. (St Petersburg: Nauka) 436~~.
Symbols
Chapter 1
displacement vector.
P density of the medium.
I elastic modulus.
P shift modulus.
t time.
radius-vector component.
Xk
acoustic operator.
ejk
acoustic operator.
2
s == symmetry operator.
Kroneker’s symbol.
6ik
r radius-vector.
E operator unity.
T, B,, PY, B,, 6, J^,, J^, J^, symmetry operators.
unit total antisymmetric tensor of third rank.
&ijk
SO(3) three-dimensional group of rotations.
E(3) three-dimensional group of all motions in the euclidian space.
so(3) Lie algebra of the SO(3) group.
43) Lie algebra of the E(3) group.
spherical radius.
icp spherical angles.
z cylindrical radius.
x, Y, z Cartesian coordinates of radius-vector r.
tension tensor.
Oij
components of the displacement vector %.
uk
v components of the velocity vector.
Y designation of the system of acoustic equation.
8 infinitesimal operator.
2 extended infinitesimal operator.
B,, bi full derivative operators.
angles of rotations with respect to different axes xi.
‘^P i
time-inversion operation.
s coordinates-inversion operation.
G’?) discrete symmetry group.
dasc
e group unity.
G”2’
cant continuous symmetry group.
G camp complete symmetry group.
I$, B, , A,, , B, irreducible representation of the discrete group.
infinitesimal operator.
xi
ix
X Symbols
Chapter 2
operator in which we are interested.
eigenvalue of A.
symmetry operators of A.
frequency of a wave.
frequency of a transversal wave.
frequency of a longitudinal wave.
wavevector. k = (k,, k,, k,).
vector amplitude of a transversal wave.
vector amplitude of a longitudinal wave.
velocity of transversal sound.
velocity of longitudinal sound.
quantum number of the angular momentum operator.
quantum number of jZ.
spherical vectors.
Bessel function of kr.
Neumann function of kr.
spherical Bessel function of kr.
radial differential operator.
radial differential operator.
state of the system.
Whittaker function.
a(2 - A, 4; kr) degenerated hypergeometrical function.
restricted cylindrical radius.
;*cs, . . . ) $) regular function of symmetry operators.
$(D) group operation of dilatation.
0 = &((cpr,( p2, q3) matrix of a three-dimensional rotation of angles (cpr, (p2, (p3)
et4 group operation of rotation.
Cj” . Clebsch-Gordan coefficients.
Jlml’j2mz
e x3 ey’ ez unity vectors directed along x, y, z-axes.
Chapter 3
a typical size of non-homogeneity.
& small parameter.
a( = &/pO c2) and p( = pO/pO c”) dimensionless parameters of order of unity.
S(R) the phase (action integral), that depends only on R.
ct local velocity of a transversal sound wave.
Cl local velocity of a longitudinal sound wave.
C velocity of an incoming plane sound wave.
c,0 ) C0l limit values of ct , c, .
u, = U,(R) acoustic potential for a transversal sound wave.
UII = u,,(R) acoustic potential for a longitudinal sound wave.
I, = I,(r), I, = 12(r) motion integrals.
9 designation of the phase equation.
H generating function of contact transformation.