The concept of energy velocity for linear dispersive waves is usually given for a normal mode solution of the system as the ratio between the mean energy flux and the mean energy density. In the absence of dissipation this velocity is known to coincide with the corresponding group velocity. When dispersion is accompanied by dissipation, this interpretation is not correct since the group velocity loses its original meaning and can assume nonphysical values. In this note the relation between energy velocity and group velocity is derived for dissipative, uniaxial waves, governed by a linear hyperbolic system. An example is provided where the energy velocity is compared with the phase and group velocities

Mainardi, F.

van Groesen, E.

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van Groesen, Embrecht W.C.

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Energy propagation in linear hyperbolic systems

University of Twente Research Information

IL NUOVO CIMENTO NOTE BREVI VOL. 104 B, N. 4 Ottobre 1989 Energy Propagation in Linear Hyperbolic Systems. F. MAINARDI Dipar t imento  di F i s ica  dell 'Universi ta - Bologna,  I ta l ia  I N F N  - Sezione di Bologna, I ta l ia  E. VAN GROESEN Depar tment  o f  Appl ied  Mathemat ics ,  Universi ty  of  Twente  - Twente,  The Nether lands  (ricevuto il 24 Agosto 1989) Summary. - -  The concept of energy velocity for linear dispersive waves is usually given for a normal mode solution of the system as the ratio between the mean energy flux and the mean energy density. In the absence of dissipation this velocity is known to coincide with the corresponding group velocity. When dispersion is accompanied by dissipation, this interpretation is not correct since the group velocity loses its original meaning and can assume nonphysical values. In this note the relation between energy velocity and group velocity is derived for dissipative, uniaxial waves, governed by a linear hyperbolic system. An example is provided where the energy velocity is compared with the phase and group velocities. PACS 03.40 - Classical mechanics of continuous media: general mathe- matical aspects. The concept of ene rgy  velocity for l inear dispersive waves  can be s ta ted  as the rat io be tween  the mean ene rgy  flux and the mean ene rgy  densi ty  of a monochromatic  wave  (normal mode solution of the system),  following a classical a rgument  originally due to Reynolds (1) and Rayleigh (2) for surface wa te r  waves .  In  the absence of dissipation this velocity is known to coincide with the group (1) O. REYNOLDS: Nature, 16, 343 (1877). (2) j .  W. STRUTT (Lord RAYLEIGH): Proc. London  M a t h  Soc., 9, 21 (1877). 33 - II Nuovo Cimento B. 487 488 F. MAINARDI and E. VAN GROESEN velocity computed at the frequency (or wavelength) of the wave; this general property of linear systems can be proven in different wayes (see e.g. (3.8)). When there is dissipation this coincidence is manifestly broken since the group velocity (defined by taking the real part in the usual formula) loses its original meaning and can assume, in certain ranges of frequency or wavelength, nonphysical values (e.g., exceeding the wave-front velocity), as pointed out by Sommerfeld and Brillouin(9) for electromagnetic waves. In other words, at variance with conservative systems, for dissipative systems the concept of group velocity may become meaningless and appears not easily related to energy propagation, so that specific treatments are necessarily required. Recently Mainardi(1~ has provided an interesting relation between the energy velocity and the phase velocity for dispersive and dissipative waves that are governed by a linear hyperbolic system. In this note, in view of the importance of the concept of group velocity for conservative waves (in both linear and nonlinear system, see e.g. (1~)), we will show how, in the presence of dissipation, the energy velocity of hyperbolic waves can be related to the group velocity as well. Furthermore an example of physical interest is provided where the energy velocity is compared with the phase and group velocity. As in (10) we restrict ourselves to uniaxial waves and we agree to write the governing hyperbolic system in the form (1) 5u D ~U + M u = O  - ~  + ~x where u = u(x, t) is a real n-vector function which represents the field variable and D and M are constant real n • n matrices, with D symmetric. Furthermore we assume that the variable u in eq. (1) is chosen in such a way that the functional (2) ~[u] - u -  u 2 is the energy density for the physical system, where 9 denotes the inner product in R ~. (~) C. (4) L. (5) M. (6) G. (~) G. (8) M. G. RossBY: J. Meteorol., 2, 187 (1945). J. F. BROER: Appl. Sci. Res. A, 2, 329 (1951). A. BIOT: Phys. Rev., 105, 1129 (1957). S. WHITHAM: Commun. Pure Appl. Math., 14, 675 (1961). S. WHITHAM: J. Fluid Mech., 22, 273 (1965). J. Sir LIGHTttILL: J. In8t. Math. Applic., 1, 1 (1965). (~) L. BRILLOUIN: Wave Propagation and Group Velocity (New York, N.Y., 1960), Chapt. 2, 5. (lo) F. MAINARDI: Wave Motion, 9, 201 (1987). (11) G. B. WHITHAM: Linear and Nonlinear Waves (New York, N.Y., 1974), Chapt. 5, 15. E N E R G Y  P R O P A G A T I O N  IN L I N E A R  H Y P E R B O L I C  SYSTEMS 4 8 9  The system is supposed to be nonconservative in the sense that  4,' satisfies the following balance law: (3) 5 6' ~_ ~,~7- _ j , ,  3t ~x where ,~7-= ,~7-[u] is the (energy) flux density and d ' =  d'[u] is the rate of loss (if positive), or of supply (if negative). We agree to refer to ~1" as to the loss density. For  a detailed discussion of the energy balance law in nonconservative systems we refer to recent papers of the authors (12.19, where also the notion of centrovelocity of energy is introduced. For  the system (1) we obtain particularly simple expressions for the flux and for the loss (1~ in the present notation they read (4) ,~Y-[u] - u .  Du 2 (5) J~[u] = u .  M~u, where M S (M a) denotes the symmetric (antisymmetric) part  of the constant matrix M. Let  us now recall the relevant concepts related to energy propagation. Denoting by ( f l u ] )  the space mean value of a d e n s i t y f = f ( x ,  t) for a prescribed solution u, computed in a fixed interval of length L (say, in xo ~< x ~< x0 + L), and assuming ,~Y-(Xo, t) -- ,~7-(Xo + L, t) Vt, we introduce the following (time depend- ent) functionals, the energy-flux velocity: (6) VJu]- and the dissipation rate ( 7 )  (,r[u]) " The velocity Ve is sometimes called the mean velocity of energy transport;  here, we will simply refer to it as the energy velocity. The importance of the rate  ~ can be recognized after  integrating (3) over x. I t  follows that  d (8) ~ ( 6' ) = - ~ ( ,': ) ,  (12) E. VAN GROESEN and F. MAINARDI: Wave Motion, 11, 201 (1989). (18) E. VAN GROESEN and F. MAINARDI: Balance Laws and Centrovelocity Dissipative Systems, to be published. in 490 F. MAINARDI and  E. VAN GROESEN from which it is found that [J ] (9) ( ~ [ u ] )  = ( & )  exp - ~[u(,)]dr , where ~0 is the energy density corresponding to the initial data u(x, 0). As a consequence of eqs. (2) and (4)-(7) we get (10) Ve[u] - (u .  Du) <u.u> ' (11) :r = 2 (u.  M~u) <u.u> From eqs. (2), (5) we note that, if the matrix M ~ is proportional to the identity matrix, then, for any solution u(x, t) of (1) and for all x, t, the loss density is proportional to the energy density. This peculiar fact, referred to as uniform damping, can also be expressed by (12) a = - -  = where now the dissipation rate ~ is a characteristic constant of the system, indipendent of the solution and of the space domain (12). In this paper we consider particular space-periodic solutions, referred to as normal modes, from which usually the analysis of linear dispersive waves starts. These solutions are of the form (13) u(x, t; k) = Re [V(k) exp [i[kx - ~(k) t]]], with k real and ~ complex in general, and V a complex n-vector. Henceforth we will denote a given normal mode simply by u(k), so dropping the dependence on x, t; furthermore we will use the notation ue(k) to denote the complex solution whose real part provides u(k). These normal modes represent (pseudo) monochromatic waves, since they are sinusoidal in space with period (wavelength) ~ = 2=/k, but not necessarily sinusoidal in time, since ~o may be complex. Only if oJ is real, these solutions represent effectively monochromatic waves with a time period T = 2=/~ and with a constant amplitude. In general these waves are decaying or growing exponentially in time if the quantity (damping factor) (14) y =  - Im[~] ENERGY PROPAGATION IN LINEAR HYPERBOLIC SYSTEMS 491 is positive or negative, and they propagate with speed (phase velocity) Re [~] (15) Vp - k For u(k), and hence uC(k), to be a solution of (1), ~ and V depend on k in such a way that they satisfy the following eigenvalue problem: (16) [ -  ioJ + ikD + M] V = 0, namely they should be an eigenvalue and a corresponding eigenvector of the matrix (2(k) = kD - iM. In particular ~o must satisfy the dispersion relation that follows from (16), i.e. (17) ~(~o, k) - det [ -  i~oI + ikD + M] = O. In general at a given k there may be several branches oJ(k) of (17), providing distinct modes. For a given branch ~o = o~(k), the dispersive properties are illustrated by the functions (14) and (15). Furthermore we introduce the group velocity as (18) Vg = d Re [~(k)]. In the conservative case o~ is real and Vg has the kinematic meaning of speed of the group (wave packet). In the nonconservative case, Im [o(k)] does not vanish in general so that Vg is expected to lose its usual meaning (see e.g. (9)), while Vp does keep it. The energy velocity and the attenuation factor associated with the waves (13) can easily be obtained from eqs. (10), (11), respectively, inserting the solution (13) and averaging over the wavelength. For this purpose we recall the following properties concerning two normal modes with a given k and o~(k) but with different amplitude: (19) 2 < / t l ( k ) "  u2(k)> --  Re [<u~(k). u~(k)}] and (20) < u~(k), u~(k) > = u](k), u~(k) = exp [ -  2~,t] (VI" V2), where the inner product is in R n or C ** as appropriate. The importance of relations (19)-(20) is that they enable us to compute Ve and without carrying out the averages present in eqs. (10)-(11); we obtain (21) Ve - V.  DV V . V  492 F.  M A I N A R D I  and E. V A N  G R O E S E N  and (22) a = 2 - -  V .  M s V  V . V  " From (21)-(22) we recover the property that for normal-mode solutions the energy velocity and the dissipation rate are constant ( t ime independen t ) ,  depending uniquely on the wavenumber. Furthermore, the energy velocity turns out to be bounded from above and below by the extreme eigenvalues of the matrix D, i.e. by the extreme characteristic velocities of the hyperbolic system. Recently Mainardi (io) has proven the following identities: i V . M a V  (23) V e = Vp 4 k V.V and V .  M s V  (24) Y= V.V These equations can readily be obtained taking the On-inner product of (16) with V, and considering the imaginary and real parts, respectively. From (22) and (24), we may incidentally note that ~ = 2•. Previously, Broer and Peletier (14) have used the eigenvalue equation (16) to prove that, in the absence of dissipation (M s = 0), the r.h.s, of (21) equals the group velocity. Now we would like to extend the analysis in (lo.14) to show how the energy velocity can be related to the group velocity, taking into account the s y m m e t r i c  part M S of the matrix M, that is responsible for dissipation. This relation, which is known to be an identity when M is purely a n t i s y m m e t r i c  (it), can be obtained from the eigenvalue equation (16) after some manipulations. For  this purpose we differentiate (16) with respect to k and take the C n inner product with V so obtaining (25) ~ o ' V . V - V . D V + V . ( o I - k D + i M ) V ' = O ,  ~ ' -  d~ V' = d__V_V dk ' dk ' and hence (26) ~ * ' V .  V - V .  D V  + V' . (~o*I - k D  - i M  t) V = 0, where * and t denote complex and Hermitian conjugate, respectively. Noting that (27) ~o* = oJ + 2 i y ,  M t = - M + 2 M  s , (~4) L. J. F. BROER and L. A. PELETIER: Appl. Sci. Res. A,  17, 65 (1967). E N E R G Y  PROPAGATION IN L I N E A R  HYPERBOLIC SYSTEMS 493 and, using (16), eq. (26) reads (28) (o * ' V .  V - V "  D V  - 2 i V '  9 ( M  s - , f I )  V = O, namely (29) V .  D V  _ _  - -  0 9 ~  t - -  V ' V  V2.V [V 9 (M s - TI) V']*. Using (21) and noting that  oJ' = V~ - i1", it is a simple mat te r  to get from (29) 2 Im [V. (M ~ - 1"1) V'] (30)  v~ = v ~  - v . - - v  and (31) dl" 2 dk V. V - -  R e  [ V .  ( M  s - 1"I) V ' ] .  We recognize from (30) the fact that  the identification of energy velocity with group velocity is valid not only in the absence of dissipation (y = 0 r M s = 0), but also for uniform damping (M s = 1"1). In the particular case of uniform damping without dispersion (M = M ~ = 14) we recover from (23), (30) the trivial equality of the three velocities V~, V~, Vg. We conclude our analysis with recalling the instructive example of the linear Klein-Gordon equation with dissipation (KGD equation) that  rules wave propa- gation in an elastic string, anchored elastically to its equilibrium position by a t ransverse restoring force and damped by air friction. I t  is a second-order hyperbolic equation that  reads (32) eft + 2aCt + b2r = c '~ Cxx, where r = r  t) denotes the transverse displacement and a, b, c are non- negative constants related with air damping, restoring force and string tension, respectively (1~). The corresponding dispersion relation reads (33) (co + i a f i  = c2k  2 + (b 2 - a2) , and indicates that  different cases can be expected, depending on the sign of (b 2 - a2). If  0 ~< a < b, then for any k the real part  of ~ does not vanish, i . e .  there is propagation for each k; if 0 ~< b < a, there is propagation only for wavenumbers (15) G. R. BALDOCK and T. BRIDGEMAN: T he  M a t h e m a t i c a l  T h e o r y  o f  W a v e  M o t i o n  (Chichester, U.K., New York, N.Y., 1981), Chapt. 5. 4 9 4  F.  M A I N A R D I  a n d  E. VAN G R O E S E N  k such that  Ikl ~>V r ~ -  b 2. The propagation regimes are characterized by a constant damping factor ~, = a and ~/c2k 2 + (b 2 - a 2) c 2 (34) Vp-  k , Vg = ~-~p. In particular we get n o r m a l  d i spers ion  (0 <~ Vg <~ c <~ Vp) if 0 ~< a < b, a n o m a l o u s  d i spers ion  (0 <- Vp <~ c <~ Vg) if 0 ~< b < a, and no d i spers ion  (Vp = Vg = c) if a = b. The energy velocity has been computed in(1~ start ing from the energy balance for a single-mode solution. The terms 3, ~ -  and ~ entering this balance are, respectively, (35) = (r 2 + c2r + b2r ~J-= _ c2r162 ~f = 2aCt 2 , and the energy velocity turns out to be (36) Ve = Re [co] c2 k b 2 + c 2 k 2 ' or, using (33)-(34), a 2 b2::2k2] Vg[1 b2+c2k21 Formulae (36) and (37) can also be found after some elementary calculations from eq. (21) or (23) or (30) as well, thus providing a check of our present  theory. For  convenience we quote the relevant passage required to get the energy velocity. Choosing u = col(r cCx, be), the KGD equation (32) can be wri t ten as a first- order hyperbolic system of type (1)-(2) with n = 3, where [0c01 (38) D =  - c  0 0 , M =  0 0 . 0 0 0 - 0 0 After  deriving the dispersion relation (33) from the eigenvalue problem (16) using (1), (13), (38) (and neglecting the spurious eigenvalue oJ = 0 tha t  arises because a three-vector u is introduced to describe a second-order equation), we obtain the eigenvector for the progressive mode, to be inserted in eqs. (21), (23), (30), (39) V = col (~, - ck,  ib) .  E N E R G Y  P R O P A G A T I O N  IN L I N E A R  H Y P E R B O L I C  SYSTEMS 495 2 1 ~  1LI 0 = ~ <  b " - e L .  1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ~ ~=~ J 0 1 2 k 2 3~V3g) O < b < r  i 1 2 k V p 2 )  0 < a <  b 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  -~- 9 --'?- - .- ' : : .  - - -"::. - . . . . . . . . . . .  ..= 0 k 1 2 k 3 Fig. 1. - Comparison among the energy velocity Vo, the phase velocity Vp and the group velocity Vg for the linear KGD equation Ctt + 2aCt + b2r = C2r with c = 1 and 1) a = 0, b = l ; 2 )  a = l / V ~ ,  b = l ; 3 )  a = l ,  b = l / V ~ ; 4 )  a = l ,  b=0.  In particular, after  simple algebra, we obtain the relevant  expressions (40) V .  V =  ~ *  + b 2+ c2k 2 = 2 ( b  2+c2k2), V .  D V  = c2k(~  + oJ*) = 2 Re [oJ] c2k = 2c2k2Vp,  V .  M " V  = ib2(co + oJ*) = 2ib ~ Re [~o] = 2 ib2kVp ,  V .  ( M  s - ],I) V'  = a(oJ*~' - c2k) = ia2Vg.  We note that  Ve as given by (36) and (37) is an increasing function of k which does not exceed the wave front velocity c (either Vp or Vg), and that  it reduces to the phase velocity only if b = 0, and to the group velocity only if a = 0. In the particular case a = b, it remains distinct from c in spite of the absence of dispersion (dissipation is however  acting not uniformly on the energy since M S is not proportional to I).  For  the sake of clearness, in fig. 1 we show pictures that  compare the energy velocity Ve with the kinematic velocities Vp, Vg in the following relevant  cases: 1) O = a < b ,  2) 0 < a < b ,  3) 0 < b < a ,  4) O = b < a .  496 F. MAINARDI and E. VAN GROESEN This work was supported in part by C.N.R., Gruppo Nazionale per la Fisica Matematica, by I.N.F.N., Sezione di Bologna, and by Italian Ministry of Education (MPI: 60% grants). EVG wishes to thank C.N.R. for providing the opportunity to visit the University of Bologna. We also thank G. Vitali for his assistance with computing. 9 by Societfi Italiana di Fisica Proprietfi letteraria riservata Direttore responsabile: RENATO ANGELO RICCI Questo fascicolo ~ stato realizzato in fotocomposizione dalla Monograf, Bologna e stampato dalla tipografia Compositori, Bologna nel mese di novembre 1989 Questo periodico iscritt0 alrUnione Stampa Periodica Italiana 

Energy propagation in linear hyperbolic systems

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