Energy velocity for hyperbolic dispersive waves

WAVE MOTION 9 (1987) 201-208 NORTH-HOLLAND

ENERGY VELOCITY FOR HYPERBOLIC

201

DISPERSIVE

WAVES

Francesco M A I N A R D I Department of Physics, University of Bologna, 1-40126 Bologna, Italy Received 14 July 1986, Revised 4 November 1986

In this note the concept of energy velocity for linear dispersive waves is discussed in the uniaxial case. When energy is not conserved, the identification of energy velocity with the kinematic concept of group velocity is not valid as shown in some examples of physical interest. For dispersive waves of hyperbolic type a general expression for energy velocity is deduced, which yields the group velocityonly for conservative waves. In special cases of nonconservativewaves the energy velocity is shown to equal the phase velocity. Examples are also presented.

1. Introduction The purpose of this paper is to draw attention to the seemingly little recognized fact that for plane waves travelling in homogeneous linear isotropic media the equivalence between the energy flux velocity and the group velocity does not necessarily hold. Hayes and Musgrave [1] provided some examples of inhomogeneous plane waves for which this equivalence is not true. Further examples have been given recently by Borejko [2]. For homogeneous plane waves the equivalence is so consolidated that the fundamental condition of conservation of energy is generally understood. There are cases, however, where conservation of energy is manifestly broken (e.g. when dissipation is not negligible) so that a specific treatment is necessarily required. In this respect noteworthy works were carded out by Brillouin [3], Vainshtein [4] for electromagnetic waves, and by Bland [5] for viscoelastic waves, where the identification of energy velocity with group velocity was shown to break down. For a special case considered in [4], and for any case considered in [5], the energy velocity is proven to be the phase velocity, a (sur-

prising) result, (unfortunately) not mentioned in textbooks on wave propagation, which we intend to explain. In Section 2 we will state a definition of energy velocity which is commonly accepted for uniaxial waves when a balance equation for energy is available. Afterwards we will discuss such a concept associated with dispersive wave motions governed by hyperbolic partial differential equations. In Section 3, we will analyze a particular hyperbolic equation of second order, which occurs in a variety of wave propagation phenomena, while in Section 4 a general hyperbolic system of first order and of symmetric type will be considered. We will provide a general expression of energy velocity, which is checked in Section 5 in two physical examples previously introduced. Our approach allows us to compute the energy velocity of dispersive waves in active or dissipative media provided that the wave motion is ruled by a linear hyperbolic system. It is based on a previous analysis by Broer and Peletier [6,7], which however treated conservative waves only. The possible extension to dispersive waves not necessarily governed by a linear hyperbolic system will be considered in a later paper.

0165-2125/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)

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F. Mainardi / Energy velocity

2. Statement of the problem

In simple media (isotropic and homogeneous) which are conservative and nondispersive, the energy velocity Ve, associated with an infinite homogeneous plane wave (in an obvious notation) O(x, t) = A cos(kx - tot),

(2.1)

is in the direction of phase propagation and has magnitude equal to the phase speed, that is V e = Vp ~-

to/k.

(2.2)

In some texts this is justified by dividing the mean energy flow (~) (for electromagnetic waves it is the mean Poynting vector) by the mean energy density (~) to obtain Ve = (~')/(~).

(2.3)

The mean values are obtained by averaging the quantities ~, ~ either over a period T = 2 ~ / t o or a wavelength L = 2~r/k. When however dispersion is present and ruled by a relation ~(to, k ) = 0, the energy velocity can still be obtained from (2.3) but it turns out to be the group velocity, that is Ve = Vg = dto/dk.

(2.4)

This result is well known and it is quoted in several textbooks on wave propagation (see e.g. [8]). The proof is based on the conservation of energy which implies a~ --+--

Ot

a~ Ox

= o.

(2.5)

When energy is not conserved the left-hand side of (2.5) is expected to be different from zero and the energy balance equation becomes --+--+ Ot Ox

~e = O.

(2.6)

The term 5e expresses the interaction of the wave with the supporting medium describing whether the wave gains (Se < 0) or loses (Se > 0) energy from the medium itself.

In these cases it is usual to consider dispersion relations with: (a) complex frequencies o3 versus real k, or (b) complex wavenumbers /~ versus real w. Correspondingly we agree to write for plane waves:

O(x, t ) =

i e-~' cos(kx - tot), 03 = t o - i T ,

(2.7a)

e -~x c o s ( k x - t o t ) , k= k+i6,

(2.7b)

where y, 6 are usually referred to as the time and space attenuation factors, respectively. These waves turn out to be effectively attenuated if y, are positive, and are harmonic only in space or in time, respectively. The energy velocity associated with the waves (2.7) can still be defined by (2.3) but the averages are to be taken over a wavelength (at a fixed time) or a period (at a fixed position), respectively for the cases (a), (b). One may now expect that the energy velocity be given by the formula (2.4) but this is true only approximatively for quasi-conservative systems (see e.g. [8, 9]). There are cases, particularly those in which there is anomalous dispersion with absorption, where the group velocity (2.4) may lose any physical meaning and a specific treatment is required as first shown by Brillouin [3], with respect to electromagnetic waves in a dielectric. For stress waves in an isotropic viscoelastic solid, Bland [5] proved that the energy velocity (2.3) turns out to be the phase velocity (2.2). Another noteworthy analysis on energy velocity of damped waves was carried out by Vainshtein [4], who considered the concept of velocity of the "center of energy" of a quasi-monochromatic signal starting from the energy balance equation (2.6). He has shown that this velocity depends only weakly on the signal shape and can in many important cases be found from the simple "energy" formula (2.3). In particular he devoted this attention to electromagnetic waves in absorbing plasmas and dielectrics. In the case of the electric and magnetic fields satisfying the simple telegraph

203

F.. Mainardi / Energy velocity

equations, he proved that the energy velocity (2.3) coincides just with the phase velocity (2.2). Since the author [10-12] has recently discussed the related concept of "signal velocity" in dispersive media with dissipation, showing the conditions in which this velocity is given by the group or by the phase velocity, it is worthwhile to examine the corresponding question with respect to the "energy velocity". The (surprising) coincidence of this velocity with phase velocity, which is found in a number of cases, requires an appropriate investigation. Our attention will be restricted to dispersive motions which are governed by hyperbolic partial differential equations, assuming as usual the definition (2.3) for energy velocity.

3. A model wave equation

As a model equation for dispersive waves we consider the following one-dimensional wave equation @it + 2a@, +/32(~) = c2~xx,

order to infer the correct expressions of the energy terms entering the balance equation (2.6). In the case of a uniform stretched string anchored elastically to its equilibrium position by a transverse restoring force and damped by air friction, the equation (3.1) governs the transverse displacement ~(x, t). The constants 2a,/3 2, c 2 refer to air damping, restoring force and tension, respectively (see [13]). In this case the energy terms ~, and 5~ are given by

~; = - c 2 ~ x ~ , ,

(3.2)

6e = 2a~2,,

and the energy balance equation (2.6) may easily be verified using equations (3.1) and (3.2). In the case of a uniform transmission line, characterized per unit length by shunt capacitance C, self-inductance L, resistance R and shunt conductance G, the equation (3.1) individually governs the voltage V and the current I with

(3.1)

where @ = ~(x, t) and a,/3, c are constants. This equation occurs in a variety of wave phenomena such as waves on an elastically supported string, waves in electrical transmission lines, electromagnetic waves in plasmas. In (3.1) a,/3 denote two nonnegative parameters which have dimension of frequency ([ T] -1) and c denotes a characteristic velocity. When a/3 = 0 we obtain particular equations referred to as: D ' A l e m bert (0 = a =/3), K l e i n - G o r d o n (0 = a
c2 = 1 LC' ___ l_

ot

2\L

(3.3)

C]'

/32=R LC"

From (3.3) it is easy to infer that 0<~13~
~,/-~L-_/32,

R / L = p+ = o~+x/t~2-/3 2,

(3.4)

so that0<~p <~p+<~2a. In order to express the energy terms by means of a unique variable satisfying (3.1), let us introduce ~ = ~(x, t) by putting V(x, t)=-¢'x,

I ( x , t) = C ~ , + G ~ .

(3.5)

204

F. Mainardi / Energy velocity

It is easy to verify that q~ satisfies (3.1) by considering the original first-order system of telegraph equations (see e.g. [14])

The averaged quantities can easily be expressed in terms of to, k by recalling that (cos 2 0) = (sin 2 0) = ½, (3.10)

CV, + I~ + G V =O,

(3.6) L I , + Vx+ R I =O,

(sin O cos 0) = 0, where 0 = k x - wt, and by using the complex dispersion relation associated with (3.1). For time/space damped waves (2.7(a)/(b)) the dispersion relations read, respectively

and that the energy terms read = I( C V 2 + L I2)

__(~92q_ c2 k 2 _ 2iao5 +/32 = 0

= s1p , ( ~ ,2+ c 2 ~ 2+ p _ 2 ~ 2 +2p_~,~), ~ - = wr = _ # ¢ 2 ( ¢ , f l ,

+ p_C,fl,),

/ - t o 2 + c 2 k 2 + / 3 2 = 2 a y - y 2, " T = oq

,9° = G V 2 + R I 2

- t o 2 + c 2 ~ - 2 i a t o +f12 = 0

= ~ [ p_(c~x)2 + p+(~, + p_ ~)2],

f--to2q- ¢2k2 q-/32---: c262, where /x = L C 2. We can also verify the energy balance equation (2.6) by using (3.1) and (3.7) with (3.4). It is worthwhile to note the difference between the equations (3.2), (3.7) which provide the energy terms in the mechanical and electromagnetic cases, respectively. We get energy similarity only when /3 = 0, so that p_=0, p+ = 2a from (3.4), namely when the transmission line has no shunt conductance ( G = 0 ) and the string is not elastically anchored. Both cases, of course, exhibit dissipation of energy since 6e/> 0. In order to obtain the energy velocity (2.3) for damped plane waves (2.7), which are solutions of our equation (3.1), we now proceed by calculating the averages of the quantities ~: and ~ by distinguishing the two physical cases previously considered. For the mechanical string we get from (3.2)

(3.8)

while for the electrical transmission line we get from (3.7) - 2 c2 ( tyr~xCr#, + p _ Cl-)xCP) 2 ~x+p_~ 2 2 2 +2p_~,~)'

V ~ = ( ~ ,2+ c

(3.9)

(3.1 lb)

( 8 = a t o / c2k.

For the mechanical case we get from (3.8) and (3.10), (3.11), after simple manipulations, (see also [13]) tokc 2 tokc 2 V e - ot 2 + to~------~-/32+ c 2 k 2.

(3.12)

We recognize that the energy velocity turns out to be not greater than c (the wavefront velocity) and that it reduces to the phase velocity Vp = t o / k if /3 = 0, and to the group velocity Vg kc2/to if a = 0. For the electrical case the algebra is less easy but the final result is simpler. Using (3.9) and (3.10) we get for time/space damped waves: =

2tokc 2 Ve = y 2 + t o 2 + c E k 2 + p 2 _ _ 2 p _ y

Ve -- 2¢2 ( (T2)xCla,) Ve = ((I)2 -b ¢2 (j~2 q_/32~2) ,

(3.11a)

(3.7)

:

2[ tokc2 + p_6] c2 ~2..F to Ec2 k2..}_ p2- .

,

(3.13a)

(3.13b)

Eliminating the attenuation coefficients y, 6 by using the dispersion relations (3.11) and p_ by the equation (3.4), we get finally Ve = to~k,

(3.14)

which states the identification of energy velocity with phase velocity.

205

F.. Mainardi / Energy velocity

The difference between the results (3.12) and (3.14) is of course a consequence of the different energy balances of the corresponding phenomena, no matter if they exhibit the same dispersion relation. On the other hand, the dispersion relation is expected to essentially govern the kinematics of wave propagation. These results will also be obtained in Section 5 as an application of a general approach to energy velocity which is introduced in the next section.

with ~ :½(~U),

~ = ½(t)AU),

and ~ = (UMSU),

MS = ½(M + hT/).

A large class of waves in physics can be described by a hyperbolic system of n first-order partial differential equations and of symmetric type. In the linear and one-dimensional case, we refer, without loss of generality, to the normal form. We write:

Ut+AUx+MU=O,

U=U(x,t),

(4.1)

where subscripts denote partial differentiations, U is a real column vector that is U = col(u~, u 2 , . . . , u,), and A, M are constant real n x n matrices. The matrix A is assumed to be symmetric, that is A = ,4, and its eigenvalues, which provide the characteristic velocities, are denoted by Ai. Systems of this type have received much attention in physics as well as in mathematics, particularly in the quasilinear case (see e.g. [15-20]). The present linear case has been investigated by Broer and Peletier [6, 7] with respect to energy propagation of harmonic waves. They have provided an interesting and alternative proof of the identification o f energy velocity with group velocity when energy is conserved. We will extend their analysis to nonconservative waves. Indeed a balance equation of the type (2.6) can easily be obtained from (4.1) using the property of symmetry of A and the energy velocity can still be defined by (2.3). Multiplying (4.1) to the left by U and the transpose of (4.1) to the right by U we obtain ~ , + ~x +S¢= 0,

(4.2)

(4.4)

We assume that, by an appropriate choice of U, the equations (4.2)-(4.4) may just describe the energy balance law introduced in (2.6). The dispersion properties of the waves can be derived by considering the following solutions

U(x, t) : Re{V(k) exp i[kx-o3(k)t]},

4. Symmetric hyperbolic systems

(4.3)

(4.5)

if we choose the wave number k as an independent variable. The quantity o3 denotes the complex frequency, which reads, adopting the same notation as for scalar waves (2.7a), o3 : to - i y.

(4.6)

As a consequence the wavemode o3 = o3(k) and the corresponding amplitude V(k) can be found from the following eigenvalue problem

[ka - i M - O3I]V(k) = 0.

(4.7)

The energy velocity associated with the waves (4.5) can be obtained from equations (2.3) and (4.3), i.e. from

Vo : (I]AU)/(I]U),

(4.8)

by inserting the solution (4.5) and averaging over the wavelength L = 2xr/k. Recalling the following formula 2(Re[a] Re[b]) = R e [ a ' b ] = Re[ab*], where a and b are two periodic functions of x having the same wavelength and * denotes complex conjugate, it is easy to prove from (4.5) and (4.6) that

(fJu): x( vt v), (4.9)

(I~IAU) = X ( V t A V ) , where x = ½ e x p ( - 2 y t ) and f denotes the Hermitian conjugate. As a consequence the energy velocity (4.8) reads

Ve = ( V t A V ) / ( V t V).

(4.10)

206

F. Mainardi / Energy velocity

As a corollary to (4.10) (see also [6, 7]) it is seen that, since Xmin( V t V) ~ ( V t A V ) <~Xmax(Vt V)

notation. We remind that the proof is valid only if the energy is conserved in all the modes. Differentiating the equation (4.7) with respect to k yields after some rearrangement

we have hmin<~ Ve ~< hmax.

(4.11)

Hence the energy velocity of any mode is bounded from above and below by the extreme characteristic velocities. Now we are going to relate for any mode the energy velocity (4.10) to the dispersive properties of that mode which can be deduced from (4.7). Multiplying (4.7) to the left by Vt and its Hermitian conjugate to the right by V we deduce

= (k(VtAV) -i( VtMV))/(Vt

V), (4.12)

~* = ( k ( V t A V ) + i( VtiC/IV))/( V t V),

-d-~ I - A

V+(toI-kA+iM)-~=O. (4.16)

Multiplying (4.16) to the left by Vt and using the properties: M =-hT/, to real, results in

v*(dto-A) (4.17) dV + [ ( t o / - kA + iM) V]t ~ = 0. Because of (4.7) the second term in (4.17) is equal to zero, whence

dto/ dk = ( V t A V)/ ( V t V),

so that Re[o3 ] =

k(VtAV) (vtv)

i [Vt(M-1C4)V] 2

(vtv)

(4.18)

which implies through (4.10) the identification of the group velocity with the energy velocity.

'

(4.13)

1 [Vt(M+I~I)V]

Im[a3 ] =

2

(vtv)

5. Examples

Using (4.6) and (4.10) we get from (4.13) the following equations for the energy velocity and the attenuation factor:

to

i (VtMaV) (VtV) '

Ve=k +k

M ° = ½(M - ~4),

(4.14)

(VtMSV) ~-

(vtv)

'

M s = ½(M + hT/).

(4.15)

We recognize from (4.14) that when the matrix M in (4.1) is symmetric, the energy velocity of any mode equals the phase velocity of that mode, a general result that has not, to the author's knowledge, been remarked on previously. From (4.15) we recover the fact that any mode is conservative if the matrix M in (4.1) is antisymmetric. In this case Broer and Peletier have proven the classical result of the equivalence between energy and group velocities. For the sake of completeness we quote their proof hereafter, using our

To elucidate and check the results of Section 4, we return to the physical phenomena described in Section 3 by the same wave equation (3.1), which, however, exhibit different energy velocities ((3.12) and (3.14)). First let us consider the mechanical case of a uniform stretched string. We have to transform the second-order wave equation (3.1) into an equivalent first-order system which preserves the energy properties stated in (3.2). Choosing U = col(~t, c ~ x , / 3 4 ) we obtain the required system of type (4.1) with n = 3, where A=

c

0 0

, (5.1)

M=

0 0

.

207

F. Mainardi / Energy oelocity

It is easy to check that the energy relations, derived from (5.1) according to (4.3) and (4.4), agree with the expressions (3.2) stated previously for the string. Now we intend to check the validity of the following qualities:

V~

( V t A V)

(vtv) ~kc 2 /32+ ¢2k2,

oJ i ( V t M ' V )

k+ k (VtV)

(5.2)

Equations (5.7), (5.10) just prove the equalities in (5.2). Now let us consider the telegraph case. We have to transform the system (3.6) into the appropriate normal form compatible with the energy properties stated in (3.7). Choosing U=coI(x/-CV, x/LI) we obtain the required system of type (4.1) with n = 2 , where, using the notation introduced in (3.3) and (3.4), A=

thus confirming (4.10), (4.14) and (3.12). From the eigenvalue problem (4.7)-(5.1) we get ~[(03+ia)2-(c2k2+/32-a2)]=O.

(5.4)

After simple algebra we get ( V t V) = 03~* +/32 + c2k 2

= 2(/32+ c2k2),

(5.5)

and (VtAV)

= c 2 k ( ~ + 03*) = 2toc2k,

1/#L--C

(5.3)

Neglecting the spurious eigenvalue 03 = 0, we just obtain the same dispersion relation as in (3.11a), and the wave modes with amplitude V ( k ) = co1(03, -ck, i/3).

( 0 0 0

/

c 0'

(o o)

It is easy to check that the energy relations, derived from (5.11) according to (4.3) and (4.4), agree with the energy relations (3.7) previously stated for transmission lines. Now, since M is symmetric, the energy velocity is expected to equal the phase velocity according to (4.18), thus confirming the result (3.14). From the eigenvalue problem (4.7)-(5.11) we derive the dispersion relation

(5.6) (a3 + i o t ) 2 = c2k2-1t-/3 2 - a 2,

so that (VtAV)

mkc 2

(VtV)

/32+c2k2"

(o0 )

(5.7)

0 0

V ( k ) = col(ck, 03+ip_).

,

(5.8)

(5.12)

in agreement with (3.1 l a), and the corresponding wavemodes with amplitude

Noting from (5.1) that M"=

(5.11)

(5.13)

After simple algebra we get ( V t V) = 2c2k 2 + (/3 2 - 2otp_ + p2_) = 2c2k 2,

we get

(5.14)

( V t M ~ V ) = i/32(03 + 03*) = 2i/32to,

(5.9)

so that

and (VtAV)=c2k(N+03*)=2c2kzo,

o~ + i ( V t M a V ) k k (VtV)

a~( /32) ~kc 2 = ~ 1 /32~--c2k2 =/32+c2k2.

(5.15)

so that

(5.10)

(VtAV) lie-- ( V t V------T- -k =- Vp .

(5.16)

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F. Mainardi / Energy velocity

Acknowledgment This work was performed in the framework of the "Gruppo Nazionale per la Fisica Matematica", C.N.R. (Italy). The author would like to thank E.W.C. van Groesen and G. Vitali for helpful criticism.

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[8] G.B. Whitham, Linear and Non Linear Waves, Wiley, New York (1974). [9] M.J. Lighthill, "Group velocity", J. Inst. Math. Appl. 1, 1-28 (1965). [10] F. Mainardi, "Signal velocity for transient waves in linear dissipative media", Wave Motion 5, 33-41 (1983). [11] F. Mainardi, "'On signal velocity for anomalous dispersive waves", I1 Nuovo Cimento 74 B, 52-58 (1983). [12] F. Mainardi, "On linear dispersive waves with dissipation", in: C. Rogers and T.B. Moodie, eds., Wave Phenomena: Modern Theory and Applications, NorthHolland, Amsterdam (1984) 307-317. [ 13] G.R. Baldock and T. Bridgeman, The Mathematical Theory of Wave Motion, Ellis Horwood, Chichester (1981) Ch. 5. [14] H. Levine, Unidirectional Wave Motions, North-Holland, Amsterdam (1978) 95. [15] P.D. Lax, "Hyperbolic systems of conservation laws, I I ' , Comm. Pure Appl. Math. 10, 537-566 (1957). [16] K.O. Friedrichs, "Differential equations of symmetric type", in: J.B. Diaz and S.I. Pai, eds., Fluid Dynamics and Applied Mathematics, Gordon and Breach, New York (1962) 51. [17] G. Boillat, La Propagation des Ondes, Gauthier-Villars, Paris (1965). [18] A. Jeffrey, Quasilinear Hyperbolic Systems and Waves, Pitman, London (1976). [19] T. Taniuti and K. Nishihara, Nonlinear Waves, Pitman, London (1983). [20] T.A. Ruggeri, "Entropy principle, symmetric hyperbolic systems and shock waves", in: C. Rogers and T.B. Moodie, eds., Wave Phenomena: Modern Theory and Applications, North-Holland, Amsterdam (1984) 211-220.