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Energy and dissipation of inhomogeneous plane waves in thermoelasticity
ELSEVIER
Wave Motion 23 ( 1996) 393406
Energy and dissipation of inhomogeneous plane waves in thermoelasticity N.H. Scott School of Mathematics,
Universiry of East Anglia, Norwich NR4 7TJ, UK
Received 29 September 1995; revised 28 November 1995
Abstract Inhomogeneous small-amplitude plane waves of (complex) frequency ware propagated through a linear dissipative material. For thermoelasticity we derive an energy-dissipation equation that contains all the quadratic dependence on the field quantities, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. The terms quadratic in the small quantities in the energy density, energy llux and dissipation give rise to inhomogeneous plane waves of frequency 2w and to (attenuated) constant terms. Usually these quadratic quantities are time-averaged and only the attenuated constant terms remain. We derive a new result in thermoelasticity for these terms, see Eq. (54). The present innovation is to retain the terms of frequency 2w, since they are comparable in magnitude to the attenuated constant terms, and a new result, see Eq. (44), is derived for a general energy-dissipation equation that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2w. Furthermore, for dissipative waves or inhomogeneous conservative waves the (complex) group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.
1. Introduction We consider the propagation of inhomogeneous small-amplitude plane waves through a continuous linear dissipative material with special reference to the linearized theory of thermoelasticity in which heat conduction provides the mechanism for dissipation. For such waves the particle displacement field takes the complex exponential form u(x, t) = (U exp io(S.x
- r)}+,
(1)
where U = U+ + ilJ- is the complex wave amplitude (or polarization) vector, w = w+ + iw- is the complex frequency and S = S+ + iS- is the complex slowness vector. These are constant quantities with the variables x and t denoting position and time, respectively. Throughout, the superscripts + and - refer to the real and imaginary parts of a complex quantity. If the planes of constant phase S+.x = constant and the planes of constant amplitude S-.x = constant are not parallel then the complex exponential solution Eq. (1) is said to represent an inhomogeneous plane wave. It is assumed that every other field quantity has the form of Eq. (1) with some other complex scalar, vector or tensor replacing lJ. Inhomogeneous plane waves arise in many different areas of mechanics both for conservative and for dissipative media. Examples include Rayleigh and Stoneley waves in linearized elasticity, electromagnetic radiation in wave 01652125/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved PIISOl65-2125(96)00003-O
394
N.H. Scott/Wave
Motion 23 (1996) 393406
guides and surface and body waves in viscous fluids, viscoelastic solids and thermoelastic solids. We study inhomogeneous waves both for their intrinsic interest and because all solutions of linear problems may be written as linear superpositions (whether as finite sums or integrals) of waves of the form of Eq. (1). In this paper we are particularly interested in energy density, energy flux and dissipation. Apart from linear terms, which we consider separately, these quantities are made up of sums of products of two field quantities each of the form of Eq. (1) and therefore consist of two parts, one an attenuated harmonic part with frequency 2w and the other an attenuated constant, with both parts suffering the same degree of attenuation. Until now attention has been focused in the literature mainly on the attenuated constant amplitudes, which may be obtained very simply from the partial differential equations governing the particle displacements and any other field quantities by time-averaging these equations over a cycle. This time-averaging process removes the amplitudes of the attenuated harmonic parts which has led to their being ignored in the literature on the grounds that they make no net contribution. However, the energy-dissipation equation is valid for all x and t, without time-averaging, and the neglected terms are of the same order of magnitude (before time-averaging) as the retained attenuated constant terms. It therefore seems worthwhile to explore the consequences of keeping these attenuated harmonic terms and this enterprise forms a major part of the present paper. Boulanger and Hayes [ 1] have given a lucid account of the theory of homogeneous and inhomogeneous plane waves [ 1, Ch. 61, energy flux [ 1, Ch. 81 and dissipation [ 1, Section 11.51, the last containing some general results on wave propagation in linear dissipative media though it is set in the context of incompressible viscous fluids. Although the amplitudes of the attenuated harmonic terms are mentioned in Ref. [ 11 no specific results are derived. Noda [2], on the other hand, argues from a general energy-dissipation equation, similar to Eq. (lo), and obtains specific equations involving the amplitudes of the attenuated harmonic terms, see [2, Eqs. (2.12) and (2.24)], which reduce to our Eqs. (44) and (64), respectively, when there are no slow variations in the amplitudes, see [2, Section 21. However, Noda makes no use of an explicit dispersion relation and so is unable to derive the connection between the complex group velocity, energy density and energy flux given in Eq. (69) in the context of thermoelasticity. In Section 2 the basic equations of linear thermoelasticity are set out and an energy-dissipation equation involving the quadratic terms in the energy density, energy flux and dissipation, is derived from them, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. In Section 3 the propagation condition for inhomogeneous plane waves in thermoelasticity is derived. In Section 4 the amplitudes of the energy density, energy flux and dissipation associated with the attenuated harmonic parts (of frequency 20), which are associated with terms quadratic in the field quantities, are denoted by the complex constants E,R and D, respectively, and the amplitudes - associated with the attenuated constant parts are denoted by the real constants e, r and d, respectively. Arguing from an energy-dissipation equation of general validity we are able to deduce the result wE - K.R + i D = 0 with K = WS denoting the complex wave vector. For thermoelastic materials we show further that 6.~2- K? + id = 0. In Section 5 we show that for inhomogeneous plane waves in thermoelasticity or isothermal elasticity the (complex) group velocity is related to the attenuated harmonic energy density and flux by R = E&o/aK. This is in contrast with the relation r = Z aw/aK, well known in the theory of homogeneous waves in conservative media. The relationship between these two equations is discussed in Section 6 in the context of isothermal elasticity.
2. Basic equations and energy dissipation For a thermoelastic and heat supply are
body the linearized equations of momentum
.. aij,j
=
PUi,
Td
=
-qj,j,
and energy balance in the absence of body force
(2)
respectively, see [3, Section 71 or [4, Eqs. (16) and (17)], in which ~7 denotes the Cauchy stress and q the heat flux. The particle displacement vector U(X, t) and entropy increment 4(x, t) denote, as functions of particle position x
395
N. H. Scott/ Wave Motion 23 (1996) 393406
and time I, the small-amplitude departures of the particle position and entropy per unit volume, respectively, from their constant values measured in a spatially uniform, time-independent, stress-free equilibrium state. The constant equilibrium values of the density and absolute temperature are denoted by p and T, respectively. The notation ( ),j denotes the spatial partial derivative a( )/axj and the superposed dot denotes the partial derivative with respect to time t. Twice-occurring roman suffixes are summed over. The constitutive equations appropriate to linearized thermoelasticity are oij
=
dijkiuk.1
-
(3)
Bij@y
for the stress, and Fourier’s law of heat conduction CJi =
-K,>O,j
for the heat flux, with 4
=
BijUi,j
+
p”6,
T
supplying the connection between the entropy per unit volume increment 4(x, t) and the temperature increment 0(x, t), see [3, Section 71. Definitions of the isothermal elasticity tensor d, the temperature coefficient of stress 0, the thermal conductivity tensor K,’and the specific heat at constant deformation c may be found in [3] or [4]. These quantities here take on constant values determined by the underlying equilibrium state. We may now show that the field equations of linearized thermoelasticity take the form: dijkiuk,j/
-
/%je,j
=
K;jO,ij
Pii,
-
T#?ijtii,j =
PCd,
(6)
in the absence of body force and heat supply, see [3, Section 71, [4]. The first is obtained by substituting for u from (3) into (2)t and the second by substituting for q and C#J from (4) and (5) into (2)~. Eqs. (6) comprise a system of four linear, constant-coefficient partial differential equations for the four unknowns Ui, i = 1,2, 3, and 8. We take the material body whose equilibrium state is subjected to the disturbances Ui and 8 to be infinite in extent, so that boundary conditions need not be considered. In order to derive an energy-dissipation equation from Eq. (6) we first multiply Eq. (6)i by tii, sum over i, and rearrange to obtain +
idijklUi,jUk,l
in which the symmetry
I
+
{-dijk[Uk,lki + Bijetii},j = Bijlii,je,
property
(8)
dijki = &ij has been used. We then multiply Eq. (6)~ by 0/T
and rearrange to obtain (9)
On eliminating g
+ rj,j
Bijtii,jQ between Eqs. (7) and (9) we obtain an energy-dissipation
equation
= -2d,
in which the quantity
(11)
N.H. Scott/Wave
396
Motion 23 (1996) 3934%
is an energy density per unit volume, the components Yj
=
-dijklUk,lLii
are the components
+
/$jeci
-
e TUjie,i
(12)
of an energy flux vector and we shall see that the quantity (13)
may be interpreted as a dissipation (per unit volume) of energy. Without loss of generality the components the thermal conductivity tensor have been replaced in Eq. (13) by those of its symmetric part
K;~
of
We do not assume that the thermal conductivity itself is symmetric. Eq. (10) is the field equation, in the absence of body force and heat supply, corresponding both to Nowacki’s fundamental energy theorem [5, p. 3 11, Eq. (1 l)] and to Carlson’s power and energy theorem [3, pp. 331,332]. The first term of Q. (11) represents the kinetic energy density, the second represents the elastic strain energy density and the third represents the heat energy density. The first two terms of the energy flux Eq. (12) are derived from the rate of working of the stress tensor Eq. (3), the first term due to purely mechanical stresses and the second due to the temperature dependence of the stress. The last term of the energy flux Eq. (12) derives purely from the flow of heat within the body. Turning attention now to the dissipation Eq. (13) we recall that the components of heat flux are given by Eq. (4) and that the second law of thermodynamics, see Ref. [3], requires there to be no component of heat flux against the temperature gradient, i.e. qje,j 5 0, which in turn requires the dissipation Eq. (13) to be non-negative:
so that K is positive semi-definite. In fact it is commonly assumed in thermoelasticity that K. is positive definite, so that d > 0 unless the temperature is spatially homogeneous. The fact that d is non-negative justifies its interpretation in Eq. (10) as a dissipation of energy. In Eqs (1 I)-( 13) the displacement components ui, the temperature increment 8 and all their derivatives are to be regarded as small quantities so that all terms occurring in these equations are quadratic in the small quantities. It follows that the energy-dissipation equation Eq. (10) also involves only quadratic terms. However, Chadwick [6] has pointed out that the total energy density in thermoelasticity is not given by Eq. (11) but contains also terms linear in the small quantities. This observation will eventually lead us to an energy-dissipation equation, alternative to Eq. (lo), that involves linear terms in both the energy density and the energy flux, see Eq. (22). First, we reproduce Chadwick’s [6] derivation of an expression for the internal energy density which is correct up to and including quadratic terms in the small quantities. From [7, Eq. (3.17)] we find that, to quadratic order in the small quantities ui,j and 0, the Helmholtz free energy per unit volume is given by A = A0 -
see+ idijklUi,jUk,l
- /$jUi,je
-
g02
(14)
in the absence of pre-stress. The constant equilibrium values of the Helmholtz free energy and the entropy are denoted by A0 and So, respectively, whilst that of the absolute temperature continues to be denoted by T. From [3, Eqs. (4.7), (5.4) and (7.4)] we see that the internal energy, is to quadratic order, given by
in which EO= A0 + TSo is the equilibrium
internal energy.
N.H. Scott/ Wave Motion 23
(1996)393-406
397
The total energy density 2 is defined to be the sum of the kinetic and internal energy densities: e=k+&,
k = ;ptiiLi;,
(16)
where k is the kinetic energy density. From Eqs. (1 l), (15) and (16) we find that the total energy density is given to quadratic order by ? = Fg + e’ + e,
e’ =
TBijUi,j
+
(17)
PCO,
in which e’ gives the linear and e the quadratic terms. From Eq. (5) we see that Eq. (17)~ may be written as e’ = T@ showing that both terms on the right-hand side of Eq. (17)~ may be regarded as heat energy, the first being partly mechanical in origin and the second purely thermal arising from the heat capacity of the material. From Eqs. (17) and (1 1) we obtain the explicit expression
for the total energy density. Notwithstanding rather that k + E, as the total energy. A total energy flux may be defined by Fj = r,j + rj ,
Eq. (IS), Carlson [3, p. 3321 refers to an integrated
rj = -/cjliH,;
form of k + P,
(19)
in which t-J gives the linear and rj, defined in Eq. (12), the quadratic terms. From Eq. (4) we see that rJ is simply the heat flux. From Eqs. (19) and (12) we obtain the explicit expression
for the total energy flux. By an appeal to Eq. (6)~ it may be verified that the linear quantities conservation equation
defined by Eqs. ( 17)~ and ( 19)t satisfy the
Thus there is no dissipation of the linear terms. Adding together Eqs. (10) and (21) and appealing to the definitions ( I7)t and (19)t allows us to deduce the energy-dissipation equation
involving the total energy density 2 of Eq. (18), the total energy flux fj of Eq. (20) and the dissipation d of Eq. (13). The phenomenon of separate energy balances for the linear and quadratic terms has been encountered previously, by Lighthill [8, p. 1.51in the context of acoustics and by Chadwick et al. [9] in the context of constrained elasticity. The energy-dissipation equation Eq. (22) is to be regarded as more fundamental than Eq. ( 10) because it involves the actual energy density rather than just its quadratic terms. Nevertheless, in the remainder of this paper we concentrate exclusively on Eq. (10) because we are concerned here with quantities arising as the product of field quantities and these are necessarily of the second order. All terms quadratic in the small quantities are included in Eq. (IO).
3. Inhomogeneous
plane waves and propagation
conditions
We now assume the small disturbance in the thermoelastic body to have the most general complex exponential plane wave form possible, so that the particle displacement u and the temperature increment 8 take the forms
398
N.H. Scott/Wave Motion 23 (1996) 393406
24 =
(UeiX)+,
(23)
I9 =
{Oe’q+,
(24)
in which the phase factor x is defined by x = w(Scc - t).
(25)
Eqs. (23) and (1) are the same and all quantities @=@++i@-.
were defined in Section
1 except the temperature
amplitude
For the most part we shall prefer to write the phase factor x in terms of the complex wave vector
K = WS
(26)
so that X=K.~-wt=x~+ix-,
(27)
where
x+ =K+.x-o+t,
x-
= K-.x
- w-t
(28)
are the real and imaginary parts of x expressed in terms of those of K and w. From Eq. (28) we see that the wave amplitudes (23) and (24) may be written * =
{Ue’X+}+e-X-,
0 =
(eeix+
(29)
)+e-xm,
(30)
from which it is clear that these wave amplitudes
represent a sinusoidal
travelling
wave of frequency w+ and wave
vector K+ which is attenuated by the real exponential factor e-x-. On substituting the forms (23) and (24), with x given by Eq. (27), into the four partial differential we find that they are reduced to the following system of four algebraic equations: PW2Ui - dijklKjUkKl
-
y@
+iwBijuiKj
+ itCijKiKj@
for the four amplitudes
n4
ZJi, i = 1,2,3,
[I g
ipijKj@
equation (6)
= 0, = 0
(31)
and 0. Eqs. (31) may be written more compactly
as
=o,
(32)
0
where the 4 x 4 complex symmetric
matrix M is defined by
_ &(K) M(w,
K) =
Q22CK) 031 (K)
-
032(K)
i&j Kj
C&(K) pm2
Q23 (K) 033(K)
-
i&j Kj
$lj Kj iP2j
w2
i&j
!E+-
Kj Kj
i/c(K) WT
1
(33)
N.H. Scott/Wave
and [U) , U2, lJ3, @IT is the wave amplitude are given, as function of K, by &k(K)
vector. In Eq. (33) the components
399
of the isothermal acoustic tensor 0
(34)
= dijkl Kj KI
and the scalar conductivity K(K)
Motion 23 (1996) 393406
is given, again as a function of K, by
= KijKiKJ.
(35)
In order for Eq. (32) to deliver non-trivial
wave amplitudes
we require
K) = 0.
detM(w,
(36)
Eqs. (32) and (36) are often termed the propagation conditions of linear thermoelasticity. We see from Eq. (33) that these propagation conditions depend upon both K and w explicitly. Eq. (36) furnishes an implicit form of the dispersion relation for thermoelasticity, see Section 5. For the most part we shall work from Eqs. (3 1) directly and so refer to them also as the propagation conditions.
4. Energy density, energy flux and dissipation 4. I. General
results
We see from Eqs. ( 1 l)-( 13) that the energy density, energy flux and dissipation that occur in Eq. ( 10) are quadratic in the small quantities and so, for inhomogeneous plane waves, are expressible as linear combinations of products of the form f(x, t) = (Ae’X]+{Beix]f
(37)
in which A and B are complex constants.
Using Eq. (27) we may evaluate the product (37) to obtain
.f’(x, t) = ( Fe2iX ]+ + fern’“= (Fe2’X+] +,e2x-
+ f-e-2xm (
(38) (39)
where F = $AB.
,f=
($AB*)+,
(40)
in which F is a (usually) complex constant and f is a real constant. Here and throughout * denotes the complex conjugate. The first term of Eq. (38) represents an inhomogeneous plane wave with phase factor 2x, whilst the second consists of the real constant f attenuated by the real exponential factor e -2x1. From Eq. (39) we see that the inhomogeneous plane wave is attenuated by the same factor. In order to interpret the quantity f we follow Boulanger and Hayes [ 1, Section 1 I .5] and integrate Eq. (39) over a cycle of x + at constant x- to show that the mean value of f is fee’x- , a quantity that depends on x and t through x -. The real constant f is then regarded as a weighted mean of f(x, t). We have already observed that the energy density, energy flux and dissipation that occur in Eq. ( 10) may all be expressed as linear combinations of products of the form (37) and so, utilizing an approach of Hayes 1lo], we may appeal to Eq. (38) in order to write each in turn as r=(Ee
2ix ]+ + ee-2x-
rJ = (Rje2iX)+ +
Fjep2x-,
d = ( De2’X )+ + aep2x-,
(41) (42) (43)
400
N.H. Scott/Wave Motion 23 (1996) 393406
in which E, Rj, D are (usually) complex constants and 2, yj, d are real constants. These latter constants are the weighted means of e, rj, d as discussed above. In the past, see, for example, Ref. [I], discussion of energy and dissipation has focused on the weighted means at the expense of terms involving the complex quantities E, Rj, D, often on the grounds that these terms do not equation Eq. (10) remains valid for contribute when averaged over a cycle of x +. However, the energy-dissipation all x and t, without averaging, and the neglected terms are of the same order of magnitude (before averaging) as the retained terms. It is our chief purpose in this paper to explore the consequences of retaining the attenuated harmonic terms on an equal footing with the weighted means. On substituting Eqs. (41)-(43) into Eq. (10) and equating the coefficients of the attenuated harmonic terms, and those of the purely attenuated terms, we obtain oE-K.RfiD=O,
(44)
w-2-K-?+d=O,
(45)
respectively. Eqs. (44) and (45) are equivalent to Eqs. (2.12) and (2.13) of Ref. [2], respectively, and Eq. (45) has been given also by Boulanger and Hayes [ 1, Eq. (11.5.8)] who derived it, in the general context, from Eqs. (41)-(43) by averaging Eq. (10) over a cycle of xf at fixed x- and then cancelling the attenuation factor ee2X-. All the quantities occurring in Eq. (44) are (usually) complex whilst all those occurring in Eq. (45) are real. Eqs. (44) and (45) have a wide range of validity, not only in thermoelasticity, since they are valid for any system that has an energy-dissipation equation of the form of Eq. (10) with energy density, energy flux and dissipation taking the forms given by Eqs. (41)-(43). Boulanger and Hayes [ 111 and Scott [ 121 have illustrated Eq. (45) for the special cases of incompressible and compressible viscous fluids, respectively. For the special case of real frequencies, Buchen [ 13, Eq. (34)] derived Eq. (45) in the context of isotropic viscoelasticity and Hayes [14, Eq. (5.9)] derived it for any linear dissipative system. We may conclude from Eq. (45) that if w and K are both real then d = 0 and there is no weighted mean dissipation. Alternatively, if there is dissipation (2 # 0) then we may draw the important conclusion that not both of w and K may be real. 4.2. Thermoelasticity - For thermoelasticity we may obtain explicit expressions for the quantities E, Rj, D and e, rj , d occurring in Eqs. (41)-(43) by substituting the inhomogeneous plane wave forms of Eqs. (23) and (24) into Eqs. (I 1)-(13) for the energy density, energy flux and dissipation and using Eqs. (38) and (40): E =-~Pw’U.~JRj =-~~dijktlJiU~K~
idij,iU,KjUkK, - ~io,PijlJi@
+ go’,
(46)
- &KijKie2,
(47)
(51) in which ]z12 = zz*, the squared modulus of the complex quantity z. As has already been remarked, the quantities E, Rj, D are, in general, complex. From the symmetry property (8) we may verify that the weighted mean energy
N.H. Scott/ Wave Motion 23 (I 996) 393406
401
density Z, given by Eq. (49), is real. If, in addition, we make the common assumption of the positive definiteness of d then it follows that Z is strictly positive for a non-trivial wave. It is easily verified that the weighted mean energy flux Fj, given by Eq. (50), is real. The reality of the weighted mean dissipation d, given by Eq. (51), may be verified using the symmetry of K. The common assumption of the positive definiteness of K implies that d is positive, vanishing only if 0 = 0 in which case the wave carries no temperature fluctuation. As in the general case, the non-vanishing of d implies that not both of w and K may be real. It is instructive to verify the general equation (44) in the special case of thermoelasticity. On substituting for E, R and D from Eqs. (46)-(48) into Eqs. (44) we obtain, after some manipulation, WE - K’R + iD =-:{Pw’U~
- dijklKjUkK[ - ipijKjO]Ui
+$ y@ +ftCijKi KjO
+ i@ijUi
Kj
I
0,
each curly bracket on the right-hand side vanishing on account of the propagation conditions (3 1). Thus we have verified the general equation (44) in the special case of thermoelasticity. In the same way that Eq. (52) was derived we may use Eqs. (49)-(51) for the weighted means 2, r, CC? to derive the identity
y@ Again, we find that the right-hand
+ fKijKiKj@
+i@ijUiKj
side vanishes on account of the propagation
(53) conditions
oe - KT + id = 0.
(31) leaving us with (54)
This equation is new to the theory of thermoelasticity though it has been derived for incompressible viscous fluids by Boulanger and Hayes [ 11, Eq. (6.16)] and for compressible viscous fluids by Scott [ 12, Eq. (86)]. Eqs. (44) and (54) have the same form, the first involving the amplitudes of the attenuated harmonic terms and the second involving the weighted means. However, it should be remembered that whilst Eq. (44) has general validity Eq. (54) has been demonstrated here only for thermoelasticity. -Bearing in mind that e, r, d are real we may take real and imaginary parts of Eq. (54) to obtain FK-
f.K+ = w+z,
= w-Z + 2.
(55)
The first of these is another result new to the theory of thermoelasticity though equations of this form have been derived previously in the theory of dissipative media, for incompressible viscous fluids by Boulanger and Hayes [ 11, Eq. (6.17)1], for compressible viscous fluids by Scott [12, Eq. (87)t] and for isotropic linear viscoelastic materials by Buchen [ 13, Eq. (38)] on the assumption of real frequency. As might be expected from the absence of d, Eq. (55)t is valid also for conservative media, which we shall see later. Eq. (55)~ simply verifies for thermoelasticity the general result Eq. (45). We conclude this section by employing Eq. (26) to write Eqs. (44) and (54) in terms of the slowness S rather than K: R.S = E + io-‘D,
(56)
?.S = 2 + iw-‘J.
(57)
Since Eq. (57) has been demonstrated isothermal elasticity, FS = e,
&Sf = 2,
r.s-
for thermoelasticity,
= 0,
taking the non-dissipative
limit (2 -+ 0) gives, for
(58)
402
N.H. Scott/Wave Motion 23 (1996) 393-406
which are equivalent to Eqs. (54) and (55) with 2 = 0. Eqs. (58), and hence Eq. (55)t, have been proved for general conservative systems by Hayes [ 15, Eqs. (4.7) and (4.8)]. Restoring the dissipation d to Eq. (57) and taking real and imaginary parts yields r.s+
in place of (58)2,3. For real frequency $Sf
r.s-
= e + &Y/]w]2,
= 2,
o taking real and imaginary ;.s-
(59)
parts of Eq. (57), or putting w- = 0 in Eq. (59), gives
= w-12.
The first of these equations, thermoelasticity. The second directly from the universally Boulanger and Hayes [ 1, Eq.
5. The dispersion
= &+/]0#
(60)
well known for conservative media, see Hayes [15, Eq. (4.8)2], is here extended to is valid for all dissipative systems since, for real frequency, it can also be derived valid Eq. (45) with the aid of Eq. (26). Such a derivation of Eq. (60)~ was given by (11.5. lo)] in the general context.
relation, group velocity and energy velocity
In theories of continuous media one typically frequency as a function of the wave vector: w = w(K),
derives from the propagation
conditions
an equation
giving the
(61)
known as the dispersion relation. In thermoelasticity the dispersion relation is given in implicit form by Eq. (36). In non-dissipative media the complex wave vector K is replaced by a real one k and the frequency o also is real. Here we wish to explore, in the context of thermoelasticity, the consequences of allowing both wave vector and frequency to be complex. We note from Eqs. (32) and (36) that U and 0, as well as w, are functions of K but not of its complex conjugate K*, and it follows that the complex quantities E, Rj, D, defined by Eqs. (46)-(48), are functions of K but not of K*. Clearly, w*, U* and O* are functions of K*, but not of K, and it follows that the real quantities 2, ?j, d, defined by Eqs. (49)-(5 l), depend upon both K and K*. At this point we break off our discussion of energy and dissipation in thermoelasticity to consider what general results may be available. For any dissipative system that obeys an energy-dissipation equation of the form of Eq. (10) we have shown that the amplitudes of the energy density, energy flux and dissipation satisfy Eqs. (44) and (45). Since for any system the dispersion relation Eq. (61) involves K, but not K*, we would expect the conclusions of the previous paragraph to hold, namely, that the real quantities 2, ?, d, occurring in Eq. (45) depend upon both K and K*, whilst the complex quantities occurring in Eq. (44) depend upon K only. Now Eq. (44) holds for all possible complex wave vectors K and so may be regarded as an identity in K. Allowing the operator a/a Kp to act upon this equation therefore gives
am
-E-R,=-
aKP
Kj
(62)
In the general case we do not have explicit expressions for E, R and D and so can make no further progress. Eq. (45) also holds for all possible wave vectors K but since each of its terms depends explicitly upon K* as well as K it is to be regarded as an identity in each of the six quantities K,,+ , K;, p = 1,2,3. Equivalently, Eq. (45) is an identity in each of the six components of K and K*, with K* now regarded as being independent of K. Accordingly, we may rewrite Eq. (45) as (O - w*)? - (Kj - KT)fj + 2id = 0
(63)
403
N.H. Scott/Wave Motion 23 (1996) 393406
and allow a/aK, obtain
to act upon it, bearing in mind that w depends upon K only and o* depends upon K* only, to
&!!_z_F
we
p-
aK,,
_-2i
ae
afj
--KF-
i aKP
J aK,
+g.
P
I
(64)
Note that the K-derivatives on the right-hand side are evaluated at constant K*. The identity obtained by allowing a/aKi to act on Eq. (63) is easily shown to be equivalent to Eq. (64) because of the reality of ?. r, 2. As with Eq. (62) we do not have explicit expressions for Z, ? and d and so can make little further progress. There is, however, one deduction we can make from Eq. (64). In the case of homogeneous waves in a dissipationless system the complex wave vector K is replaced by the real one k, the frequency w also is real and the dissipation d vanishes, so that o-
KJ: = 0,
= 0,
Li=o
and Eq. (64) reduces to _ am
(65)
r”=ei)ki,
valid for homogeneous waves in a general linear dissipationless system. Equation (65) was first proved by Hayes [ 10, Eq. (20)], also for homogeneous waves in a general linear dissipationless system, by a different method. Hayes’ proof of Eq. (65) proceeds by adding a small, but otherwise arbitrary, imaginary part to the real wave vector k, computing from the dispersion relation the resulting small imaginary addition to the otherwise real frequency w and employing the results of these calculations in the energy-dissipation equation, Eq. (lo), with d = 0. Applying a similar technique to Eq. (10) with complex K and w leads to Eq. (45) instead of Eq. (65). An alternative derivation of Eq. (64) is provided by applying the same technique to Eq. (45) itself. This, in fact, corresponds to the procedure adopted by Noda in proving his main theorem, see Ref. [2, Eq. (2,24)], but because he assumed the dispersion relation only in implicit form, see Ref. [2, Eqs. (2.6) and (2.14)], his result is less explicit and more difficult to interpret than our Eq. (64). We return to our discussion of energy and dissipation in thermoelasticity and seek a connection between the complex group velocity &o/aK and quantities already defined using a method first employed by Fedorov [ 16, pp. 122, 1241 in the theory of elasticity. We apply the operator a/aK, to Eq. (3l)t, contract the resulting equation with Vi and use the symmetry (8) of d to obtain alJi - ZLdipkl Ui UkK/ - $ip Ui O = -pW2 Ui +dijk/UiKjsKi 8KiJ On contracting i/lijUiKjO
+i/IijUiKjs. I’
I’
(66)
Eq. (3 1)t with Ui we find that = PW~U’U - dijklUiKjUkK/.
(67)
now apply the operator a/aK, to Eq. (31)2, multiply the resulting from the coefficient of &/a Kp by means of Eq. (67) to obtain we
I
PO~U,U - dijkl(li KjUkK/ + $e2 aUi
= -iw/$j Kj 0 T
P
-
y@
equation
by 8 and eliminate
i/!lijU; Kj (7)
. y I
$
”
+ i@3ipUiO + $KipKi@’
+ J+/cijKiKj@
From this equation we subtract a multiple w of Eq. (66), divide the resulting (47) for E and R in the left-hand side to obtain
equation by 4 and use Eqs. (46) and
404
N.H. Scott/ Wave Motion 23 (1996) 393406
aw
E-
- Rp = :(PW2Ui
- dijk/KjUkK[
- ip,IKIc)Js
aKP
P
(68) The right-hand side of Eq. (68) is very similar to those of Eqs. (52) and (53) and indeed we note from the propagation conditions (3 1) that it vanishes leaving R,=E*.
(69)
a
KP
This is a new result in the theory of inhomogeneous waves in dissipative media, here demonstrated for thermoelasticity. It connects the complex group velocity with the attenuated harmonic terms in the energy-dissipation equation. Eq. (69) may also be deduced from Eq. (62) by verifying that its right-hand side reduces to zero upon substitution for E, R and D of the explicit forms given by Eqs. (46)-(48). Both these methods of proof depend on a knowledge of the propagation conditions (31) of thermoelasticity, not just on the existence of an energy-dissipation equation of the form of Eq. (10). Noda assumes only an energy-dissipation equation, see Ref. [2, Eq. (2.4)], and so is unable to derive a result as explicit as Eq. (69). Previous work on energy flux and group velocity has largely been for homogeneous waves propagating through a conservative medium, with real frequency w and real wave vector k, for which Hayes [ 10, Eq. (20)] has proved a general result, namely Eq. (65) above. In dissipative media we have deduced from Eq. (45) that not both of o and K can be real and the same is true for inhomogeneous waves in conservative media. Since C?and 7 are real it follows that Eq. (65) cannot hold either for waves in dissipative media or for inhomogeneous waves in conservative media. At least in the theory of thermoelasticity we have shown that Eq. (65) is replaced by Eq. (69). One might expect Eq. (69) to have a wider validity in the theory of dissipative media but this has not been demonstrated. In the next section we shall pursue the connection between Eqs. (69) and (65) for isothermal elasticity, a non-dissipative limit of thermoelasticity. We may define a complex energy velocity vector G associated with the attenuated harmonic terms of the energydissipation equation by G=
energy flux associated with attenuated
harmonic terms
energy density associated with attenuated harmonic
terms
=-
R (70)
E ’
provided E # 0, which from Eq. (69) may be written G=g.
(71)
In terms of G, Eqs. (44) and (56) may be written as G.K = o + iDIE,
GS = 1 f iD/wE,
respectively. The energy velocity vector more usually considered quantities, is defined by weighted mean energy flux g=
weighted mean energy density
(72)
in the literature,
that associated
with the weighted mean
i = =, e
(73)
a purely real vector, well defined since 2 # 0. Written in terms of g, Eqs. (54) and (57) become
g.K = w -I- id/?,
g.S = 1 + iLi/&,
(74)
N.H. Scott/Wave Motion 23 (1996) 393406
respectively.
The first has real and imaginary
g.K+ = o+,
g.K-
405
parts
= w- + d/2.
(75)
These new results in thermoelasticity could have been obtained directly from Eq. (55) on division by t?. For real frequency w the real and imaginary parts of Eq. (74)~ yield g.S+ = I,
g.s-
= d/we.
The first of these equations, well known for conservative media, is here extended to thermoelasticity and the second gives the generalization to a dissipative medium of the result gK = 0, also well known for conservative media, see [ 1, Eq. (8.3.2O)l.
6. Inhomogeneous
plane waves in isothermal
elasticity
The propagation conditions appropriate to isothermal elasticity may be obtained from those of thermoelasticity Eq. (3 1) by equating to zero the temperature coefficient of stress Bij and the temperature increment 0 to obtain Pm2 Ui - d/jk/ Kj UkK/ = 0,
(76)
in which u, K and U may be complex. The propagation det(pw2Z - o(K)}
= 0.
condition
Eq. (36) reduces to (77)
Eqs. (46) and (47) reduce to E =-~Po~U.U
- idijk/UiKjUkKl,
Rj =-$wdijklUiUkK/, respectively, ?=
(78) (79)
and Eqs. (49) and (50) reduce to $/J~wI*U.U* + idijk/UiKjUiK;,
‘i = ~dijk/(W*U~UkK, fwUiU:K;),
(80) (81)
respectively. Eqs. (48) and (5 1) reduce to D = 0, d = 0, respectively, confirming the well-known fact that isothermal elasticity is dissipationless. By its method of proof Eq. (69) remains valid for isothermal elasticity. Alternatively, Eq. (69) may be obtained directly for isothermal elasticity by applying a/a K, to Eq. (76) and using Eqs. (78) and (79). On utilizing in Eqs. (80) and (81) the simple fact that the complex conjugate of any quantity q may be written as q* = q - 2iq-, and bearing in mind Eqs. (78) and (79), we find that t? = -E Y; =-Rj
- ~{PwU~(WU)- + dijk[Ui Kj(UkK/)-}, - idijk/((WUi)pUkK/
+wUi(UkK/)-).
(82) (83)
In general, w and K, and therefore U, are complex and so the right-hand sides of Eqs. (82) and (83) do not simplify. However, in the case of homogeneous plane waves in isothermal elasticity o, K and U are real with the result that Eqs. (82) and (83) simplify to F=--E,
Yj = -Rj,
(84)
406
N.H. Scott/Wave Motion 23 (1996) 393-406
respectively.
Thus the two energy velocities,
defined by Eqs. (70) and (73), are equal:
G =g, and from Eq. (71):
g=-g
aw
in which k is the real wave vector and w(k) is the real frequency given, in implicit form, by the dispersion (77). Of course, Eq. (86) is equivalent to Eq. (65) from the definition Eq. (73).
relation
Acknowledgements The author wishes to thank several referees for their helpful comments and suggestions especially the one who drew Noda’s work to his attention and even supplied a copy of the paper. In addition, the author is greatly indebted to Professor P Chadwick for pointing out that the energy density in thermoelasticity necessarily contains terms linear in the field quantities and for supplying the reference to Nowacki’s work. References 111Ph. Boulanger and M. Hayes, &vectors and Waves in Mechanics and Optics, Chapman and Hall, London (1993). 14 A. Noda, ‘A general relationship between group velocity and fluxes quadratic in wave amplitude”, J. Met. Sot. Japan 64, 3 19-327 (1986). 131 D.E. Carlson, Linear Thermoelasticity, in Encyclopaedia of Physics, Vol. Via/2, Springer, Berlin (1972). [41 P. Chadwick, “Basic properties of plane harmonic waves in a prestressed heat-conducting elastic material”, J. Thermal Stresses 2, 193-214 (1979). [I W. Nowacki, Dynamic Problems ofThermoelasticity, Noordhoff, Leyden (1975). [61 P. Chadwick, Private communication. 171 P Chadwick and L.T.C. Seet, “Second-order thermoelasticity theory for isotropic and transversely isotropic materials”, in: R.E. Czarnota-Bojarski, M. Sokolowski and H. Zorski, eds., Trends in Elasticity and Thermoelasticity, Wolters-Noordhoff, Groningen (1971). PI J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge (1978). [91 P. Chadwick, A.M. Whitworth and P. Borejko, “Basic theory of small-amplitude waves in a constrained elastic body”, Arch. Ration. Mech. Analysis 87,339-354 (1985). [lOI M. Hayes, “A note on group velocity”, Proc. Roy. Sot. London A 354.533-535 (1977). 1111 Ph. Boulanger and M. Hayes, “Inhomogeneous plane waves in viscous fluids”, Continuum Mech. Thermodyn. 2, l-16 (1990). L121N.H. Scott, “Inhomogeneous plane waves in compressible viscous fluids”, Wave Motion 22,335-347 (1995). [I31 P.W. Buchen, “Plane waves in linear viscoelastic media”, Geophys. J.R. Asts Sot. 23,531-542 (1971). [I41 M. Hayes, “Viscoelastic plane waves”, in: F. Mainardi, ed., Wave Propagation in Viscoelastic Media, Pitman Research Notes, Vol. 52, Pitman, London (1982) 2840. [ISI M. Hayes, “Energy flux for trains of inhomogeneous plane waves”, Proc. Roy. Sot. London A 370,4 17-429 ( 1980). [I61 EI. Fedorov, Theory of Elastic Waves in Crystals, Plenum Press, New York (1968).
Wave Motion 23 ( 1996) 393406
Energy and dissipation of inhomogeneous plane waves in thermoelasticity N.H. Scott School of Mathematics,
Universiry of East Anglia, Norwich NR4 7TJ, UK
Received 29 September 1995; revised 28 November 1995
Abstract Inhomogeneous small-amplitude plane waves of (complex) frequency ware propagated through a linear dissipative material. For thermoelasticity we derive an energy-dissipation equation that contains all the quadratic dependence on the field quantities, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. The terms quadratic in the small quantities in the energy density, energy llux and dissipation give rise to inhomogeneous plane waves of frequency 2w and to (attenuated) constant terms. Usually these quadratic quantities are time-averaged and only the attenuated constant terms remain. We derive a new result in thermoelasticity for these terms, see Eq. (54). The present innovation is to retain the terms of frequency 2w, since they are comparable in magnitude to the attenuated constant terms, and a new result, see Eq. (44), is derived for a general energy-dissipation equation that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2w. Furthermore, for dissipative waves or inhomogeneous conservative waves the (complex) group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.
1. Introduction We consider the propagation of inhomogeneous small-amplitude plane waves through a continuous linear dissipative material with special reference to the linearized theory of thermoelasticity in which heat conduction provides the mechanism for dissipation. For such waves the particle displacement field takes the complex exponential form u(x, t) = (U exp io(S.x
- r)}+,
(1)
where U = U+ + ilJ- is the complex wave amplitude (or polarization) vector, w = w+ + iw- is the complex frequency and S = S+ + iS- is the complex slowness vector. These are constant quantities with the variables x and t denoting position and time, respectively. Throughout, the superscripts + and - refer to the real and imaginary parts of a complex quantity. If the planes of constant phase S+.x = constant and the planes of constant amplitude S-.x = constant are not parallel then the complex exponential solution Eq. (1) is said to represent an inhomogeneous plane wave. It is assumed that every other field quantity has the form of Eq. (1) with some other complex scalar, vector or tensor replacing lJ. Inhomogeneous plane waves arise in many different areas of mechanics both for conservative and for dissipative media. Examples include Rayleigh and Stoneley waves in linearized elasticity, electromagnetic radiation in wave 01652125/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved PIISOl65-2125(96)00003-O
394
N.H. Scott/Wave
Motion 23 (1996) 393406
guides and surface and body waves in viscous fluids, viscoelastic solids and thermoelastic solids. We study inhomogeneous waves both for their intrinsic interest and because all solutions of linear problems may be written as linear superpositions (whether as finite sums or integrals) of waves of the form of Eq. (1). In this paper we are particularly interested in energy density, energy flux and dissipation. Apart from linear terms, which we consider separately, these quantities are made up of sums of products of two field quantities each of the form of Eq. (1) and therefore consist of two parts, one an attenuated harmonic part with frequency 2w and the other an attenuated constant, with both parts suffering the same degree of attenuation. Until now attention has been focused in the literature mainly on the attenuated constant amplitudes, which may be obtained very simply from the partial differential equations governing the particle displacements and any other field quantities by time-averaging these equations over a cycle. This time-averaging process removes the amplitudes of the attenuated harmonic parts which has led to their being ignored in the literature on the grounds that they make no net contribution. However, the energy-dissipation equation is valid for all x and t, without time-averaging, and the neglected terms are of the same order of magnitude (before time-averaging) as the retained attenuated constant terms. It therefore seems worthwhile to explore the consequences of keeping these attenuated harmonic terms and this enterprise forms a major part of the present paper. Boulanger and Hayes [ 1] have given a lucid account of the theory of homogeneous and inhomogeneous plane waves [ 1, Ch. 61, energy flux [ 1, Ch. 81 and dissipation [ 1, Section 11.51, the last containing some general results on wave propagation in linear dissipative media though it is set in the context of incompressible viscous fluids. Although the amplitudes of the attenuated harmonic terms are mentioned in Ref. [ 11 no specific results are derived. Noda [2], on the other hand, argues from a general energy-dissipation equation, similar to Eq. (lo), and obtains specific equations involving the amplitudes of the attenuated harmonic terms, see [2, Eqs. (2.12) and (2.24)], which reduce to our Eqs. (44) and (64), respectively, when there are no slow variations in the amplitudes, see [2, Section 21. However, Noda makes no use of an explicit dispersion relation and so is unable to derive the connection between the complex group velocity, energy density and energy flux given in Eq. (69) in the context of thermoelasticity. In Section 2 the basic equations of linear thermoelasticity are set out and an energy-dissipation equation involving the quadratic terms in the energy density, energy flux and dissipation, is derived from them, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. In Section 3 the propagation condition for inhomogeneous plane waves in thermoelasticity is derived. In Section 4 the amplitudes of the energy density, energy flux and dissipation associated with the attenuated harmonic parts (of frequency 20), which are associated with terms quadratic in the field quantities, are denoted by the complex constants E,R and D, respectively, and the amplitudes - associated with the attenuated constant parts are denoted by the real constants e, r and d, respectively. Arguing from an energy-dissipation equation of general validity we are able to deduce the result wE - K.R + i D = 0 with K = WS denoting the complex wave vector. For thermoelastic materials we show further that 6.~2- K? + id = 0. In Section 5 we show that for inhomogeneous plane waves in thermoelasticity or isothermal elasticity the (complex) group velocity is related to the attenuated harmonic energy density and flux by R = E&o/aK. This is in contrast with the relation r = Z aw/aK, well known in the theory of homogeneous waves in conservative media. The relationship between these two equations is discussed in Section 6 in the context of isothermal elasticity.
2. Basic equations and energy dissipation For a thermoelastic and heat supply are
body the linearized equations of momentum
.. aij,j
=
PUi,
Td
=
-qj,j,
and energy balance in the absence of body force
(2)
respectively, see [3, Section 71 or [4, Eqs. (16) and (17)], in which ~7 denotes the Cauchy stress and q the heat flux. The particle displacement vector U(X, t) and entropy increment 4(x, t) denote, as functions of particle position x
395
N. H. Scott/ Wave Motion 23 (1996) 393406
and time I, the small-amplitude departures of the particle position and entropy per unit volume, respectively, from their constant values measured in a spatially uniform, time-independent, stress-free equilibrium state. The constant equilibrium values of the density and absolute temperature are denoted by p and T, respectively. The notation ( ),j denotes the spatial partial derivative a( )/axj and the superposed dot denotes the partial derivative with respect to time t. Twice-occurring roman suffixes are summed over. The constitutive equations appropriate to linearized thermoelasticity are oij
=
dijkiuk.1
-
(3)
Bij@y
for the stress, and Fourier’s law of heat conduction CJi =
-K,>O,j
for the heat flux, with 4
=
BijUi,j
+
p”6,
T
supplying the connection between the entropy per unit volume increment 4(x, t) and the temperature increment 0(x, t), see [3, Section 71. Definitions of the isothermal elasticity tensor d, the temperature coefficient of stress 0, the thermal conductivity tensor K,’and the specific heat at constant deformation c may be found in [3] or [4]. These quantities here take on constant values determined by the underlying equilibrium state. We may now show that the field equations of linearized thermoelasticity take the form: dijkiuk,j/
-
/%je,j
=
K;jO,ij
Pii,
-
T#?ijtii,j =
PCd,
(6)
in the absence of body force and heat supply, see [3, Section 71, [4]. The first is obtained by substituting for u from (3) into (2)t and the second by substituting for q and C#J from (4) and (5) into (2)~. Eqs. (6) comprise a system of four linear, constant-coefficient partial differential equations for the four unknowns Ui, i = 1,2, 3, and 8. We take the material body whose equilibrium state is subjected to the disturbances Ui and 8 to be infinite in extent, so that boundary conditions need not be considered. In order to derive an energy-dissipation equation from Eq. (6) we first multiply Eq. (6)i by tii, sum over i, and rearrange to obtain +
idijklUi,jUk,l
in which the symmetry
I
+
{-dijk[Uk,lki + Bijetii},j = Bijlii,je,
property
(8)
dijki = &ij has been used. We then multiply Eq. (6)~ by 0/T
and rearrange to obtain (9)
On eliminating g
+ rj,j
Bijtii,jQ between Eqs. (7) and (9) we obtain an energy-dissipation
equation
= -2d,
in which the quantity
(11)
N.H. Scott/Wave
396
Motion 23 (1996) 3934%
is an energy density per unit volume, the components Yj
=
-dijklUk,lLii
are the components
+
/$jeci
-
e TUjie,i
(12)
of an energy flux vector and we shall see that the quantity (13)
may be interpreted as a dissipation (per unit volume) of energy. Without loss of generality the components the thermal conductivity tensor have been replaced in Eq. (13) by those of its symmetric part
K;~
of
We do not assume that the thermal conductivity itself is symmetric. Eq. (10) is the field equation, in the absence of body force and heat supply, corresponding both to Nowacki’s fundamental energy theorem [5, p. 3 11, Eq. (1 l)] and to Carlson’s power and energy theorem [3, pp. 331,332]. The first term of Q. (11) represents the kinetic energy density, the second represents the elastic strain energy density and the third represents the heat energy density. The first two terms of the energy flux Eq. (12) are derived from the rate of working of the stress tensor Eq. (3), the first term due to purely mechanical stresses and the second due to the temperature dependence of the stress. The last term of the energy flux Eq. (12) derives purely from the flow of heat within the body. Turning attention now to the dissipation Eq. (13) we recall that the components of heat flux are given by Eq. (4) and that the second law of thermodynamics, see Ref. [3], requires there to be no component of heat flux against the temperature gradient, i.e. qje,j 5 0, which in turn requires the dissipation Eq. (13) to be non-negative:
so that K is positive semi-definite. In fact it is commonly assumed in thermoelasticity that K. is positive definite, so that d > 0 unless the temperature is spatially homogeneous. The fact that d is non-negative justifies its interpretation in Eq. (10) as a dissipation of energy. In Eqs (1 I)-( 13) the displacement components ui, the temperature increment 8 and all their derivatives are to be regarded as small quantities so that all terms occurring in these equations are quadratic in the small quantities. It follows that the energy-dissipation equation Eq. (10) also involves only quadratic terms. However, Chadwick [6] has pointed out that the total energy density in thermoelasticity is not given by Eq. (11) but contains also terms linear in the small quantities. This observation will eventually lead us to an energy-dissipation equation, alternative to Eq. (lo), that involves linear terms in both the energy density and the energy flux, see Eq. (22). First, we reproduce Chadwick’s [6] derivation of an expression for the internal energy density which is correct up to and including quadratic terms in the small quantities. From [7, Eq. (3.17)] we find that, to quadratic order in the small quantities ui,j and 0, the Helmholtz free energy per unit volume is given by A = A0 -
see+ idijklUi,jUk,l
- /$jUi,je
-
g02
(14)
in the absence of pre-stress. The constant equilibrium values of the Helmholtz free energy and the entropy are denoted by A0 and So, respectively, whilst that of the absolute temperature continues to be denoted by T. From [3, Eqs. (4.7), (5.4) and (7.4)] we see that the internal energy, is to quadratic order, given by
in which EO= A0 + TSo is the equilibrium
internal energy.
N.H. Scott/ Wave Motion 23
(1996)393-406
397
The total energy density 2 is defined to be the sum of the kinetic and internal energy densities: e=k+&,
k = ;ptiiLi;,
(16)
where k is the kinetic energy density. From Eqs. (1 l), (15) and (16) we find that the total energy density is given to quadratic order by ? = Fg + e’ + e,
e’ =
TBijUi,j
+
(17)
PCO,
in which e’ gives the linear and e the quadratic terms. From Eq. (5) we see that Eq. (17)~ may be written as e’ = T@ showing that both terms on the right-hand side of Eq. (17)~ may be regarded as heat energy, the first being partly mechanical in origin and the second purely thermal arising from the heat capacity of the material. From Eqs. (17) and (1 1) we obtain the explicit expression
for the total energy density. Notwithstanding rather that k + E, as the total energy. A total energy flux may be defined by Fj = r,j + rj ,
Eq. (IS), Carlson [3, p. 3321 refers to an integrated
rj = -/cjliH,;
form of k + P,
(19)
in which t-J gives the linear and rj, defined in Eq. (12), the quadratic terms. From Eq. (4) we see that rJ is simply the heat flux. From Eqs. (19) and (12) we obtain the explicit expression
for the total energy flux. By an appeal to Eq. (6)~ it may be verified that the linear quantities conservation equation
defined by Eqs. ( 17)~ and ( 19)t satisfy the
Thus there is no dissipation of the linear terms. Adding together Eqs. (10) and (21) and appealing to the definitions ( I7)t and (19)t allows us to deduce the energy-dissipation equation
involving the total energy density 2 of Eq. (18), the total energy flux fj of Eq. (20) and the dissipation d of Eq. (13). The phenomenon of separate energy balances for the linear and quadratic terms has been encountered previously, by Lighthill [8, p. 1.51in the context of acoustics and by Chadwick et al. [9] in the context of constrained elasticity. The energy-dissipation equation Eq. (22) is to be regarded as more fundamental than Eq. ( 10) because it involves the actual energy density rather than just its quadratic terms. Nevertheless, in the remainder of this paper we concentrate exclusively on Eq. (10) because we are concerned here with quantities arising as the product of field quantities and these are necessarily of the second order. All terms quadratic in the small quantities are included in Eq. (IO).
3. Inhomogeneous
plane waves and propagation
conditions
We now assume the small disturbance in the thermoelastic body to have the most general complex exponential plane wave form possible, so that the particle displacement u and the temperature increment 8 take the forms
398
N.H. Scott/Wave Motion 23 (1996) 393406
24 =
(UeiX)+,
(23)
I9 =
{Oe’q+,
(24)
in which the phase factor x is defined by x = w(Scc - t).
(25)
Eqs. (23) and (1) are the same and all quantities @=@++i@-.
were defined in Section
1 except the temperature
amplitude
For the most part we shall prefer to write the phase factor x in terms of the complex wave vector
K = WS
(26)
so that X=K.~-wt=x~+ix-,
(27)
where
x+ =K+.x-o+t,
x-
= K-.x
- w-t
(28)
are the real and imaginary parts of x expressed in terms of those of K and w. From Eq. (28) we see that the wave amplitudes (23) and (24) may be written * =
{Ue’X+}+e-X-,
0 =
(eeix+
(29)
)+e-xm,
(30)
from which it is clear that these wave amplitudes
represent a sinusoidal
travelling
wave of frequency w+ and wave
vector K+ which is attenuated by the real exponential factor e-x-. On substituting the forms (23) and (24), with x given by Eq. (27), into the four partial differential we find that they are reduced to the following system of four algebraic equations: PW2Ui - dijklKjUkKl
-
y@
+iwBijuiKj
+ itCijKiKj@
for the four amplitudes
n4
ZJi, i = 1,2,3,
[I g
ipijKj@
equation (6)
= 0, = 0
(31)
and 0. Eqs. (31) may be written more compactly
as
=o,
(32)
0
where the 4 x 4 complex symmetric
matrix M is defined by
_ &(K) M(w,
K) =
Q22CK) 031 (K)
-
032(K)
i&j Kj
C&(K) pm2
Q23 (K) 033(K)
-
i&j Kj
$lj Kj iP2j
w2
i&j
!E+-
Kj Kj
i/c(K) WT
1
(33)
N.H. Scott/Wave
and [U) , U2, lJ3, @IT is the wave amplitude are given, as function of K, by &k(K)
vector. In Eq. (33) the components
399
of the isothermal acoustic tensor 0
(34)
= dijkl Kj KI
and the scalar conductivity K(K)
Motion 23 (1996) 393406
is given, again as a function of K, by
= KijKiKJ.
(35)
In order for Eq. (32) to deliver non-trivial
wave amplitudes
we require
K) = 0.
detM(w,
(36)
Eqs. (32) and (36) are often termed the propagation conditions of linear thermoelasticity. We see from Eq. (33) that these propagation conditions depend upon both K and w explicitly. Eq. (36) furnishes an implicit form of the dispersion relation for thermoelasticity, see Section 5. For the most part we shall work from Eqs. (3 1) directly and so refer to them also as the propagation conditions.
4. Energy density, energy flux and dissipation 4. I. General
results
We see from Eqs. ( 1 l)-( 13) that the energy density, energy flux and dissipation that occur in Eq. ( 10) are quadratic in the small quantities and so, for inhomogeneous plane waves, are expressible as linear combinations of products of the form f(x, t) = (Ae’X]+{Beix]f
(37)
in which A and B are complex constants.
Using Eq. (27) we may evaluate the product (37) to obtain
.f’(x, t) = ( Fe2iX ]+ + fern’“= (Fe2’X+] +,e2x-
+ f-e-2xm (
(38) (39)
where F = $AB.
,f=
($AB*)+,
(40)
in which F is a (usually) complex constant and f is a real constant. Here and throughout * denotes the complex conjugate. The first term of Eq. (38) represents an inhomogeneous plane wave with phase factor 2x, whilst the second consists of the real constant f attenuated by the real exponential factor e -2x1. From Eq. (39) we see that the inhomogeneous plane wave is attenuated by the same factor. In order to interpret the quantity f we follow Boulanger and Hayes [ 1, Section 1 I .5] and integrate Eq. (39) over a cycle of x + at constant x- to show that the mean value of f is fee’x- , a quantity that depends on x and t through x -. The real constant f is then regarded as a weighted mean of f(x, t). We have already observed that the energy density, energy flux and dissipation that occur in Eq. ( 10) may all be expressed as linear combinations of products of the form (37) and so, utilizing an approach of Hayes 1lo], we may appeal to Eq. (38) in order to write each in turn as r=(Ee
2ix ]+ + ee-2x-
rJ = (Rje2iX)+ +
Fjep2x-,
d = ( De2’X )+ + aep2x-,
(41) (42) (43)
400
N.H. Scott/Wave Motion 23 (1996) 393406
in which E, Rj, D are (usually) complex constants and 2, yj, d are real constants. These latter constants are the weighted means of e, rj, d as discussed above. In the past, see, for example, Ref. [I], discussion of energy and dissipation has focused on the weighted means at the expense of terms involving the complex quantities E, Rj, D, often on the grounds that these terms do not equation Eq. (10) remains valid for contribute when averaged over a cycle of x +. However, the energy-dissipation all x and t, without averaging, and the neglected terms are of the same order of magnitude (before averaging) as the retained terms. It is our chief purpose in this paper to explore the consequences of retaining the attenuated harmonic terms on an equal footing with the weighted means. On substituting Eqs. (41)-(43) into Eq. (10) and equating the coefficients of the attenuated harmonic terms, and those of the purely attenuated terms, we obtain oE-K.RfiD=O,
(44)
w-2-K-?+d=O,
(45)
respectively. Eqs. (44) and (45) are equivalent to Eqs. (2.12) and (2.13) of Ref. [2], respectively, and Eq. (45) has been given also by Boulanger and Hayes [ 1, Eq. (11.5.8)] who derived it, in the general context, from Eqs. (41)-(43) by averaging Eq. (10) over a cycle of xf at fixed x- and then cancelling the attenuation factor ee2X-. All the quantities occurring in Eq. (44) are (usually) complex whilst all those occurring in Eq. (45) are real. Eqs. (44) and (45) have a wide range of validity, not only in thermoelasticity, since they are valid for any system that has an energy-dissipation equation of the form of Eq. (10) with energy density, energy flux and dissipation taking the forms given by Eqs. (41)-(43). Boulanger and Hayes [ 111 and Scott [ 121 have illustrated Eq. (45) for the special cases of incompressible and compressible viscous fluids, respectively. For the special case of real frequencies, Buchen [ 13, Eq. (34)] derived Eq. (45) in the context of isotropic viscoelasticity and Hayes [14, Eq. (5.9)] derived it for any linear dissipative system. We may conclude from Eq. (45) that if w and K are both real then d = 0 and there is no weighted mean dissipation. Alternatively, if there is dissipation (2 # 0) then we may draw the important conclusion that not both of w and K may be real. 4.2. Thermoelasticity - For thermoelasticity we may obtain explicit expressions for the quantities E, Rj, D and e, rj , d occurring in Eqs. (41)-(43) by substituting the inhomogeneous plane wave forms of Eqs. (23) and (24) into Eqs. (I 1)-(13) for the energy density, energy flux and dissipation and using Eqs. (38) and (40): E =-~Pw’U.~JRj =-~~dijktlJiU~K~
idij,iU,KjUkK, - ~io,PijlJi@
+ go’,
(46)
- &KijKie2,
(47)
(51) in which ]z12 = zz*, the squared modulus of the complex quantity z. As has already been remarked, the quantities E, Rj, D are, in general, complex. From the symmetry property (8) we may verify that the weighted mean energy
N.H. Scott/ Wave Motion 23 (I 996) 393406
401
density Z, given by Eq. (49), is real. If, in addition, we make the common assumption of the positive definiteness of d then it follows that Z is strictly positive for a non-trivial wave. It is easily verified that the weighted mean energy flux Fj, given by Eq. (50), is real. The reality of the weighted mean dissipation d, given by Eq. (51), may be verified using the symmetry of K. The common assumption of the positive definiteness of K implies that d is positive, vanishing only if 0 = 0 in which case the wave carries no temperature fluctuation. As in the general case, the non-vanishing of d implies that not both of w and K may be real. It is instructive to verify the general equation (44) in the special case of thermoelasticity. On substituting for E, R and D from Eqs. (46)-(48) into Eqs. (44) we obtain, after some manipulation, WE - K’R + iD =-:{Pw’U~
- dijklKjUkK[ - ipijKjO]Ui
+$ y@ +ftCijKi KjO
+ i@ijUi
Kj
I
0,
each curly bracket on the right-hand side vanishing on account of the propagation conditions (3 1). Thus we have verified the general equation (44) in the special case of thermoelasticity. In the same way that Eq. (52) was derived we may use Eqs. (49)-(51) for the weighted means 2, r, CC? to derive the identity
y@ Again, we find that the right-hand
+ fKijKiKj@
+i@ijUiKj
side vanishes on account of the propagation
(53) conditions
oe - KT + id = 0.
(31) leaving us with (54)
This equation is new to the theory of thermoelasticity though it has been derived for incompressible viscous fluids by Boulanger and Hayes [ 11, Eq. (6.16)] and for compressible viscous fluids by Scott [ 12, Eq. (86)]. Eqs. (44) and (54) have the same form, the first involving the amplitudes of the attenuated harmonic terms and the second involving the weighted means. However, it should be remembered that whilst Eq. (44) has general validity Eq. (54) has been demonstrated here only for thermoelasticity. -Bearing in mind that e, r, d are real we may take real and imaginary parts of Eq. (54) to obtain FK-
f.K+ = w+z,
= w-Z + 2.
(55)
The first of these is another result new to the theory of thermoelasticity though equations of this form have been derived previously in the theory of dissipative media, for incompressible viscous fluids by Boulanger and Hayes [ 11, Eq. (6.17)1], for compressible viscous fluids by Scott [12, Eq. (87)t] and for isotropic linear viscoelastic materials by Buchen [ 13, Eq. (38)] on the assumption of real frequency. As might be expected from the absence of d, Eq. (55)t is valid also for conservative media, which we shall see later. Eq. (55)~ simply verifies for thermoelasticity the general result Eq. (45). We conclude this section by employing Eq. (26) to write Eqs. (44) and (54) in terms of the slowness S rather than K: R.S = E + io-‘D,
(56)
?.S = 2 + iw-‘J.
(57)
Since Eq. (57) has been demonstrated isothermal elasticity, FS = e,
&Sf = 2,
r.s-
for thermoelasticity,
= 0,
taking the non-dissipative
limit (2 -+ 0) gives, for
(58)
402
N.H. Scott/Wave Motion 23 (1996) 393-406
which are equivalent to Eqs. (54) and (55) with 2 = 0. Eqs. (58), and hence Eq. (55)t, have been proved for general conservative systems by Hayes [ 15, Eqs. (4.7) and (4.8)]. Restoring the dissipation d to Eq. (57) and taking real and imaginary parts yields r.s+
in place of (58)2,3. For real frequency $Sf
r.s-
= e + &Y/]w]2,
= 2,
o taking real and imaginary ;.s-
(59)
parts of Eq. (57), or putting w- = 0 in Eq. (59), gives
= w-12.
The first of these equations, thermoelasticity. The second directly from the universally Boulanger and Hayes [ 1, Eq.
5. The dispersion
= &+/]0#
(60)
well known for conservative media, see Hayes [15, Eq. (4.8)2], is here extended to is valid for all dissipative systems since, for real frequency, it can also be derived valid Eq. (45) with the aid of Eq. (26). Such a derivation of Eq. (60)~ was given by (11.5. lo)] in the general context.
relation, group velocity and energy velocity
In theories of continuous media one typically frequency as a function of the wave vector: w = w(K),
derives from the propagation
conditions
an equation
giving the
(61)
known as the dispersion relation. In thermoelasticity the dispersion relation is given in implicit form by Eq. (36). In non-dissipative media the complex wave vector K is replaced by a real one k and the frequency o also is real. Here we wish to explore, in the context of thermoelasticity, the consequences of allowing both wave vector and frequency to be complex. We note from Eqs. (32) and (36) that U and 0, as well as w, are functions of K but not of its complex conjugate K*, and it follows that the complex quantities E, Rj, D, defined by Eqs. (46)-(48), are functions of K but not of K*. Clearly, w*, U* and O* are functions of K*, but not of K, and it follows that the real quantities 2, ?j, d, defined by Eqs. (49)-(5 l), depend upon both K and K*. At this point we break off our discussion of energy and dissipation in thermoelasticity to consider what general results may be available. For any dissipative system that obeys an energy-dissipation equation of the form of Eq. (10) we have shown that the amplitudes of the energy density, energy flux and dissipation satisfy Eqs. (44) and (45). Since for any system the dispersion relation Eq. (61) involves K, but not K*, we would expect the conclusions of the previous paragraph to hold, namely, that the real quantities 2, ?, d, occurring in Eq. (45) depend upon both K and K*, whilst the complex quantities occurring in Eq. (44) depend upon K only. Now Eq. (44) holds for all possible complex wave vectors K and so may be regarded as an identity in K. Allowing the operator a/a Kp to act upon this equation therefore gives
am
-E-R,=-
aKP
Kj
(62)
In the general case we do not have explicit expressions for E, R and D and so can make no further progress. Eq. (45) also holds for all possible wave vectors K but since each of its terms depends explicitly upon K* as well as K it is to be regarded as an identity in each of the six quantities K,,+ , K;, p = 1,2,3. Equivalently, Eq. (45) is an identity in each of the six components of K and K*, with K* now regarded as being independent of K. Accordingly, we may rewrite Eq. (45) as (O - w*)? - (Kj - KT)fj + 2id = 0
(63)
403
N.H. Scott/Wave Motion 23 (1996) 393406
and allow a/aK, obtain
to act upon it, bearing in mind that w depends upon K only and o* depends upon K* only, to
&!!_z_F
we
p-
aK,,
_-2i
ae
afj
--KF-
i aKP
J aK,
+g.
P
I
(64)
Note that the K-derivatives on the right-hand side are evaluated at constant K*. The identity obtained by allowing a/aKi to act on Eq. (63) is easily shown to be equivalent to Eq. (64) because of the reality of ?. r, 2. As with Eq. (62) we do not have explicit expressions for Z, ? and d and so can make little further progress. There is, however, one deduction we can make from Eq. (64). In the case of homogeneous waves in a dissipationless system the complex wave vector K is replaced by the real one k, the frequency w also is real and the dissipation d vanishes, so that o-
KJ: = 0,
= 0,
Li=o
and Eq. (64) reduces to _ am
(65)
r”=ei)ki,
valid for homogeneous waves in a general linear dissipationless system. Equation (65) was first proved by Hayes [ 10, Eq. (20)], also for homogeneous waves in a general linear dissipationless system, by a different method. Hayes’ proof of Eq. (65) proceeds by adding a small, but otherwise arbitrary, imaginary part to the real wave vector k, computing from the dispersion relation the resulting small imaginary addition to the otherwise real frequency w and employing the results of these calculations in the energy-dissipation equation, Eq. (lo), with d = 0. Applying a similar technique to Eq. (10) with complex K and w leads to Eq. (45) instead of Eq. (65). An alternative derivation of Eq. (64) is provided by applying the same technique to Eq. (45) itself. This, in fact, corresponds to the procedure adopted by Noda in proving his main theorem, see Ref. [2, Eq. (2,24)], but because he assumed the dispersion relation only in implicit form, see Ref. [2, Eqs. (2.6) and (2.14)], his result is less explicit and more difficult to interpret than our Eq. (64). We return to our discussion of energy and dissipation in thermoelasticity and seek a connection between the complex group velocity &o/aK and quantities already defined using a method first employed by Fedorov [ 16, pp. 122, 1241 in the theory of elasticity. We apply the operator a/aK, to Eq. (3l)t, contract the resulting equation with Vi and use the symmetry (8) of d to obtain alJi - ZLdipkl Ui UkK/ - $ip Ui O = -pW2 Ui +dijk/UiKjsKi 8KiJ On contracting i/lijUiKjO
+i/IijUiKjs. I’
I’
(66)
Eq. (3 1)t with Ui we find that = PW~U’U - dijklUiKjUkK/.
(67)
now apply the operator a/aK, to Eq. (31)2, multiply the resulting from the coefficient of &/a Kp by means of Eq. (67) to obtain we
I
PO~U,U - dijkl(li KjUkK/ + $e2 aUi
= -iw/$j Kj 0 T
P
-
y@
equation
by 8 and eliminate
i/!lijU; Kj (7)
. y I
$
”
+ i@3ipUiO + $KipKi@’
+ J+/cijKiKj@
From this equation we subtract a multiple w of Eq. (66), divide the resulting (47) for E and R in the left-hand side to obtain
equation by 4 and use Eqs. (46) and
404
N.H. Scott/ Wave Motion 23 (1996) 393406
aw
E-
- Rp = :(PW2Ui
- dijk/KjUkK[
- ip,IKIc)Js
aKP
P
(68) The right-hand side of Eq. (68) is very similar to those of Eqs. (52) and (53) and indeed we note from the propagation conditions (3 1) that it vanishes leaving R,=E*.
(69)
a
KP
This is a new result in the theory of inhomogeneous waves in dissipative media, here demonstrated for thermoelasticity. It connects the complex group velocity with the attenuated harmonic terms in the energy-dissipation equation. Eq. (69) may also be deduced from Eq. (62) by verifying that its right-hand side reduces to zero upon substitution for E, R and D of the explicit forms given by Eqs. (46)-(48). Both these methods of proof depend on a knowledge of the propagation conditions (31) of thermoelasticity, not just on the existence of an energy-dissipation equation of the form of Eq. (10). Noda assumes only an energy-dissipation equation, see Ref. [2, Eq. (2.4)], and so is unable to derive a result as explicit as Eq. (69). Previous work on energy flux and group velocity has largely been for homogeneous waves propagating through a conservative medium, with real frequency w and real wave vector k, for which Hayes [ 10, Eq. (20)] has proved a general result, namely Eq. (65) above. In dissipative media we have deduced from Eq. (45) that not both of o and K can be real and the same is true for inhomogeneous waves in conservative media. Since C?and 7 are real it follows that Eq. (65) cannot hold either for waves in dissipative media or for inhomogeneous waves in conservative media. At least in the theory of thermoelasticity we have shown that Eq. (65) is replaced by Eq. (69). One might expect Eq. (69) to have a wider validity in the theory of dissipative media but this has not been demonstrated. In the next section we shall pursue the connection between Eqs. (69) and (65) for isothermal elasticity, a non-dissipative limit of thermoelasticity. We may define a complex energy velocity vector G associated with the attenuated harmonic terms of the energydissipation equation by G=
energy flux associated with attenuated
harmonic terms
energy density associated with attenuated harmonic
terms
=-
R (70)
E ’
provided E # 0, which from Eq. (69) may be written G=g.
(71)
In terms of G, Eqs. (44) and (56) may be written as G.K = o + iDIE,
GS = 1 f iD/wE,
respectively. The energy velocity vector more usually considered quantities, is defined by weighted mean energy flux g=
weighted mean energy density
(72)
in the literature,
that associated
with the weighted mean
i = =, e
(73)
a purely real vector, well defined since 2 # 0. Written in terms of g, Eqs. (54) and (57) become
g.K = w -I- id/?,
g.S = 1 + iLi/&,
(74)
N.H. Scott/Wave Motion 23 (1996) 393406
respectively.
The first has real and imaginary
g.K+ = o+,
g.K-
405
parts
= w- + d/2.
(75)
These new results in thermoelasticity could have been obtained directly from Eq. (55) on division by t?. For real frequency w the real and imaginary parts of Eq. (74)~ yield g.S+ = I,
g.s-
= d/we.
The first of these equations, well known for conservative media, is here extended to thermoelasticity and the second gives the generalization to a dissipative medium of the result gK = 0, also well known for conservative media, see [ 1, Eq. (8.3.2O)l.
6. Inhomogeneous
plane waves in isothermal
elasticity
The propagation conditions appropriate to isothermal elasticity may be obtained from those of thermoelasticity Eq. (3 1) by equating to zero the temperature coefficient of stress Bij and the temperature increment 0 to obtain Pm2 Ui - d/jk/ Kj UkK/ = 0,
(76)
in which u, K and U may be complex. The propagation det(pw2Z - o(K)}
= 0.
condition
Eq. (36) reduces to (77)
Eqs. (46) and (47) reduce to E =-~Po~U.U
- idijk/UiKjUkKl,
Rj =-$wdijklUiUkK/, respectively, ?=
(78) (79)
and Eqs. (49) and (50) reduce to $/J~wI*U.U* + idijk/UiKjUiK;,
‘i = ~dijk/(W*U~UkK, fwUiU:K;),
(80) (81)
respectively. Eqs. (48) and (5 1) reduce to D = 0, d = 0, respectively, confirming the well-known fact that isothermal elasticity is dissipationless. By its method of proof Eq. (69) remains valid for isothermal elasticity. Alternatively, Eq. (69) may be obtained directly for isothermal elasticity by applying a/a K, to Eq. (76) and using Eqs. (78) and (79). On utilizing in Eqs. (80) and (81) the simple fact that the complex conjugate of any quantity q may be written as q* = q - 2iq-, and bearing in mind Eqs. (78) and (79), we find that t? = -E Y; =-Rj
- ~{PwU~(WU)- + dijk[Ui Kj(UkK/)-}, - idijk/((WUi)pUkK/
+wUi(UkK/)-).
(82) (83)
In general, w and K, and therefore U, are complex and so the right-hand sides of Eqs. (82) and (83) do not simplify. However, in the case of homogeneous plane waves in isothermal elasticity o, K and U are real with the result that Eqs. (82) and (83) simplify to F=--E,
Yj = -Rj,
(84)
406
N.H. Scott/Wave Motion 23 (1996) 393-406
respectively.
Thus the two energy velocities,
defined by Eqs. (70) and (73), are equal:
G =g, and from Eq. (71):
g=-g
aw
in which k is the real wave vector and w(k) is the real frequency given, in implicit form, by the dispersion (77). Of course, Eq. (86) is equivalent to Eq. (65) from the definition Eq. (73).
relation
Acknowledgements The author wishes to thank several referees for their helpful comments and suggestions especially the one who drew Noda’s work to his attention and even supplied a copy of the paper. In addition, the author is greatly indebted to Professor P Chadwick for pointing out that the energy density in thermoelasticity necessarily contains terms linear in the field quantities and for supplying the reference to Nowacki’s work. References 111Ph. Boulanger and M. Hayes, &vectors and Waves in Mechanics and Optics, Chapman and Hall, London (1993). 14 A. Noda, ‘A general relationship between group velocity and fluxes quadratic in wave amplitude”, J. Met. Sot. Japan 64, 3 19-327 (1986). 131 D.E. Carlson, Linear Thermoelasticity, in Encyclopaedia of Physics, Vol. Via/2, Springer, Berlin (1972). [41 P. Chadwick, “Basic properties of plane harmonic waves in a prestressed heat-conducting elastic material”, J. Thermal Stresses 2, 193-214 (1979). [I W. Nowacki, Dynamic Problems ofThermoelasticity, Noordhoff, Leyden (1975). [61 P. Chadwick, Private communication. 171 P Chadwick and L.T.C. Seet, “Second-order thermoelasticity theory for isotropic and transversely isotropic materials”, in: R.E. Czarnota-Bojarski, M. Sokolowski and H. Zorski, eds., Trends in Elasticity and Thermoelasticity, Wolters-Noordhoff, Groningen (1971). PI J. Lighthill, Waves in Fluids, Cambridge University Press, Cambridge (1978). [91 P. Chadwick, A.M. Whitworth and P. Borejko, “Basic theory of small-amplitude waves in a constrained elastic body”, Arch. Ration. Mech. Analysis 87,339-354 (1985). [lOI M. Hayes, “A note on group velocity”, Proc. Roy. Sot. London A 354.533-535 (1977). 1111 Ph. Boulanger and M. Hayes, “Inhomogeneous plane waves in viscous fluids”, Continuum Mech. Thermodyn. 2, l-16 (1990). L121N.H. Scott, “Inhomogeneous plane waves in compressible viscous fluids”, Wave Motion 22,335-347 (1995). [I31 P.W. Buchen, “Plane waves in linear viscoelastic media”, Geophys. J.R. Asts Sot. 23,531-542 (1971). [I41 M. Hayes, “Viscoelastic plane waves”, in: F. Mainardi, ed., Wave Propagation in Viscoelastic Media, Pitman Research Notes, Vol. 52, Pitman, London (1982) 2840. [ISI M. Hayes, “Energy flux for trains of inhomogeneous plane waves”, Proc. Roy. Sot. London A 370,4 17-429 ( 1980). [I61 EI. Fedorov, Theory of Elastic Waves in Crystals, Plenum Press, New York (1968).