WAVE MfJ~nMl

ELSEVIER Wave Motion 25 (1997) 5 l-60

Energy velocity of complex harmonic plane waves in viscous fluids

M. Deschamps a.e, B. Poirke b, 0. Poncelet a a Laboratoire de kfe’canique Physique, Universite' Bordeam I, 11101 CNRS 867, 351, Cours de la Lib&ration, 33405-Talence Cede-u,

France b DRET/STRDT/G63, 26 Boulevard Victor; 00460 AmPer, France

Received 2 October 1995

Abstract

This paper presents a theoretical investigation of the energy velocity of complex harmonic plane waves in viscous fluids. The complex harmonic plane wave, which is characterized by a complex wave vector and a complex frequency, may propagate in absorbing fluids. The initial nonconservative energy balance equation is modified into another energy equation, in which the new loss density is, on average, nil. It appears that the energy velocity, which is defined from this system that is conservative on average only, is not always oriented along the phase velocity direction. More precisely, the energy velocity may be interpreted as the phase velocity in the direction of the real part of the slowness bivector.

1. Introduction

Most studies of the inhomogeneous plane waves which have been conducted till date have assumed, either explicitly or implicitly, that the frequency is real [l-9]. However, in order to explain peculiar physical phenomena it is of interest to hypothesize in the frequency to be complex in electromagnetism [ 10,l l] and in acoustics [ 121. The physical interpretation of such waves is that they present locally transient waves.

In contrast to earlier works [ 10,13,14], which treat the problem in terms of complex wave vectors, our emphasis will be placed on an approach in terms of complex slowness vectors. In this way, the different roles played by the source, i.e., the complex frequency, and the medium, i.e., the complex slowness vector, will be dissociated. Beyond this mathematical aspect, the interest of this approach will be made clear through the interpretations of physical phenomena.

For isotropic media, both the phase and group velocities, which are, by definition, directly related to the phase of the wave [ 151, propagate orthogonally to the phase front. Regarding the propagation of the energy, the acoustic Poynting vector, which is the energy flux density, may be used in order to quantify the energy velocity vector. The energy flux vector has a dynamical definition and consequently, polarization of the wave is taken into account. From this point of view, all quantities being complex, ellipitically polarized waves with complex frequency can lead to unexpected results. Particular attention will be then focused on the energy velocity of such waves in a viscous fluid.

* Corresponding author. Tel.: 33 05 56 84 62 26; fax: 33 05 56 84 69 64; e-mail: deschamp@lmp.u-bordeaux.fr

0 165-2 125/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved

PIZSO165-2125(96)00032-7

52 M.Deschamnps et al./Wuve Motion 25 (1997) 51-60

2. Energy balance equation in a viscous fluid

The fluid is supposed to be viscous, homogeneous, at rest in the absence of sound, with no external forces and

no internal sources. In such conditions, the linearized equation of propagation can be written in the form

&d = c; Ad + b A&d, (2.1)

where A is the Laplacian, a, and a,, denote the simple and double partial derivatives with respect to time. In the above equation, d represents the acoustic displacement, co denotes the wave velocity (for ideal fluids

and homogeneous waves only) and the quantity b is related to absorption. In the case of a viscous fluid, b is such that b = pO’(k + 2~), where pu is the density and ,u and h are the shear and the second viscosity coeffcients, respectively.

Multiplying (2.1) by p&d, leads to the following energy balance equation:

a,~+v.,f=z2. (2.2)

The parameters E, .I and 27 stand for the volume, flux and loss densities of energy, respectively. They are expressed by:

E = ;po(&d . &d + c;(V . d)‘), (2.3)

f = -po{bV .ard + CT; V . d}&d, (2.4)

V = -pub(V . &d)“. (2.5)

It is to be noted that the energy propagation equation (2.2j is strictly the same as the equation given in [14]. However, the quantities J and 27 are not similarly defined. In comparison with the result given in [14], the form (2.2) has two advantages. On the one hand, the vectorJ can be interpreted as the energy flux vector, i.e., the product between the pressure and the displacement velocity. On the other hand, the term D, associated with the viscous losses, is always strictly negative in the present paper.

3. Complex harmonic plane wave solution

Solutions of wave equation (2.1) is presented for the complex harmonic plane wave. The more general plane wave is defined, at any point of space M and time t, by the acoustic displacement

d = Re(*a*Pexpi(*wt -* K. M)). (3.lj

In the latter expression, *a = ]*a ( exp iB is the complex amplitude, *K = K’ - i K” is termed the wave bivector, *P represents the normalized polarization vector and *w = w’ + i w” is the complex frequency. Note that the wave bivector, which is a complex vector, has been studied in detail in [3]. The notation Re( } stands for the real part. The star superscript index on the left hand side indicates that the quantities are complex. All other parameters are

real. The complex phase of the wave defined by Eq. (3.1), may be written as

*wt -* K . M = (a’ + iqo”,

where PO’ = w’t - K’ . M and cp” = d’t + K” M.

(3.2)

The propagation vector is denoted by K’ and the attenuation (or damping) vector is represented by K”. The real scalar w’ (with w’ > 0) stands for angular frequency. The parameter w ” is the extinction coefficient (with W” > 0) or the switching on of the source coefficient (with w” < 0). This coefficient describes the time dependent exponentially transient part of the wave. In experimental practice, such quasi-harmonic waves might be observed for

MDeschmps et al. /Wave Motion 2.S (1997) 51-60 53

a ratio lo”/w’I less than unity. Considerations on the real and imaginary phases suggest to define both the amplitude and phase velocities, respectively, by

(3.3)

(3.4)

c, is defined as the velocity of the point for which the amplitude remains constant. For further analysis let us introduce the slowness bivector *S such that

*K E* w*S

with *S = S’ - i S”.

(3.5)

The complex frequency *w might be directly related to the source, and *S to the medium considered. The real

and imaginary parts of *S are expressed by the following form:

S” = a - * [w’K’ _ w”,“],

S’ = a! - ’ [ m”K’ + co/K" 1.

where (Y = LIP + d2.

(3.6)

(3.7)

From the equation of motion, the dispersion equation is expressed as

*02 - (ci + i b*w)*K .* K = 0.

This equation may be rewritten in terms of S’ and S” by

(c; - bw”) M - bto’N = 1, b&M + (c; - bw”)N = 0,

where M = S’ . S’ - S” . S” and N = -2 S’ . S”.

(3.8)

It should be noted that the above system depends on the imaginary component of the frequency only for absorbing fluids. Solving the system (3.8) in terms of M and N yields

M = (c; - bw”)A-‘,

N = -bw’A-‘,

where A = co4 - 2bw”ci + ab2.

(3.9)

(3.10)

The normalized polarization vector, associated to lamelar wave, that satisfies

V x”d=O, V .‘d #O and *P-*P= 1, (3.11)

is then given by:

*p = (*s .* s)-‘/’ “S. (3.12)

Let us now give more details concerning the slowness bivector *S. In [3], it has been shown that if two bivectors *A and *B are such that *A = *cY*B where *a is a complex scalar, then the ellipses associated to *A and *B are similar. In other words, the major and minor axes of the ellipse of *A are parallel, respectively, to the major and minor ellipse of *B, and also the aspect ratio is the same for both ellipses. The two bivectors are then said to be parallel. As an example, Figs. 1 and 2 show the relation between *S and *K, for absorbing and nonabsorbing media. The solid and dashed lines correspond to the wave bivector associated to w” > 0 and W” < 0, respectively. Notice that, for nonabsorbing media the real wave vectors of these two situations are symmetric with respect to x-axis

51 MDeschamps et al. /Wave Motion 2.5 (1997) 5140

Fig. 1. Relation between “S and *K for absorbing media: (-_) W” z 0 and (- - -)w” < 0.

Fig. 2. Relation between *S and *K for nonabsorbing media: (-) w” > 0 and (- - -1,” < 0.

(or S’), while the imaginary wave vectors are symmetric with respect to y-axis (or S”). Furthermore, for real plane waves, i.e., 0” = 0, and for both type of medium, S’ and S” are, respectively, collinear to K’ and K”.

As can be pointed out, even for nonabsorbing media, the damping and propagation vectors K” and K’ are not necessarily orthogonal, which was the usual definition of evanescent plane waves that propagate in nonabsorbing fluids [SJ. Consequently, for transient plane waves, it seems that the relative directions of these two vectors are not directly related to the wave structural behaviour. A more detailed discussion concerning the plane wave differentiation will be found later.

M.Descharnps et al. / Wave Motion 25 (1997) 514.0 55

4. Instantaneous volume, flux and loss densities of energy

Before calculating the energy velocity, let us examine first the instantaneous energy, flux and loss densities. For simplicity, let us note

A = K’ . K’ _ K” . K”, B = -2K’ . K”, C=K’.K’+K”.K”. (4.1)

By introducing the displacement field defined by the Eq. (3. I) in relation to (2.3) and by setting *k = K exp i 0 (K

modulus of *k) where *k = (*K .* K)‘l’, it is found that

& = ;G exp -2q”[(aC + ci K’) - Re{*k*(*w’ + co2 *k*) exp2i!P)], (4.2)

where @ = rp’ + /? - 8, G = &,~*a~“/2~~.

Expanding the above expression leads to the instantaneous volume density of energy

& = $G exp -2p”[(arC + COKE) + (2dd’A + (CO’* - w”*)B + 2ABci) sin 2!P

- ((w’2 - w”*)A - 2w’w”B + &A’ - B*)) cos 2P]. (4.3)

In a similar way, from (2.4), the instantaneous energy flux density is given by

J = G exp -Zp”Re([(ci - i bw*)k** - (ci + ib*w)*k* exp 2iP]*w*K}, (4.4)

where the star superscript index on the right hand side indicates that the quantities are complex conjugate. Noting that the expression in square brackets contains terms of the dispersion equation and of its complex conjugate, J can be simplified by

J = Gexp -2rp”Re{[cro* -* w3 exp2i@]*K].

Finally, the energy flux vector becomes

J = G[E+K’ - E-K”], (4.5)

where

E+ = exp -2q~‘![aw’ - (w” - 30’0”~) cos 2W + (3~0’~~” - df3) sin 291

and the expression of E_ is given by E+ on inverting o1 and w” (except in p” and ly). The instantaneous loss density can be expressed as

V = -bG exp -2#[((o” - d’*)(A* - B*) - 4dw”AB)cos2’P

- (20’wN(A2 - B’) + 2AB(w’* - CO”*)) sin 2W + a~“]. (4.6)

At this stage, let us notice that all the above results are not approximate expressions and they do not depend harmonically on space and time. Accordingly, special attention should be paid in determining the mean quantities.

5. A new equivalent conservative system

From an experimental point of view. it is more interesting to define the velocity from averaged quantities. In [ 141, the energy velocity and the mean loss are, respectively, defined by

ce = (J)/(E), I- = P)l(a,

(5.1)

(5.2)

56 M.Desclumps et al. / Wave Motion 25 (1997) 5140

where (f) denotes the mean value over a time period or wavelength of functions f such that:

r+T

W=f / fdt, (5.3)

T = 2n/w’ and t stands for an arbitrary origin. It should be noted that in using this definition the means of volume, flux and loss densities, defined in Eqs. (5.1)

and (5.2), depend on the position M and time t. This is due to the space and time exponential decay. In fact, even for the limit case of a conservative energy equation, i.e., b = 0, Eq. (5.1) does not always provide a correct expression of the energy velocity, since the mean values depend on position and time.

A possible way to avoid this problem consists in finding another system, expressed by new variables. In order to find this other form of the system (2.2) let us assume that the energy, flux and loss densities are such that

J = f(t, M) J’, E = f(r, My?‘, D = f(t. M) 0, (5.4)

where f(t. M) is related to the wave amplitude. Inserting (5.4) into (2.2) then leads to

&&‘+V.J’=D”, (5.5)

where D” = D’ - [a,f(t, M)I’ + Vf(t, M) . J’][f(t, M)]-‘. Usually, a nonconservative system is transformed into an equivalent conservative system by considering com-

pensated quantities [14,16]. The system under consideration (5.5) is generally nonconservative. However, for a harmonic wave, a system shall be called conservative with respect to I’, if the mean loss density D”, defined by integral (5.3), is zero.

Let us now examine the solution of (5.5) for the inhomogeneous plane wave (3.1). The expressions of f(t, M) and D’ are given by

f(t, M) = exp -2Cw”t + K” . M), 73” = 27 + 2 (w”E’ + K” . J') . (5.6)

Remark that, except for homogeneous and nonattenuated plane waves, i.e., w” = 0 and b = 0, the loss density is generally nonzero.

The mean value of the compensated loss density is such that

(D”) = $,0~‘(2,S’ . S” - &d/A) - o”(Sf2 - Slr2 - (c; - bw”)/A)], (5.7)

A being defined in (3.9) and (3.10). However, by virtue of the dispersion equation (Eqs. (3.9) and (3.10)), it can be immediately shown that

(0’) = 0. (5.8)

It is noteworthy that exactly the same form of (D”), Eq. (5.7), could have been obtained by averaging over a wavelength. This consideration is not restricted to this expression. In fact, for further calculations of all averaged quantities, the choice of the variable of integration does not change the results.

At this point, it would be quite interesting to discuss a few advantages of the change of variable providing the system (5.5) in comparison to other works. In [15], the author is compelled to use either a spatial or a temporal integration following the problem treated; to a time (or space) exponential decay corresponds a space (or time) integral. Obviously, except for real harmonic plane waves, i.e., 0” = 0 and S” = 0, both integrations cannot lead to

the same results. In the present paper, referring back to system (5.5) as alreaday mentioned, no particular attention must be paid to the choice of the variable of integration. This is of great importance, mainly for complex harmonic

M.DescRamps et al. / Wnl*e Motion 25 (1997) 51-60 57

plane waves, since these waves are neither purely harmonic in time nor in space, i.e., w” # 0 and S” # 0. In addition, for this kind of wave, observe that Eq. (5.5) does not depend on the actual amplitude f(t, M). Then, the averages are performed on periodic terms stemming from the real phase of the wave, for which the intrinsic nature physically different from the exponential behaviour, as intuitively indicated by Warenghem [IO]. In other words, energy equation must be independent of the value of the wave amplitude, i.e., f(t, M), but not of its variation, i.e., K”.

From a thermodynamic point of view, let us consider the variation of the integral of E’ over a volume Q. Introduce 6/6t the derivative with respect to time, obtained when a point is followed in its own motion defined by the velocity field W = W(t, 44). This derivative of an integral of a quantity A over a volume 52 is classically given by

where 3~2 is the closed regular boundary of L?, do an elementary surface element and rz is normal to da. Applying this to the compensated energy E’, for which the associated velocity field is assumed to be W = J’/&‘, leads to

(5.10)

It can immediately be shown from (5. lo), that if the integral is taken over a wavelength, then

; _/- &‘dL’ = (D”) = 0.

Incidentally. this equation shows the conservation of quantity E’. This reinforces the fact that Eq. (5.5) using Eq. (5.8) can be considered as a conservative system on the whole with respect to E’ and J’, although it is not locally conservative. Another observation to be made is that, by using the new system (5.5), the mean values of the compensated stored energy and flux densities, E’ and J’, can be easily obtained for transient wave, i.e. for o” # 0. Finally, it is of interest to notice that this last result holds always true even for an absorbing fluid.

The above results are different from those obtained in [ 14,161. As a matter of fact, the authors obtain an equivalent conservative system from a nonconservative system. To illustrate the difference with the present work, let the fluid be nonabsorbing. In such a circumstance, the initial equation (2.2) is conservative and the equivalent system (5.5) is a nonconservative system, 27” being a nonvanishing function; nevertheless, its integral over a period is zero. This consequence is of importance since using this system, the energy velocity can be clearly defined. This is the purpose of Section 6.

6. Energy velocity

In this section, the energy velocity is calculated from the means of resealed energy E’ and flux densities J’. In an agreement with the conservation of the resealed energy E’, the energy velocity can be defined by

c, = (J’)l@).

Clearly, the means of both energy and flux densities E’ and J’ are given by

(6.1)

(E’) = $G[~(K’ . K’ + K”. K” + &c’I, (6.2)

(J’) = aG[o’K’ - d’K”] = a’GS’. (6.3)

58 M. De&amps et al. /Wave Motion 25 (1997) 51-60

Fig. 3. Relative orientations of S’. K’, C~ and c,.

After cancellations and simplifications, the energy velocity defined by (6.1) is

(6.4)

Note from now on that energy propagates following the direction of the real part of the slowness bivector *S. By introducing the expression of b/A obtained from (3.10) in the energy velocity (6.4), it is immediately apparent

that

S’ ce=w. (6.5)

where S, = K’lw’ stands for the phase slowness vector. Except in a few instances, such as the case of homogeneous waves, the energy does not propagate following

the phase slowness vector. Comparing the expression of the phase velocity (3.4) with the energy velocity given by Eq. (6.9, it is clear that the directions as well as the moduli of these two velocities are different.

To fix ideas, Fig. 3 shows the relative directions and moduli of both phase and energy velocities. Clearly, as seen above, for transient inhomogeneous waves the energy and the phase velocities are not oriented in the same direction while for harmonic waves they are. In addition, the energy velocity is always greater than (or equal to) the phase velocity.

If the phase velocity in an arbitrary direction x is defined by c,+,* = (S, . x)-‘x, it is noticeable that the energy velocity is, in fact, the phase velocity in the direction S’.

A final inspection is to search and follow the moving point P for which the instantaneous energy remains constant. Obviously, the intersection point of both planes of constant amplitude and phase satisfies this condition for nonabsorbing media. The energy velocity may be then interpreted as the speed of this point.

Fig. 4 shows the planes of constant amplitude and phase and their associated velocities c, and cV, respectively. Let cP be the speed of the point P. Since this point belongs to both planes, its speed must satisfy the following relations:

yK’=d, cp . K” = -_w”. (6.6j

In other words, this systems indicates that, as time increases, the specific point P, which conserves both its phase and amplitude, moves with velocity cp.

To solve the system (6.6), it is more convenient to rewrite it in terms of S’ and S”. It is then found that

cp.S” 1, CP . s” = 0. (6.7)

Remark first, that from the previous equations, the velocity cP is orthogonal to the slowness damping vector S”. Second, assuming a nonabsorbing medium, it is obtained S’ . S” = 0 and S’ S, = S’ . S’. Clearly, from these two observations, cP = ce for an initial conservative system (2.2).

M.Deschantps et al/Wave Motion 25 (1997) 51-60 59

plane of constant phase t- cu

-t

Fig. 4. Propagation of the planes of constant phase and amplitude.

Of particular interest are these results since they suggest to propose a different definition of the homogeneous, purely inhomogeneous (or evanescent) and inhomogeneous plane waves based on the relative slowness damping direction S” and the energy velocity direction ce.

Waves are said to be evanescent if their slowness damping vector S” and energy velocity vector c, are orthog- onal. This wave propagates only in nonabsorbing media. This is a more general definition since such behaviours characterize transient surface waves as well as surface waves that propagate on anisotropic interface. For this last case, the energy flux vector is along the surface and orthogonal to the slowness damping direction, although the propagation vector K' may not be necessarily oriented along the surface.

For isotropic media, waves are said to be homogeneous if their slowness damping vector S” and energy velocity vector c, are parallel. Finally, in other situations, waves are so-called inhomogeneous (or heterogeneous). Note that these definitions include those already well known for harmonic inhomogeneous plane wave that propagate in isotropic as well as in anisotropic media [ 171.

7. Particular cases

The purpose of this section is to briefly present the two limit cases of the real harmonic inhomogeneous plane wave and of the propagation in nonabsorbing media.

(ij The harmonic irzhomogeneous plane wave: w” = 0. This permanent inhomogeneous plane wave has been studied in detail in [3,5,6,8]. From the calculations under consideration, for this particular case, the velocity of energy becomes

dK’ ce=K’. (7.1)

This result in accordance with those obtained in [S]. In addition it is interesting to note that, for any values of absorption, the energy and phase velocities are the same; c P = cc. When an inhomogeneous plane wave is reflected on a plane interface, it has theoretically and experimentally been shown that, at certain angles of incidence, the phase velocity of transmitted waves cq may be oriented toward the incident medium [1X]. To explain this phenomenon, it is reported in this reference that, although the refracted angle can be greater than 90”, the associated energy velocities might be oriented to the refracted medium. Since c rp = ce, the results of the present paper show that this intuition does not hold true. This peculiar refraction is therefore not yet completely understood.

(ii) Nonabsorbingfluid: b = 0. By assuming b = 0, Eq. (6.5) is reduced to

S' ce=gT-yt. (7.2)

60 MDeschmrzps et al/Wave Motion 25 (1997) 51-60

In such a case the energy velocity is different from the phase velocity. From consideration on planes of constant amplitude and of constant phase, this result has been obtained graphically in Section 6.

8. Conclusion

From the particular example of energy velocity calculation, we may state that for a better understanding of physical phenomena, it is interesting to differentiate the effects due to the frequency from those due to wave propagation. For real harmonic inhomogeneous plane waves, the differentiation between slowness and wave bivectors is not necessary since the collinearity coefficient of both bivectors is real scalar. In contrast, for complex.harmonic inhomogeneous plane waves it is necessary to differentiate these two bivectors. As an example, the energy velocity of these last waves is oriented following the real part of the slowness bivector, which actually is the propagation direction of the waves, while the real part of the wave bivector does not represent this direction. Finally, we remark that this work can be expanded without difficulty to isotropic solids. However, it should be noted that there is still a substantial need for further works. Additional efforts are required for the problem of propagation of transient waves in anisotropic media, particularly with regard to relation between the group and energy velocities of the complex harmonic inhomogeneous plane waves. This remains to be investigated.

Acknowledgements

This work was supported by DRET contract no. 94/088.

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