# Inhomogeneous Waves in Solids and Fluids

Inhomogeneous Waves in Solids and Fluids

Submit an Article. Shopping Cart. Buy Now. Issue 1 Issue 2 Issue 3 Issue 4. DOI: Barak, M. Tomar, Ashish Arora. Reinisch, Vladimir G. If the viscoelastic solid is isotropic then the relaxation tensor function G is represented by two scalar functions only. Upon substitution we see that, for isotropic solids, the equation of motion 2. Although thermal effects are disregarded, thermodynamics plays an important role in the characterization of the model for viscoelastic solids.

The functional 7" E' , expressed by 4. Upon the statement of the second law through the Clausius inequality 2. As shown in [70], the relations 4. The inequality 4. The physical meaning of 4. For isotropic bodies 4. Upon substitution in 4.

Then 4. There seem to be motivations for models of viscoelastic solids with G' non-integrable and Go infinite cf. Here we disregard such possibilities also because they rule out wave propagation. S D i s s i p a t i v e fluids The most natural models of dissipative fluids are the viscous and viscoelastic fluids. We regard the viscous fluid model as satisfactory whenever memory properties are inessential.

Both models are usually applied in the linear approximation. A fluid at rest cannot sustain shear stresses and differences in normal stresses. Of course, by Galilean invariance the same occurs if the fluid undergoes a rigid motion. The symmetry of T implies that r too is symmetric. The splitting 5. Examine first viscous fluids. Of course it may well happen that p. In the particular case of fi, A constant, the last equation is usually referred to as NavierStokes equation.

Often the linearly viscous, incompressible fluid is called Newtonian. Here we give an outline of the model and its derivation by following the lines of [61]. The subscript t means that the current placement is taken as reference. Modelling of Dissipative Media 35 which means that the extra-stress r of the particle at x at the time t depends on the whole history of the Lagrangian strain at that particle.

To derive a linear model we proceed as follows. Even for fluids, thermodynamic restrictions provide inequalities which prove of fundamental importance in the analysis of wave propagation. Really, by 2. Then p d — p 0 if and only if the integral of t r D on [0,d vanishes. Any function p p is compatible with the second law 5. Let the subscript c denote the half-range Fourier-cosine transform, e. The case when the fluid is Newtonian, namely r is given by 5.

Hence the second law holds if and only if the viscosity coefficients satisfy p. The assignment of Y is a highly non-trivial task. Indeed, sometimes the question may be ill-posed, as we show in a moment. To fix ideas, take the simplest model of solid body, viz. To determine Y we should know the constitutive equation for Y. But 6. As a first step it is natural to have recourse to non-linear elasticity and examine the corresponding linearization procedure. Quite generally, let the stress T be a function of the deformation gradient F.

Since H is small we disregard o H terms. Finally, letting b be a constant vector, typically the gravity acceleration, and substituting into the equation of motion 2. Equation 6. The corresponding analysis of the perturbation of the traction proceeds along similar lines. Modelling of Dissipative Media 39 The second equality in 6. Once we know the tensor B we can apply 6. Of course to determine B we go back to the non-linear stress-strain relation.

Yet, it may well happen that we have reasons for linearizing with respect to the superimposed motion while we have no or not enough information on the stress-strain relation. In such a case it is reasonable to assume that the incremental part B H is isotropic and this assumption is termed incremental isotropy. Then 6. This scheme is applied in Ch. Accordingly the equation of motion 6.

Further, use Cartesian coordinates with z the upward vertical axis and denote by u,v,w the components of u. Multiply 6.

### Introduction

Pergamon, Amsterdam Based on this theory, numerous problems have been studied. The interpenetrating motion of a viscoelastic component and a liquid is considered as a fluid motion in a deformable porous medium. Ewing, W. The comma notations denote partial differentiation with respect to space.

Substitution into 6. Obviously, if the fluid is nondissipative, i. Solutions in the form of inhomogeneous waves are sought. The use of the scalar and vector potentials for the displacement u proves advantageous in both dissipative solids and fluids. The equation of energy may be viewed as the balance between the time rate of the energy density and the net contribution of the energy flux, along with the energy supply.

This balance shows interesting features also because of non-uniqueness of the energy density. Within a general setting for this topic, as non-trivial applications the detailed expressions of the energy flux are determined for the pertinent waves in solids and fluids. T h e propagation condition and the propagation modes are significantly affected by the occurrence of internal constraints and body forces. For internal constraints, such as inextensibility in one direction and incompressibility, results arise which generalize the analogous ones for non-dissipative media.

## Special Topics & Reviews in Porous Media: An International Journal

A similar conclusion holds for the effect of body forces. They have a twofold role. First, they put the body in equilibrium in a placement where the stress cannot vanish. Wave propagation is then viewed as superimposed on a prestress and a predeformation. That is why prestressed media are examined in some detail.

The displacement field as the gradient of a scalar potential plus a solenoidal field was introduced by Poisson in Next Clebsch proved that every solution of the equation of motion admits the aforementioned representation; a rigorous proof of this result was provided by Duhem. A result and procedure, strictly analogous to those for elastic solids, can be exhibited when a time-harmonic dependence is involved. Here we give the detailed proof also with a view to applications to bulk waves, reflection and refraction at plane interfaces, surface waves.

Consider a homogeneous viscoelastic solid relative to a stress-free placement. The motion of the solid is described by the displacement vector u as a function of the position vector x and the time t. Upon substitution, 1. Specialize 1. A]H, 1. To determine N2 we have to require that U meet 1. Substitution in 1.

Since N2 is harmonic, by 1. This means that N2 may depend on x, at least linearly. In a sense, equilibrium conditions seem to allow a more general representation for the potentials. The representation for the potentials is non-unique. As a non-trivial example we mention that two equivalent representations are known for the equilibrium case. This means that we have to examine the existence of scalar and vector solutions to 1.

Some geometrical aspects are examined in a moment. This is so because a physical monochromatic wave of frequency ui is the superposition of two complex-valued waves of frequency u and —w; the validity of 2. As a consequence of 2. It is worth emphasizing some consequences of thermodynamics. Because of 2.

So, in essence, thermodynamics implies that the wave amplitude decays while the wave propagates. The properties 2. We write any one of 2. It follows at once from 2. Upon obvious substitutions we obtain from 2. Rather, given the constitutive properties and the angle between kj and k 2 the wave mode is determined by 2. Then by 2. Then we are left with a single equation, 2.

There are infinitely many solutions parameterized, e. Such solutions may be expressed, e. For such wave solutions the energy flux velocity and the group velocity are different [85]. In the next chapter we show how such waves can be generated by refraction. Then k is a real vector and is determined, to within the direction, by 2. The relation 2. To each point of the circle corresponds a wave exv[iu qxx cf. As already mentioned, waves of the form 2. In the general scheme adopted here, evanescent waves are merely a particular example of inhomogeneous waves. Indeed, they correspond to k2 being orthogonal t o k i , which is easily seen to describe waves of constant amplitude in the direction of propagation.

We observe that the result 2. In view of 3. Again, we can regard 3. The consequences of thermodynamics on the wave modes follow at once. The analysis of the wave modes for the viscoelastic solid then applies, mutatis mutandis. As regards the viscous fluid, by 2. Observe that no restriction is placed by thermodynamics on the derivative pp. This is not necessarily so for transverse waves. Roughly speaking, propagation and attenuation have the same weight. In viscoelastic fluids no bound is given on fis and then we may have a 0, is true. It is often asserted that, in linearly viscous fluids, 52 Inhomogeneous Waves in Solids and Fluids the attenuation of acoustic waves is proportional to the frequency squared LJ2.

Of course, the knowledge of the attenuation is important in many respects; for example, the attenuation provides a very sharp upper cutoff frequency above which a given lens cannot be operated. To our knowledge, the standard argument for claiming that the attenuation is proportional to the frequency squared has been the one exhibited, e.

For the benefit of the interested reader, here we follow very closely the pertinent part in []. As a consequence, by 3. To our mind this conclusion is questionable. By means of the results 3. Accordingly, for transverse waves the attenuation k2 is proportional to the square root of w. Consider longitudinal waves. Substitution of 3. Jv Owing to the oscillatory behaviour, a useful measure of the power is given by its average V over a time period.

Jv T-Vdx.

ocivumyfof.tk Consider a region ft in the configuration of the body. For viscoelastic bodies, due to the thermodynamic inequalities 2. Incidentally, this is a check of consistency of the definition of Vt. Owing to some discrepancy exhibited in the literature about energy density and flux cf. Inner multiplication of the equation of motion 1. Examine separately 4. First regard the body as elastic.

To our mind this interpretation is misleading. According to the general approach to balance equations, the balance of energy can be written as cf. Otherwise no surprise should arise that some authors [17, 11], on the basis of the different splitting V. Now let the body be viscoelastic. The possibility of inequivalent choices of energy density and flux is more apparent. By following a splitting procedure and having recourse to considerations about a spring-dashpot model, some authors cf.

This splitting is highly subjective. Further, by appropriate splittings the rate of dissipation is made to vanish, which might seem quite odd due to the dissipative character of the viscoelastic Inhomogeneoua Waves in Unbounded Media 57 model. However, looking at V as the rate of dissipation is an arbitrary choice motivated by a subjective partition of the rate of working. The same is true for the energy flux which then is found to have different representations and values. A final remark seems to be in order. The literature has devoted noticeable attention cf. Needless to say, this quantity is well defined only when both the energy density and the energy flux are uniquely determined.

As we have seen, sometimes such is not the case and the ambiguities remain even though we consider the mean values. We regard the energy flux intensity I as the energy transfer per unit area in the direction of wave propagation. Substitution in 5. Things are not that simple in viscoelasticity [39]. The peculiar aspect of dissipativity consists in the decay, here described by exp —2k 2 -x. Further, the inhomogeneity of the wave results in the direction of J which is usually different than n i.

By use of 5. The first line of 5. The second line is new in the literature. Here we are content with remarking that if 3. Indeed, via suitable interpretations of the pertinent quantities, our developments apply to both viscous and viscoelastic fluids. With reference to the equation of motion 2. Correspondingly by 5.

If the fluid is viscous. Hence, by 5. Substitution in 6. This result is formally similar t o the analogous one for solids. Inner multiplication by iij provides the mean energy flux intensity I. Two common examples of internal constraints are inextensibility in one direction and incompressibility. By objectivity requirements, the value of any function 7. Then we can write 7. It is reasonable to expect that these restrictions on the motion shall not specify the strain.

The corresponding 64 Inhomogeneous Waves in Solids and Fluids which are apparently symmetric. The quantities a' Lagrange multipliers are, so iar, indeterminate. They are in fact functions of the position x and time t which are required to satisfy the equations of motion and the pertinent boundary conditions. Again on disregarding the body force, we consider the equation of motion 1. Here, for the sake of definiteness, attention is focussed on a singly constrained body and it is assumed that the inextensibility constraint 7. While the constitutive stress is given by 1. Upon substitution into the equation of motion 7.

Observe t h a t , because of the constraint, the propagation condition 7. Then, quite naturally, the internal constraint reduces the set of solutions. We examine two non-trivial cases. The wave vector is orthogonal to e, namely to the direction of inextensibility. The transverse isotropy of the body suggests that we find the same result as for waves in unconstrained media.

Such is really the case, i. The polarization is orthogonal to the direction of inextensibility. It is convenient to take the inner product of 7. In both cases we find formally the same results as for elastic bodies cf. Really, from a qualitative standpoint, the body force has a twofold effect. Second, even when b is constant, the body force enters the propagation condition thus affecting in fact, reducing the set of wave solutions.

Both aspects are now emphasised by investigating wave propagation in dissipative fluids cf. Regard the body force b as constant and then apply the equation of motion in the form 2. Incidentally, we expect that p and p p depend on the position x in the fluid. For standard experimental conditions it is then reasonable to regard p in 8. Investigate the possible solutions to 8. It follows from 8. In fact inner multiplication of 8. In the latter case, instead, 8. Two possibilities occur accordingly as k is parallel to b or is not. Yet, by 8.

Inner multiplication of 8. A simple, immediate effect of the body force appears by considering the wave vector k in the plane orthogonal to b , i. The determinantal equation 8. Of course the quantitative effect on the phase speed may not be small in other fluids. This motivates a detailed analysis of reflection and refraction of inhomogeneous waves at a plane interface. Of course, the general case occurs when the plane of the incident wave vector is not orthogonal to the interface.

The behaviour of inhomogeneous waves at a plane interface is investigated by letting the interface be the boundary of a viscoelastic solid half-space, or the common boundary of two viscoelastic solids or two layers of a multilayered solid; this last case is of remarkable interest in seismology. Relative to the particular case when the plane of the incident wave vector is orthogonal to the interface, new effects are shown in connection with the polarization of the reflected and transmitted waves.

To unify the treatment of incident longitudinal and transverse waves, the incident field is taken in the form of a conjugate pair, namely the superposition of a longitudinal and a transverse wave whose wave vectors have equal projections on the interface. In this framework the matrices describing the reflection and refraction can be derived in a straightforward way for any interface. Of course, owing to the complex nature of the polarizations of the pertinent waves, the determination of reflection and refraction coefficients deserves some attention.

As with the interface between elastic media, the refraction coefficients may become greater than unity for certain directions of the incident wave, usually around critical angles. This looks paradoxical also on the basis of an intuitive idea of energy conservation. In fact the seeming paradox is overcome by considering carefully the energy flux intensities of the pertinent waves; for definiteness this is performed in the case of a perfect fluid-viscoelastic solid interface. In essence, for any interface the energy flux intensity involves the cross section and the ratios of the cross sections approaches zero, or infinity, when the angle of the incident wave vector is around the critical value.

Inhomogeneous waves correspond to special choices of the potentials. Now, regardless of the properties of the sources, we may consider incident waves with real-valued amplitudes. The trouble is that, as shown later, the real-valuedness is not invariant under reflection and refraction and this makes the real-valuedness assumption generally inconsistent. No choice is made about the polarization of the incident wave which may be longitudinal or transverse.

Nor is the coplanarity of the vectors k i , k 2 , n assumed. Because of the discontinuity surface, four inhomogeneous waves originate at any point of V: a longitudinal and a transverse wave for each medium. A reflection-refraction problem consists in the determination of the reflected and transmitted or refracted waves in terms of the properties of the two media and the incident wave. To solve such a problem we first determine a priori geometrical restrictions on the pertinent wave vectors.

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We assume that the two half-spaces are in welded contact. Then the boundary conditions governing the process of reflection and refraction are the continuity of displacement 72 Inhomogeneous Waves in Solids and Fluids Fig. Denote by the accent " any quantity pertaining to the half-space opposite to that of the incident wave.

In particular it follows from the separate contributions of the real and imaginary parts of 2. For instance, by 3. For any wave vector k, we can combine 3. The solution of the algebraic system 2. Really, reflected waves correspond to the positive sign, while refracted waves require the negative sign. Restrict attention to the reflected wave of the same kind of the incident one i. Further consequences follow if the two media are specified. Then 2. The following possibilities occur. Then they both are negative in view of the fact that we are considering transmitted waves.

As in the elastic case, the amplitude decays with distance from the interface. If, however, also the upper medium is viscoelastic, then by 2. The previous scheme can be summarized as follows. The z-components of the incident and reflected waves coincide with the left-hand sides of 2. The wave vector is said to be vertically polarized if k i , k2 and n belong to a common vertical plane jr. In view of Snell's law, the property of being vertically polarized is inherited by the reflected and the transmitted wave vectors. Unlike the elastic case, in general the wave vector of the incident wave is not vertically polarized.

Indeed, the vectors ki and k2 depend on how the wave has been generated, and this is unrelated to the direction of n. This feature is at the origin of the formal complications, connected with the behaviour at interfaces, which cannot be avoided unless restrictive assumptions are made. Because of Snell's law, both pairs of reflected and transmitted waves still constitute conjugate pairs.

To unify the treatment of longitudinal and transverse waves we will consider reflection and refraction of a conjugate pair of downgoing waves.

### Submission history

Owing to conjugacy, there are only four, rather than eight, outgoing waves just as in the case of a single incident wave. By the linearity of the equations, the particular case of a longitudinal or transverse wave hitting the interface follows trivially. It is understood that the z-components of the wave vectors involved are non-zero.

The limit case when the z-components vanish will be examined separately. By use of 1. The analogous expressions for displacement and traction in the lower half space are simply recovered from 3. Really, addition of the left sides of 3. Of course, this condition is preserved for the reflected and transmitted pairs and, according to 2. To bring into evidence the origin of the various contributions t o the amplitudes of the reflected pair we reformulate 4. The meaning of such a matrix is apparent.

Suppose for definiteness that the incident wave is longitudinal. Similarly, the second and third columns identify the reflection coefficients corresponding to the case of an incident transverse wave. Then the system 4. T h e subsystem embodies the case of a longitudinal or a transverse vertically-polarized wave in the elastic framework, while the single equation corresponds to an incident transverse horizontally polarized wave. Along with the limits of 4. A further simplification occurs in the case when the incident wave vector is vertical i. Then the expression of t is given by 3.

We are now in a position to consider some features, e. Inspection of 4. Comparison with the definitions of D and A, namely 4. Hence 4. This result is not true in general. This shows that in general complete conversion of a transverse wave into a longitudinal one is not allowed.

Obviously, under these conditions we cannot consider an incident pair but it is appropriate to deal separately with the two allowable choices of the incident wave. In view of its inherent simplicity we study first the case when the incident wave is transverse. Suppose that the incident and reflected transverse waves are described as in 3. As is easily verified, the potential 4. Similarly, continuity of the phase is also ensured by 4. Therefore we have to consider the field obtained by superposition of the potentials 3.

Actually, the wave vector of the longitudinal wave is purely horizontal, which means that a generic reference to a reflected wave of longitudinal type is, in a sense, a slight abuse. Let us consider a scalar potential of the form 3. The continuity condition for the phase at the interface is also satisfied. Notice that the wave vector of the transverse wave is horizontal. This is ultimately due to the fact that the constant parameter 4.

First we exhibit the solution in a matrix form, which is particularly suitable for numerical evaluation. The subsystem 5. Substitution of this expression into 5. We now determine the explicit solution to 5. This allows the transformation 5. Left multiplication of 5. Taking the difference of 5. Now we write the solution in terms of the coefficients a ; , regarded as privileged parameters. Insert 5. As in the general case we arrive at 5.

Now we examine some degenerate situations where the z-components of reflected or transmitted wave vectors vanish. In principle there are four possibilities. This case is very simple and then is not considered. Rather we examine transmitted wave vectors with a vanishing z-component in which case the incident waves may still constitute a pair. Setting aside inessential details, we say that when these results are replaced into the continuity conditions 5. We only observe that the first column of the matrix 90 Inhomogeneous Waves in Solids and Fluids 5.

So, apart from this substitution, everything goes as in the general case, as far as the determination of reflected and transmitted amplitudes is concerned. With reference to 5. Reflection and Refraction 91 These relations can be used to construct the analogues of the continuity conditions 5. We multiply 5. Then we sum the results and contrast with 5.

Next we multiply 5. Moreover, the more familiar description in terms of angles is established. The columns of T and R may be interpreted as refraction and reflection coefficients. As opposed to the elastic case, two pairs of coefficients, instead of two coefficients, are required to recover the transmitted and reflected transverse waves originated by an incident longitudinal one [22]. Of course, similar remarks also apply to the case of an incident transverse wave. Actually, the form of the coefficients of reflection and refraction is rather involved, in that they depend on the characteristics of the incident pair, via kx and ky, and the material parameters of the media via KL, KT, kL, kT.

By analogy with elasticity, one might think that a qualitative information is obtained by regarding the coefficients as functions of the geometric characteristics of the incident waves, which is obtained by looking at kx as the product k' sin 9', with variable 0'. Seemingly, such is not the case in the present context in that we have to vary the two components kx and ky, both of them being complex-valued; furthermore, also the reflection and refraction coefficients are complex-valued.

A rather special case where this can be done is treated in the next section. To avoid any ambiguities it is worth recalling that if the real and imaginary parts of the incident wave vectors belong to a common vertical plane, then this plane is invariant under reflection and refraction.

That is why special attention has been drawn to wave vectors in the vertical plane, which allows considerable simplifications in the expressions of the reflection and transmission matrices. The analogy with elastic waves suggests that we consider also amplitude vectors that are vertically or horizontally polarized. Therefore, in dealing with the polarization we have to specify whether we are referring to amplitudes or wave vectors, unless this is unambiguously clear through the context. In any case the vertical plane is systematically identified with the z,z -plane. A further remark is in order.

This approach results in much simpler calculations and brings immediately into evidence mode conversion phenomena at Inhomogeneous Waves in Solids and Fluids 94 interfaces, the only drawback being that the relations between the complex amplitudes pertaining to longitudinal and transverse waves are left aside.

Of course, these relations must be considered before polarization effects are investigated. This feature is ultimately due to the fact that, as is shown later, the wave vectors of homogeneous plane waves in non-dissipative media are vertically polarized. With reference to the representation 1.

### 1 Introduction

Incidentally, by 6. To benefit from the results of the previous section, it is worth observing t h a t , upon substitution of 3. Comparison with 6. In view of 6. Accordingly, the vertical polarization for amplitudes is not preserved under refraction. In particular, 5. As a consequence of 6. In addition, according to 6. Substitution into the expressions 3.

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Then application of 6. Again, we have the same formal results as for the shear waves of horizontal polarization in linear elasticity [22]. Results on the behaviour of inhomogeneous waves at a plane interface have been described in terms of Cartesian components of wave vectors. Most often, though, incidence, reflection, and transmission angles are the parameters used in the investigation of plane waves within the framework of linear elasticity and linearized fluid dynamics [2, 3]. It is then worth establishing a connection between the two descriptions. The wave vector of the incident wave is regarded as given; those pertaining to the reflected and transmitted waves are determined through the use of Snell's law and the material properties of the medium where propagation takes place.

Moreover we denote by the angle between the unit vectors n 2 and ez, with e [0, ar , and by a and ip the angles between the i-axis and the projection of rij and n 2 on the i , y -plane. It follows that the unit vectors n! Of course, the relations 6. Further, t h e analogues of the relations 6. In general, the better understanding allowed by the use of angles has an unpleasant counterpart due to the fact that the expressions of the matrices T and R become more and more involved.

This fact is essentially the motivation for our preference to the use of Cartesian components. In special cases, though, the use of angles may be more profitable. A case in this sense occurs when the wave vectors of the incident pair are vertically polarized. By Snell's law this holds for the reflected and the transmitted pairs as well.

Reflection and Refraction 99 Substitution into the expressions of Tv and Rv obtained from 5. For definiteness and for simplicity we restrict attention to the case when the interface is the common boundary of a viscoelastic solid and an inviscid fluid. No essential effect is lost, while considerable simplifications of the expressions involved are achieved. Via straightforward changes the procedure is applicable to the other cases already examined.

In our numerical calculations the fluid is identified with water and the solid with annealed copper. It is assumed that the incident wave is coming from the fluid in the upper half-space. The displacement field of the inviscid fluid occupying the upper half-space is described through the scalar potential.

The incident wave is taken as plane and homogeneous. As an immediate consequence of 7. We examine such dependence for both longitudinal and transverse waves at the same time. Omit the subscripts l o r T and consider 3. Observe that in such a case the solution 3. In order to determine the numerical values of tkL and kT we identify the solid with a material such as annealed copper. Reflection and Refraction Fig 4. L is negligible for both waves; this behaviour closely resembles that of elastic bodies.

However the effect of dissipation becomes more and more influential for growing angles of incidence. When the incidence angle is greater than the critical one Re 0L and Re J3T vanish, which means that the transmitted waves propagate parallel to the interface and decay with distance from the interface. As shown also by the vanishing of the energy flux intensity, this behaviour corresponds to total reflection. To determine the emerging waves we need the reflection and transmission coefficients as functions of the incidence angle. Now, the results of the previous section follow from the requirement of continuity for displacement and traction at the boundary.

If one of the media is an inviscid fluid then, in addition to the continuity of the traction, we require only the continuity of the normal component of the displacement, in that the fluid can freely slip on the boundary without formation of cavitations. In the viscoelastic solid, according to 3. Accordingly, displacement and traction follow from 3. Inhomogeneous Waves in Solids and Fluids Fig. The behaviour in Fig. These results parallel those obtained by Mott [] in the analysis of incidence at a water-stainless steel interface and those of [44] under the influence of dissipation.

Figure 4. It is apparent that they are greater than unity at certain values of the angle of incidence. This looks quite paradoxical in that we have in mind that the energy of the incident wave is partitioned among the reflected and transmitted waves. The paradox is solved by the following analysis of the energy flux intensity associated with the pertinent waves.

As regards the transverse wave the wave vector is vertically polarized while the vector amplitude 4? Reflection and Refraction Fig. Therefore 3. The dependence of the intensities of reflected and transmitted, longitudinal waves on the incidence angle 9' is represented in Figs. The transmitted, transverse wave shows a behaviour very close to that of the transmitted, longitudinal wave. According to 7. This seems to contradict the energy conservation, especially because the ratio between reflected and incident densities is equal to unity.

T h e tubes have a common intersection with the interface and, for technical convenience, we regard the common area da as infinitesimal. Here the first equality is a direct consequence of Snell's law. Then, by 7. By the natural idea that energy is conserved in the reflection-refraction process, we expect that the sum of the quantities 7. The numerical check through Fig. In view of 7. This means that the amplitude of surface waves varies in planes of constant phase. This in turn shows that inhomogeneous waves are the natural framework for the investigation of surface waves.

The surface that guides the wave may be the external boundary of a body or a material discontinuity such as an interface between different materials. The existence and the form of surface waves depends on the conditions at the surface. Surface waves are easily seen to occur in incompressible viscous fluids. The corresponding propagation condition, or secular equation, determines k in terms of the frequency w and the viscosity of the fluid. The corresponding surface wave is shown to be the superposition of two inhomogeneous waves. The analysis of surface wave solutions on elastic half-spaces is a preliminary step toward the more involved case of viscoelastic half-spaces.

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In this regard emphasis is given to the role of the secular equation and the corresponding polynomial analogue, usually called Rayleigh equation. Non-trivial solutions are found to hold when the polarization is in the sagittal plane. The procedure to select the physically admissible solutions is highly non-trivial and involves both the decay condition and the thermodynamic restrictions. Obviously more complicated is the analysis of Stoneley waves, namely surface waves at the interface between two half-spaces with different material properties.

Surface waves in solids are customarily investigated by neglecting gravity effects. The smallness of these effects is not obvious at all. Indeed, the gravity acceleration induces a prestress in the pertinent body. Such prestress turns out to affect significantly the Rayleigh equation and then the surface wave solutions for any material properties of the half space. Roughly speaking, surface wave solutions are sought in the form of functions Inhomogeneous Waves in Solids and Fluids which oscillate along a direction of the free surface of the fluid and decay with depth in the fluid or distance from the surface.

Then 1. Surface Waves Equations 1. We now have to determine the "amplitudes" A,B through the boundary conditions 1. In the linear approximation, time differentiation of 1. So we have a homogeneous linear system in A, B. A natural question arises as to whether surface waves are inhomogeneous waves. This in turn indicates that inhomogeneous waves are the natural framework for the analysis of surface waves.

The x, z -plane determined by the direction of propagation x in the plane boundary surface and the orthogonal direction z is usually called sagittal plane. Now, by 2. Accordingly, complex roots of the Rayleigh equation 2. Only the real root s ' 1 ' 6 0,1 represents a surface wave. Though mathematically correct, t o our mind this statement is open t o doubts about the consistency of the model.

Complex roots for s correspond t o complex values of c. Then it Surface Waves should come as no surprise that complex roots do not represent admissible surface waves. All this applies to waves whose polarization lies in the sagittal plane which then may be viewed as a superposition of P and SV waves. The analogous problem for waves whose polarization is orthogonal to the sagittal plane, i.

SH waves, is trivial. This shows that surface waves of SH type cannot exist in elastic traction-free half-spaces. In fact SH waves may exist when a layer of elastic material is superimposed on an elastic half-space Love waves. In this regard one should perhaps conclude that, in viscoelasticity too, only the real root represents a surface wave.

Such need not be the case [54]. An appropriate investigation might be based on the structure of surface Inhomogeneous Waves in Solids and Fluids waves so that the admissibility of a root, as representative of a surface wave, m a y clearly be ascertained. Considered here are waves that are a superposition of inhomogeneous waves and, of course, are essentially confined to a neighbourhood of the boundary.

Look at a viscoelastic solid which occupies a half-space. SnelPs law is taken to hold and then kx and ky are also the common values for all wave vectors involved. Really this restriction proves unnecessary for surface waves. The characteristic features of the reflected waves, and of the secular equation that defines Rayleigh waves, are determined by the vanishing of the traction. We characterize a Rayleigh wave by the following two properties. Second, the amplitudes of the single components decay with distance from the surface i.

On squaring 3. It follows at once from 3. Of course 3. To find such solutions we indicate the following procedure [43]. The sought values of s should be determined through 3. However, the observation that it is much simpler to find the roots of a polynomial and that the roots of 3.

Obviously, 3. Inhomogeneous Waves in Solids and Fluids Evaluate fiL through 3. In the procedure so outlined the relation 3. Indeed, we might follow an alternative, equivalent procedure where the roles of 0L and 0T are interchanged and 3. It can be shown very easily that disregarding 3. Now, squaring 3. It is true that if liakx 0 might prove overly restrictive thus dropping out admissible wave solutions.

Incidentally, the value of kx occurs only at the last step of the procedure. Then it can be disregarded if a thorough characterization of the surface waves is not in order but, e. Hence, by 3. It is worth remarking that the solutions s 1 , s 2 ' , s 3 to 3. So, given the value of ui for the incident perturbation which excites the surface wave, we determine the value of g and hence the three solutions to 3.

In this sense it may be instructive to consider a particular example of viscoelastic solid. For this case we apply the procedure indicated above. The thermodynamic requirement 2. Then the roots of the Rayleigh equation 3. For reasons that become clear in a moment, such a wave is the analogue of the elastic wave; let us call it quasi-elastic wave.

This shows that if 0. It follows that, if s2 ji 0, the solution to 3. Really, a numerical check shows that no root exists in the circle 3. Consider the real root s 1 of 3. Such waves are often called Stoneley waves though sometimes Stoneley waves Inhomogeneous Waves in Solids and Fluids are meant as surface waves between two solid half-spaces. We call Stoneley waves all surface waves at the interface between two half-spaces.

Look at the interface between two elastic half-spaces. Requiring that the determinant of the coefficients vanish yields the secular equation in the unknown speed c. Here we remark that the wavenumber fc does not appear in the coefficients, and then in the secular equation. Accordingly, the corresponding solutions, namely the Stoneley waves, are not dispersive. The analysis of the secular equation is developed in the book by Cagniard [28]; the generalization to dissipative bodies seems to be a formidable problem.

Definite results are obtained here by restricting attention to the interface between a dissipative solid and a fluid. Our procedure parallels that examined for the Rayleigh waves. No assumption is made about the sign of Im kx. The absence of a transverse, horizontal wave is due to the upper medium being a fluid. This view is strengthened by the dissipative character of the viscoelastic solid.

Leaky waves seem to contradict this view or, at least, seem to be outside the realm of surface waves. Now look at the case when no incident wave occurs.

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