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I. Introduction

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Reflection and transmission responses of layered transversely isotropic viscoelastic media

Entire Site De Gruyter Online. Sign in Register. English Deutsch. Surface waves in non-homogeneous, general magnetothermo, viscoelastic media of higher order. Sethi munishsethi76 gmail. Gupta 2 and M. Rani 3. Open access. Keywords: inhomogeneous media ; variable density ; surface waves ; viscosity. International Journal of Applied Mechanics and Engineering. These results are completely analogous to those for the incident SI wave problem. They imply that for layered anelastic media the reflected SI wave and the transmitted waves will, in general, be inhomogeneous.

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A detailed discussion of the implications of this result is provided following Theorem 5. They are restated here for completeness. Results for elastic boundaries implied by Theorem 5.


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For the problem of a homogeneous P wave incident in an elastic medium Von a plane boundary with anelastic medium V 0, no critical angles exist for either reflected or transmitted waves. Results for an anelastic—viscoelastic boundary are as follows. For elastic— anelastic and anelastic—viscoelastic boundaries Theorems 5. Results for the problem of an incident general SII wave problem, as initially derived by Borcherdt , are briefly stated for completeness as a rigorous basis to establish the nature of energy flow at a viscoelastic boundary.

Parameters for the problem of a general SII wave incident on a welded boundary are illustrated in Figure 5. Diagram illustrating notation for directions and magnitude of propagation and attenuation vectors for the reflected and refracted waves due to a general Type-II S wave incident on a welded boundary between HILV media. A similar discussion is omitted here. These solutions establish that the boundary conditions for the specified problem are satisfied with the assumed form of the solutions 4. Hence, the amplitude and phase of the displacement for the reflected and refracted general SII waves may be expressed in terms of that for the given incident general SII wave.

Solutions 5. With k given by 5. Hence, from a mathematical point of view the solution for the problem of a general SII wave incident on a viscoelastic boundary is established. For brevity, only the results for the incident SII problem are given here see Borcherdt for details. For the problem of a general SII wave incident on a welded viscoelastic boundary, the reflected SII wave is homogeneous if and only if the incident SII wave is homogeneous.

[فروش کتاب] Viscoelastic Waves in Layered Media

The condition for the transmitted SII wave is as follows. They are stated here for completeness. For the problem of a homogeneous SII wave incident on a plane boundary between an elastic medium V and an anelastic medium V 0, no critical angles exist for either reflected or transmitted waves. A detailed treatment for the incident general SII wave problem provides a framework from which to consider energy flow for the more cumbersome incident SI and P wave problems.

The development presented here follows that initially derived by Borcherdt A general result which describes the intuitively evident situation for energy flow across the boundary is Theorem 5.

Viscoelastic Waves in Layered Media

For steady-state radiation fields described by 4. This result follows immediately from the fact that for the prescribed radiation fields the mean energy flux as given by 3. Hence, substitution of 5. The mean energy flux associated with the propagation of the individual incident and reflected SII waves is given in terms of the wave parameters from 3.

Substitution of the physical displacement field as specified by 3. However, if medium V is anelastic, then 5. This expression also shows that this normal component of energy flow varies sinusoidally with distance from the boundary. It shows that at some distances there is a mean flow of energy toward the boundary and at other distances the mean flow of energy due to interaction is away from the boundary.

The condition of continuity of the normal component of mean energy flux across the boundary as stated in Theorem 5. Energy partition curves showing the relative amounts of energy reflected and transmitted as various wave types as computed for elastic media Ewing, Jardetsky, and Press, , pp. For anelastic viscoelastic media the preceding results show that energy flows as a result of interaction of superimposed wave fields.

Consequently, for such media, it is reasonable to expect that energy also would be dissipated due to interaction of superimposed wave fields. The mean rate of energy dissipation per unit volume for a steady-state radiation field is given from 3. Hence, a finite amount of energy is dissipated due to the interaction of the two superimposed wave fields, a phenomena not encountered when considering elastic reflection—refraction problems. The preceding results derived to account for the energy flow and dissipation at a viscoelastic boundary are extendable in a straightforward way to problems of incident P and SI waves.

Proofs for the problems of incident P and SI waves are similar to those presented here. However, they are more cumbersome, because the number of interaction coefficients increases to six in medium V and two in medium V 0. For later reference, notation used for the incident SII problem is extended here for the problems of incident P or SI waves.

The notation is extended as follows: 1 reflection and transmission coefficients for P and SI waves are distinguished by addition of subscripts corresponding to wave type, 2 interaction coefficients corresponding to interaction of the velocity and stress fields of the incident wave with those of the reflected P and SI waves are distinguished by addition of subscripts corresponding to wave type, 3 interaction coefficients corresponding to interaction of the velocity and stress fields of the reflected P and SI waves or the transmitted P and SI waves are distinguished by addition of an ordered pair of subscripts corresponding to wave type.

Using these notational extensions for the problems of either an incident P or an incident SI wave, the condition for conservation of energy at the boundary 5. Proofs of these expressions for the problems of incident P and SI problems are similar to those for the incident SII problem.

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For brevity, they are left to the reader. As a result physical characteristics of the waves such as phase velocity, energy velocity, phase shifts, attenuation, particle motion, fractional energy loss, direction and amplitude of maximum energy flow, and energy flow due to wave interaction vary with angle of incidence. Consequently, these physical characteristics of inhomogeneous waves propagating in a stack of anelastic layers will not be unique at each point in the stack as they are for homogeneous waves propagating in elastic media.

Instead these physical characteristics of the waves will depend on the angle at which the wave entered the stack and hence the travel path of the wave through previous layers. Towards understanding the significance of these dependences of the physical characteristics on angle of incidence and inhomogeneity of the incident wave, numerical models for general SII and P waves incident on single viscoelastic boundaries are presented in this chapter. Study of this chapter, especially the first three sections, provides additional insight into the effects of a viscoelastic boundary on resultant reflected and refracted waves.

A computer code WAVES is used to calculate reflection—refraction coefficients and the physical characteristics of reflected and refracted general waves for the problems of general homogeneous or inhomogeneous plane P, SI, and SII waves incident on a plane boundary between viscoelastic media Borcherdt et al. The code utilizes general solutions in media V and V 0 as specified by 4. Physical characteristics of the reflected and refracted waves are specified in terms of the given parameters of the incident wave and the parameters of the media using equations 3.

This characterization permits interpretation of results in terms of any chosen linear viscoelastic model such as those specified in Table 1. It permits straightforward incorporation of any chosen frequency-dependent behavior corresponding to a particular viscoelastic model by specification of wave speeds and reciprocal quality factors from corresponding dispersion and absorption curves for homogeneous waves. Normalization of certain wave parameters by the appropriate power of angular frequency allows the calculations to be applicable to any viscoelastic model and frequency to which the chosen material parameters are appropriate.


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  7. This comparison helps quantify the errors associated with estimates of the physical characteristics of the waves based on low-loss models which do not account correctly for inhomogeneity of anelastic waves. This part of the computer code assumes the physical characteristics of the resultant homogeneous waves are described by low-loss expressions 3.

    It assumes the wave speeds that characterize the material are those derived from the parameters specified for the material using the low-loss expressions for homogeneous waves, namely 3. Media parameters chosen for a corresponding initial numerical model describe either water-saturated sediments Hamilton et al. The parameters for the media are given in Table 6.

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    These parameters characterize media with a moderate amount of intrinsic absorption for which the low-loss assumption is not valid. Choice of this model serves to illustrate several 6.

    Notation for the problem of a homogeneous Type-II S wave incident on a plane boundary between viscoelastic media is shown in Figure 6. Numerical estimates for the physical characteristics of the waves reflected and transmitted at the specified anelastic boundary computed as a function of angle of incidence are shown in Figures 6.

    Viscoelastic waves in layered media / Roger D. Borcherdt. - Version details - Trove

    Notation for the problem of a homogeneous Type-II S wave incident on a plane viscoelastic interface across which the material velocity and intrinsic absorption increase and decrease, respectively. Figure 6. It shows that phase propagation is not parallel to the boundary for any angle of incidence, which is consistent with results concerning critical angles as stated in Theorem 5.

    The exact anelastic model predicts that the direction of maximum mean energy flux for the transmitted wave increases with respect to that of phase propagation as the angle of incidence increases Figure 6.