How To Find Domain_3 Of A Wave

Summary

The nowadays investigation is concerned with transient wave propagation in a stone mass with a set of parallel joints past using a recursive method. Co-ordinate to the displacement field of a stone mass with a prepare of parallel joints, the interaction between four plane waves (two longitudinal-waves and 2 transverse-waves) and a joint is analysed get-go. With the displacement aperture model and the time shifting function, the wave propagation equation based on the recursive method in fourth dimension domain for obliquely longitudinal-(P) or transverse-(S) waves across a ready of parallel joints is established. The joints are assumed linearly elastic. The analytical solution obtained by the proposed method is compared with the existing results for some special cases, including oblique incidence across a single joint and normal incidence across a set up of parallel joints. By verification, it is constitute that the solutions past the proposed method lucifer very well with the existing methods. The applicability and limitations of the new method are then discussed for incident waves with unlike properties.

1 Introduction

Studying transient wave propagation across rock masses is an important topic, which has received considerable attending in geophysics, mining and cloak-and-dagger constructions. A rock mass commonly consists of multiple, parallel planar joints, known as joint sets, which not simply govern the mechanical behaviour of the stone mass only also bear upon the wave propagation in the rock mass. Considering of the discontinuity in nature, the wave propagation across stone joints becomes a complicated procedure. Therefore, it is of significance to develop an efficient and explicit method to analyse this process.

Generally, 3 methods are available for studies on moving ridge propagation across a jointed stone mass. I is the numerical modelling, which provides a user-friendly and economical approach, especially for complicated geological cases. However, representation of the joints in the numerical models presents a peachy challenge. The experimental study is the second method, which is sometimes limited by the existing exam techniques. The tertiary is the theoretical study from which the mechanism and the procedure of wave propagation beyond a jointed rock mass can be revealed for some special geological cases. Meanwhile, the analytical report can provide an guess and reference for the lab and/or the field tests to some extent.

Analytical studies for an incident P- or South-wave propagation across a discontinuous interface take been extensively conducted by many researchers. Kolsky (1953) derived the relation between the wave propagation speeds and the emergence angles of the reflected and refracted waves for a discontinuous interface betwixt 2 media, which is also termed as the Snell′s law. On the ground of the displacement discontinuity model and the Snell′s police force, propagation of oblique wave incidence beyond a planar linear slip interface was investigated by Schoenberg (1980). After, the close-form solutions for a harmonic incidence across a stone joint were obtained and expressed in a matrix form later derived by Pyrak-Nolte et al. (1990a,b), Gu et al. (1996) and Zhu et al. (2011). The in a higher place methods were based on the central solutions of the equation of motion. Based on the characteristic line theory ( Ewing et al. 1957; Bedford & Drumheller 1994) and the displacement discontinuity model (Miller 1977; Schoenberg 1980), Zhao and Cai (2001) calculated the transmission coefficient of incident P waves across a unmarried rock articulation. Considering the balance of momentum at the moving ridge front and the displacement discontinuity model, Li and Ma (2010) analysed the interaction betwixt a nail-induced moving ridge and a rock joint with arbitrary impinging angle.

Normally incident waves across a set of joints take too been investigated by a number of researchers. For example, Zhao et al. (2006a,b) adopted the feature line theory to derive a wave propagation equation in time domain, which can be applied for analysing normally incident P and Due south waves with an arbitrary waveform. With a transmission line formula, the handful matrix method (SMM; Aki & Richards 2002; Perino et al. 2010) was used to written report harmonic wave propagation beyond a set of parallel joints. By considering the rock mass every bit an equivalent viscoelastic medium, Li et al. (2010) analysed the unremarkably incident P-wave propagation across a fix of rock joints.

Compared to the normal case and the example of an oblique incidence across a single joint, the analysis for an incident wave across a number of rock joints is much more complicated, due to the new kinds of wave produced from the joint interface and multiple wave reflections among the joints. The multiple reflections have been recognized to have significant effects on the reflected and transmitted waves in jointed rock masses ( Pyrak-Nolte et al. 1990b). A reflection method (Fuchs & Müller 1971) and a propagator method (Kennett & Kerry 1979; Luco & Apsel 1983) were presented to written report an incidence travelling obliquely in a periodical layered medium. These 2 methods can establish the relation among unlike layers with respect to the reflection and transmission amplitudes and the corresponding stage shift, which were expressed in frequency domain. For an incident wave with capricious waveform, it is necessary to use the Fourier synthesis over frequency in the two methods.

This study is motivated by the need to better understand the role of a set of parallel joints on the transient wave propagation and how relevant parameters play their roles on the transmission and reflection. The transient waves are longitudinal-(P) or transverse-(S) waves, and their interaction with a linearly rubberband rock joint is outset analysed. According to the time-shifting function of the P and S waves betwixt ii adjacent joints and a recursive method in fourth dimension domain, the wave propagation equation across rock joints is established for arbitrary impinging angle. The special cases, such as the normal incidence across joints and the oblique incidence beyond a single joint are also investigated. The calculations for some special cases are compared with the existing results. Finally, the applicability and limitations of the proposed method are discussed.

two Time-domain recursive method

2.1 Trouble clarification

We consider a homogeneous, isotropic and linearly elastic rock which contains a fix of parallel joints, every bit shown in Fig. ane, where Due north denotes the joint number. Each joint is considered equally a non-welded contact interface, which is planar, large in extent and small in thickness compared to the wavelength. The joints are linearly elastic, lying in the x-z plane and extend to infinity in the ten-y aeroplane. The joints behave linearly with normal stiffness and shear stiffness , and the spacing betwixt two side by side joints is S. For the intact stone, µ and υ denote the shear modulus and the Poisson′s ratio, is the density, and and are the velocities of P and S waves, respectively.

Effigy i

Incident P wave upon a rock mass with a set of parallel joints.

In an ideally elastic stone, the displacement field for a two-D trouble is expressed in terms of a scalar potential and a vector potential (Achenbach 1973), which correspond to uncoupled longitudinal-(P-) and shear-(Due south-)waves propagation, respectively. An incident plane wave of either P- or S-wave travels in the 10-z plane. When an incident wave impinges on the interface of a discontinuity, both reflection and transmission take identify (Kolsky 1953). Define and equally the emergence angles of the incident P and S waves, respectively, and and as the critical angles of the incident P and S waves, respectively. And so, and , where and from the Snell′s law.

For the rock mass with a set of parallel joints, the rock materials between joints are identical. According to the Snell′s constabulary, the emergence angles of the reflected and transmitted waves are equal to the incident angles. In another discussion, if the emergence angle of the incident P wave is , the angles of the reflected and transmitted P waves are also equal to , so are the emergence angles of the incident, reflected and transmitted S waves. For the present problem, when the incident moving ridge propagates in a rock mass and is reflected multiple times among the joints, there exist iv waves propagating in four directions in the rock mass, that is, the right-running P and S waves and left-running P and S waves, which are shown in Fig. 2. The four waves along 4 directions across multiple parallel joints were likewise illustrated in the researches past Pyrak-Nolte et al. (1990a,b) and Gu et al. (1996). For a articulation, such as the Jthursday joint in the stone mass, the correct- and left-running P waves are symmetric with respect to the joint, so are the right- and left-running South waves.

Figure two

Schematic view of left- and right-running P and S waves in a rock mass.

two.2 Interaction betwixt stress waves and stone articulation

As illustrated in Fig. three(a), when a beam of right-running P-moving ridge impinges the left-hand side of the Jth articulation, there is a tiny element ABC delimited by the left-manus side of the joint, the wave forepart and the side of the moving ridge beam. Similarly, the other left- and correct-running P and S waves also give rise to the other tiny elements, as shown in Figs iii(b)-(h), when the waves arrive at 2 sides of the joint. In Fig. 3, and denote the normal stresses of the right- and left-running P waves on their moving ridge fronts, and denote the shear stresses of the right- and left-running S waves on their moving ridge fronts, where represents the symbols '-' and '+' which indicate the left- and right-manus sides of the Jth joint, respectively. Since the nowadays problem is a plane strain problem, the stresses on the beam sides of the right- and left-running P waves are and , respectively. If the body force is not considered, the normal and tangential components of stresses, that is, and (q= 1~four, m=-, +) on the left- and right-hand sides of the Jth joint tin be derived according to Li & Ma (2010), that is,

two

four

The normal and tangential stresses, and , on the left-hand side of the Jth articulation can be obtained from the superposition of and (q= 1~4), respectively, or and . The normal and tangential stresses, and , on the right-mitt side of the Jthursday joint can also be obtained from eqs (1) to (four). and are defined as the particle velocities of P and S waves, respectively, and and as and , respectively. According to the remainder of momentum on the wave fronts, the stresses on the wave fronts of P and S waves can be written as

When eq. (v) is substituted into eqs (1)-(4), and (m=-, +) on the 2 sides of the Jth joint tin can exist rewritten equally

seven

where and (g=-, +) are the particle velocities of right- and left-running P waves on the two sides of the joint, respectively; and and (m=-, +) are the particle velocities of right- and left-running S waves on the 2 sides of the joint, respectively.

Figure 3

Stresses and right- and left-running waves on the two sides of a joint.

It is noted from Fig. iii that the normal and tangential components of the velocity before the left-hand side of the Jth joint are

eleven

and the normal and tangential components of the velocity later the right-manus side of the Jth joint are

2.3 Wave propagation equation

For the Jth joint, the stresses and the displacements earlier and subsequently the 2 sides of the joint should satisfy the displacement discontinuous purlieus condition (Miller 1977; Schoenberg 1980), that is,

fourteen

where and are the normal and tangential stiffness of the joint; and are the normal displacement before and after the 2 sides of the joint, respectively; and are the shear displacement before and after the two sides of the joint, respectively. When eq. (fifteen) is differential with respect to time t, there is

sixteen

where Δt is a minor time interval. The relations for the P and Due south waves between ii adjacent joints must satisfy the time-shifting function, or

eighteen

where 'J-i' and 'J+i' denote the (J-one)th and (J+1)th joints in the rock mass, respectively. Eq. (17) for the P and S waves between two adjacent joints implies that the right-running P or Due south wave, or , on the left-manus side of the Jth articulation keeps zero earlier or emitted from the right-hand side of the (J-1)th joint arrives at the Jthursday joint. Similarly, eq. (18) shows that the left-running P or S wave, or , on the correct-hand side of the Jth joint is zero until the arrival of or caused from the left-paw side of the (J+1)th joint. In the following analysis, the shifting times for the P and S waves between two adjacent joints are scattered as and , respectively, which are the integers of and , or and .

When eqs (six)-(xiii) are combined with eqs (xiv) and (16), the moving ridge propagation equation beyond the Jth joint tin exist derived and expressed as a matrix course

twenty

Considering the fourth dimension-shifting functions (17) and (18), the waves between two adjacent joints are written as

where A to F are the matrix parameters shown as

Eqs (19)-(21) are the recursive equations for an incident P- or Due south-wave propagation across a ready of parallel stone joints. The moving ridge propagation eqs (19)-(21) include two portions: ane is eqs (19) and (20) for moving ridge propagation across a articulation and another is the fourth dimension-shifting role (21) for wave propagation between ii adjacent joints.

ii.four Calculating steps

If the incident P or Southward wave in Fig. ane is or , there is or in eqs (nineteen) and (xx) for the showtime articulation or J= 1, where the symbol 'T' denotes the matrix transpose. The initial condition is that the left- and right-running P and S wave are zero except equal to the incident wave.

First, at time , the left-running P and S waves, and , on the right-hand side of the first joint can be obtained from the left-running P and S waves, and , on the left-hand side of the second joint, that is, from Eq. (21). Similarly, for fourth dimension can likewise be obtained. Co-ordinate to the initial condition and the incidence, the correct-running P and S waves for time can be calculated from eqs (nineteen) and (20) for J= 1.

Secondly, when J varies from two to Northward-ane, the left- and correct-running P and Southward waves across the Jth joint tin exist obtained from the waves caused by the two adjacent joints from eq. (21), that is, and For an incident S moving ridge and J= 2, there is . Combining the results in the previous steps, the right-running P and Due south waves, (J= two~N-one), at time tin can be calculated from eqs (nineteen) and (xx) for the joints ranging from the second to the (North-1)th.

Thirdly, when J is N, there is no left-running P and Southward waves on the correct-hand side of the articulation, that is, . From eq. (21), the correct-running P and S waves, and , on the left-hand side of the Nth articulation can all the same be obtained from the waves emitted from the (N-1)thursday joint, that is, . Hence, eqs (19) and (20) can requite the reflected wave at time earlier the first joint and the transmitted waves at time after the Nthursday joint.

When the above iii steps are repeated, the wave propagation across the N stone joints tin be finally obtained in the whole time history. Past applying this recursive method, the transmitted and the reflected waves across the joints in a rock mass can be calculated progressively by eqs (19)-(21).

To depict reflection and refraction, the transmission and reflection coefficients acquired by an incident P or South wave are defined as,

where η denotes the incident P and Southward waves, that is, η = p, s; denotes the P and S waves acquired by the rock mass, that is, =p, south; the subscripts r and l are for the right- and left-running waves, respectively.

three Special cases and comparisons

In this section, comparisons are conducted between the solutions from the proposed belittling method and the existing established analytical methods, by examining several cases. This also serves as a verification of the proposed method. Here, and are defined to be the normalized normal and tangential joint stiffness, respectively, where ω is the angle frequency of the incident waves.

3.ane Oblique incidence across a single joint

It is assumed that the rock mass contains only one single articulation in this section. When the incident P or S waves obliquely impinges the stone joint, there is no left-running P and S waves on the right-paw side of the joint, that is, and . Eqs (xix) and (20) tin can be simplified as

where and denote the incident P and S waves, respectively. It is institute that eqs (29) and (30) are in the aforementioned course with those by Li & Ma (2010), who derived the blast wave propagation equation for a linear rock articulation. In social club to verify the present method, the parameters adopted in the present study are the same as those by Gu et al. (1996), that is, , and The incidence is assumed to be sinusoidal P or S waves. Fig. 4 shows the variation of the transmission and reflection coefficients with the normalized joint stiffness, where Fig. 4(a) is for the incident P moving ridge and Fig. 4(b) is for the incident S moving ridge. In Fig. iv, the continuous curves are calculated by the proposed method and the scattered points are from the method past Gu et al. (1996), who studied the two-D problem for ane linear joint in frequency domain. Past comparing, it is found that the results obtained by the ii methods are exactly the same.

Figure four

Comparing of the transmitted waves for an oblique incidence across a unmarried rock joint. The continuous curves are from the proposed recursive method and the scattered points are from the method by Gu et al. (1996).

three.two Normal incidence across a prepare of parallel joints

For incident P or S waves normally impinges a fix of parallel joints, the transmitted waves can exist calculated from eqs (19) to (21) when α→0 and β→0. Presume there are four joints with joint spacing , where is the wavelength and equal to . The incident P moving ridge is a half-cycle sinusoidal wave and the incident S wave is a one-bike sinusoidal wave, that is,

where and = 100 Hz.

The calculation results are compared with those obtained by the methods of Zhao et al. (2006a,b) and Li et al. (2010, 2011). On the basis of the deportation discontinuity model and the feature line theory, Zhao et al. (2006a,b) derived the wave propagation equation for an incident P or Due south wave across a set of parallel joints with linearly elastic or Coulomb-slip behaviour. The wave propagation equations obtained by Zhao et al. (2006a,b) were in fourth dimension domain. Li et al. (2010, 2011) proposed an equivalent viscoelastic medium model for a rock mass with a set of parallel joints, which are linearly elastic. This equivalent model includes a viscoelastic model and the concept of the virtual wave source, and can direct be adopted to analyse P- or Due south-moving ridge propagating unremarkably across parallel joints in frequency domain. Figs five(a) and (b) evidence the ciphering results by the present method and the comparisons for incident P and Due south waves. It can be seen from Fig. five that the present results are consistent with those by the methods of Zhao et al. (2006a,b) and Li et al. (2010, 2011). Therefore, the wave propagation equations derived in the present report are proven to be constructive to study plane P- or S-wave propagation across a stone mass with a set of parallel joints.

Figure v

Comparison of the transmitted waves for a normal incidence across rock joints.

4 Applicability and advantage

The time-domain recursive method has the advantage over the other existing analytical methods, reflected primarily by its wider applicability on parametric studies and computational efficiency. The one-time is of item importance equally in many cases, such every bit the furnishings of various wave or loading parameters on wave propagation in a jointed medium.

The examples given in this department is to illustrate, through the analyses on incident waveform, impinging bending and loading elapsing, the applications of this recursive methods in studying wave propagation with more complex configurations. In the examples, the normal and tangential stiffness are however assumed to exist identical and 3.five GPa g^-1, rock density is 2650 kg thousand^-iii, P-wave velocity is 5746.two m s^-1 and shear moving ridge velocity is 3071.5 1000 s^-1, the joint spacing S is λ₀/10. The applicability of the recursive method presented in Section two.3 will also be studied for soft and stiff joints.

4.one Consequence of incident waveform

For an incidence with an arbitrary waveform, the Fourier and the inverse Fourier transforms were unremarkably needed to calculate the transmitted waves (Achenbach 1973), while the recursive method can straight give the transmitted wave for an incidence with an arbitrary waveform without boosted mathematical methods. The incidences shown in Fig. ane are assumed to exist half-wheel sinusoidal, triangular and rectangular pulses with the meridian value in one unit and the same loading duration. The impinging angles are and for the incident P and S waves, respectively. The incidence is the right-running P or S wave in eqs (19) and (20). When there are 4 or 20 stone joints, that is, N= four or 20, the recursive method presented in Section 2.3 can direct be used to solve the transmitted P and S waves, as shown in Figs 6-8 for the incident P or S waves with three waveforms, respectively. Difference is observed among the transmitted P or S waves for the three incident P or S waves.

Figure vi

Transmitted waves for a one-half-cycle sinusoidal incident P or S wave.

Figure 7

Transmitted waves for a triangular incident P or S wave.

Effigy eight

Transmitted waves for a rectangular incident P or S wave.

4.2 Effect of impinging angle and loading duration

In the following exercise, the incident P and Due south waves with functions defined past eqs (31) and (32) will be adopted, and the joint number N is causeless to be four. The loading duration of the incident Southward moving ridge is , which is twice that of the incident P wave.

When and on the first joint are respectively evaluated every bit the incident P and S waves, the transmitted and reflected waves tin can be calculated from the recursive equations (19) to (21), and hence the corresponding transmission and reflection coefficients for a given impinging bending are obtained from eq. (28). Fig. 9 shows the variation of transmission and reflection coefficients with the impinging angle. Fig. ix shows that the manual and reflection coefficients change with varying incident angle until the incident angles are close to the critical angles, that is, and .

Figure 9

Consequence of the incident angle on transmission and reflection coefficients.

When the impinging angles are and , the recursive equations (19) to (21) and eq. (28) can too be applied to calculate the manual and reflection coefficients for a loading duration . The variations of the transmission and reflection coefficients with are shown in Figs 10(a) and (b) for the incident P and South waves, respectively.

Effigy 10

Upshot of the loading duration on manual and reflection coefficients.

iv.iii Limit cases of joint stiffness

The recursive method can be used to analyse the limit joint stiffness. For joints with depression stiffness, that is, and , from eq. (20), the left- and right-running P and S waves on the right-mitt side of the joints must be zero to go on the equality of equation, which means that there are no P and S waves emitted from the correct-hand side of the joints. Co-ordinate to the time-shifting part eq. (21), there are also no P and S waves given rise from the left-hand side of the joints except for the offset joint. If the incident waves in Fig. 1 are known, the left-running P and S waves on the left-mitt side of the get-go joint can be calculated from eq. (19). In another give-and-take, the incidence in Fig. 1 is completely reflected to be the left-running P and S waves when the joint stiffness is very small. For this instance, the joints deed as a free surface, from which just reflected P and Southward waves are emitted from the first joint interface.

For very stiff joints, that is, when and , from eq. (xx), there are , , and , which means all the incident waves propagate across the stone joints without whatsoever changes. For this example, the interface of a joint is considered to be welded and the rock mass is continuous.

4.4 Other applications and limitations

The proposed recursive method is used in the above given examples to analyse transient waves propagating across linearly deformable rock joints. Moreover, the method tin can be applied to other waves (e.g. earthquake waves) and to more complex joints (east.g. nonlinearly deformable and viscoelastic behaviour).

Information technology should be noted that the proposed recursive method can only be applicative to the incidence with impinging angles less than the critical angles, that is, and . When or , the interface waves will exist generated and propagate on the interface of the joints. The recursive method does not include interface wave analysis.

5 Conclusions

To efficiently investigate the effect of multiple joints on transient P- and S-wave propagation beyond a stone mass, a time-domain recursive method is adopted to express the moving ridge propagation equation when the impinging angles are less than the disquisitional impinging angles. In the present report, the interaction between the P and Southward waves with a articulation is offset analysed co-ordinate to the deportation field for plane wave propagation across a set of parallel stone joints. By comparison with some special cases, such as oblique incidence across a single joint and normal incidence beyond multiple joints, the proposed method and the derived moving ridge propagation equation are proved to be effective to report wave propagation in jointed rock masses. The proposed method can be direct applied for any incident wave with different waveform without losing precision. Meanwhile, the method can be used to calculate the transmission and reflection coefficients for incidences with different impinging angle and loading duration. By discussions on the effects of the soft and rigid joints on wave propagation, we also detect that the analytical method proposed in the newspaper is reasonable and effective. Since the other mathematical methods, such equally the Fourier and the inverse Fourier transforms, are not involved, the calculating efficiency prominently increases.

Acknowledgments

The study is supported by Chinese National Science Inquiry Fund (11072257, 51025935) and the Major State Basic Research Project of Mainland china (2010CB732001). The authors also thank Dr Jianbo Zhu of EPFL Switzerland for the useful discussions.

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Source: https://academic.oup.com/gji/article/188/2/631/586158

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How To Find Domain_3 Of A Wave

Summary

1 Introduction

two Time-domain recursive method

2.1 Trouble clarification

two.2 Interaction betwixt stress waves and stone articulation

2.3 Wave propagation equation

ii.four Calculating steps

three Special cases and comparisons

3.ane Oblique incidence across a single joint

three.two Normal incidence across a prepare of parallel joints

4 Applicability and advantage

4.one Consequence of incident waveform

4.2 Effect of impinging angle and loading duration

iv.iii Limit cases of joint stiffness

4.4 Other applications and limitations

5 Conclusions

Acknowledgments

References

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