## Abstract

The problem on a surface harmonic elastic wave propagating along the free surface of cylindrical cavity in the direction of cavity axis is considered. In the case of isotropic medium, this is the classical Biot’s problem of 1952. First, the Biot pioneer work is revisited: the analytical part of Biot’s findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. Then the new direct generalization of Biot’s problem on the case of transversely isotropic medium within the framework of linear theory of elasticity is proposed. Transition to the transverse isotropy needs some novelty—necessity of using the more complex representations of displacements through two potentials. Finally, a generalization of Biot’s problem on the case of isotropic and transversely isotropic media in the framework of linearized theory of elasticity with allowance for initial stresses is stated. This part repeats briefly the results of A.N. Guz with co-authors of 1974. The main features are using the linearized theory of elasticity and one only potential. All three parts are shown as analytical study up to the level when the numerical methods have to be used.

### Keywords

- surface harmonic cylindrical wave
- classical Biot’s problem
- generalization to the case of transversely isotropic medium

## 1. Introduction

Note first that the seismic waves include mainly the primary and secondary body waves and different kinds of surface waves. This chapter is devoted to one kind of surface waves. The problem is stated as follows: the infinite medium with cylindrical circular cavity having the symmetry axis

## 2. Main stages of solving the classical Biot’s problem on surface wave along a cylindrical cavity

### 2.1 Statement of problem and main equations in potentials

A cylindrical system of coordinates

or

Further the potentials

When Eq. (5) is substituted into Eqs. (3) and (4), then two uncoupled linear wave equations are obtained:

Here the standard notations of Laplace operator

### 2.2 Solving the wave equations in the form of Macdonald functions

The solution of Eqs. (6) and (7) is found in the form of harmonic waves in the direction of vertical coordinate:

A substitution of representations (8) into the wave Eqs. (6) and (7) gives the equations relative to the unknown amplitudes

These equations correspond to the Bessel equation for Macdonald functions

More exactly, Eqs. (9) and (10) have the solutions in the form of Macdonald functions, if the conditions.

are fulfilled. According to conditions (12), the wave number of cylindrical wave must be real, and the wave velocity must be less of the velocities of classical longitudinal and transverse plane waves.

Further the wave Eqs. (9) and (10) are considered separately. The first equation is written in the form

This equation has the solution in the form of Macdonald function:

of zeroth order and unknown argument

The second equation can be written in the form

The corresponding solution under conditions (12) is expressed by the Macdonald function

of the first order and unknown argument

Note that the Macdonald functions have the property of attenuation with increasing arguments which is shown in Figure 1. Therefore, the propagation along the vertical coordinate

This means that amplitudes can decrease essentially with increasing the distance from the surface of cylindrical cavity. In this sense, the waves (15) and (16) are the surface ones. This forms also the sense of conditions (12). The same conditions are used in the analysis of classical Rayleigh surface wave which propagates along the plane surface of isotropic elastic medium [5, 6, 7, 8, 9]. But the Rayleigh wave attenuates as an exponential function when being moved from the free surface, whereas the cylindrical surface Biot’s wave attenuates as the Macdonald functions. At that, the arguments in exponential function and Macdonald functions are identical and depend on the wave velocity.

### 2.3 Boundary conditions: equations for unknown wave number

The boundary conditions correspond to the absence of stresses on surface

The stresses

are written through the potentials

Then the boundary conditions (17) can be written in the form.

In the work [1], Biot has used the expressions.

Then the substitution of solutions (14) and (16) into the boundary conditions (21) gives two homogeneous algebraic equations relative to the unknown constant amplitude coefficients

An analysis of these equations that describe the cylindrical surface wave is very similar to the analysis that has been carried out by Rayleigh for the classical wave propagating along the plane surface. Some novelty in analysis of systems (22) and (23) is consideration of the system relative to quantities

Solving of systems (24) and (25) gives two results. First, the solution is found accurate within one amplitude factor. Second, an equation for determination of phase velocity of cylindrical surface wave can be obtained in an explicit form.

The work of Biot (1952) has demonstrated some art in handling the Macdonald functions and has written Eq. (24) through only functions of the zeroth and first orders. For that, the known formulas

have been used [3]. As a result, the equation for determination of phase velocity of cylindrical wave has the form

Let us write the corresponding equation for the Rayleigh wave [5, 6, 7, 8, 9] as

Thus, a presence of Macdonald functions in Eq. (27) complicates essentially an analysis of this equation because according to relations

If the cavity radius is not small, then the Macdonald functions can be represented by the simple formula

Strictly speaking, the analytical part of analysis is ended by obtaining Eq. (27). Further analysis can be continued with the aim of the numerical methods. Biot in [1] has shown some comments and conclusions based on resources of the 1950s.

A possibility of analytical approach is still saved in the problem on existence of the appropriate wave velocity. First of all, Eq. (27) depends on the elastic constants, and this dependence can be shown in the form of dependence on the ratio of known velocities

It seems appropriate to recall here the most known ways of proving the existence of velocity of the classical Rayleigh wave. An initial equation is always Eq. (28). Two different notations

Finding the real root of Eq. (30) is the key step in the analysis of the Rayleigh wave [5, 6, 7, 8, 9] . For more than 100 years of analysis of this wave, many methods of proving the existence of real velocity of wave were elaborated.

First of all, the sufficiently useful and exact empirical Viktorov’s formula [5].

should be shown.

Let us show further briefly some phenomenological methods. Note that the restriction on the Rayleigh wave velocity is already obtained from a statement of the problem—it is less of the velocity of plane transverse wave. This restriction can be written in the form

*Method 1* (graphical method [10, 11]). Eq. (30) is considered as a sum of two summands _{1} = z^{3} describes a cubic parabola; the lower branch of which lies in the first quadrant of the plane *c*/*cT*) ≤ 0.955.

*Method 2* (method of finding the interval, on ends of which the equation possesses the different by sign values [2, 11]). This method is based on the analysis of Eq. (30). The value of equation that corresponds to the point

*Method 3* (another method of finding the interval, on ends of which the equation possesses the different by sign values [5]). This method is based on the analysis of Eq. (31). The right point is chosen as

*Method 4* (method based on assumption relative to the Poisson ratio [7]). This assumption consists in the choice of value of Poisson ratio that is often used in the analysis of seismic waves in Earth’s crust

The main conclusion from the shown above methods is that they really allow to establish an existence of real root of Rayleigh equation (the real value of velocity of harmonic Rayleigh wave). They give the positive answer on the question whether the Rayleigh wave exists. In the case of other surface waves including the cylindrical wave under consideration, the experience of the classical Rayleigh wave analysis can be quite useful.

## 3. Cylindrical wave propagating along the surface of the cylindrical cavity in the direction of vertical axis: The case of transversal isotropy of medium

Let us return to the initial statement of problem and consider an infinite medium with cylindrical circular cavity that has the symmetry axis

### 3.1 Some information on transversally isotropic medium

Let us consider the case when

Then the constitutive relations

Also, five independent elastic technical constants are often used.

The longitudinal Young modulus that corresponds to tension along the symmetry axis

The transverse Young modulus that corresponds to tension in the isotropy plane

The shear modulus that corresponds to the shear along the isotropy plane

The shear modulus that corresponds to the shear along the symmetry axis

The Poisson ratio that corresponds to the shear along the symmetry axis

Sometimes, the corresponding Lame moduli are used.

The Poisson ratio (40) is determined by the known formula of isotropic theory

The Poisson ratio

The constants

Let us comment briefly some features of transversally isotropic materials. They can be divided on the natural and artificial ones. An example of the classical natural material is the rock. An example of the modern material is a family of fibers “Kevlar®.” Kevlar® KM2 [15] is characterized by elastic constants

An example of composite materials can be four fibrous composites of micro- and nanolevels, which are described in [15]. The corresponding elastic constants for some variants of these materials are as follows [15]:

10% of carbon microfibers

10% of graphite microwhiskers

10% of zig-zag carbon nanotubes

10% of chiral carbon nanotubes

The shown above values are typical for the transversally isotropic materials, and therefore they are briefly commented below.

*Comment* 1. The Young modulus in the direction along the symmetry axis

*Comment* 2. The Lame moduli

*Comment* 3. The Poisson ratio

*Comment* 4. The shear moduli

### 3.2 The basic formulas for elastic transversely isotropic medium with axial symmetry

Let us write the basic formulas for the case of symmetry axis

The substitution of constitutive equations.

into the motion Eqs. (43) gives the motion equations in displacements

Note that Eqs. (45) and (46) include only four constants (the constant

### 3.3 Three classical ways of introducing the potentials in transversely isotropic elasticity

The basic equations of the theory of transversely isotropic elasticity are frequently analyzed by the use of potentials. The potentials are introduced in theory of elasticity mainly for static problems. Transition to the dynamic problems is associated with complications that are sometimes impassable. Because the problem on waves is related to the dynamic ones, let us show further the possible complications with introducing the potentials.

*Way 1* [12]. It is proposed for the axisymmetric problems of equilibrium (not motion) and is based on introducing one only potential

The next step consists in substitution of formulas (47) and (48) into the first equation of equilibrium and the equations that are obtained from the Cauchy relations and formulas for the strain tensor. This permits to determine the unknown parameters through the elastic constants represented in the equilibrium equations. Further, the second equation of equilibrium gives the biharmonic equation for finding the potentials

where

Thus, a transition from the isotropic case to the transversally isotropic one complicates the procedure of solving the static problems. Here a necessity of solving the classical biharmonic equation is changed on necessity of solving some generalization of this equation in the form (49).

*Way 2* [12, 16]. This way is also proposed for the static problems. Here, two potentials are introduced which are linked immediately with displacements

A substitution of representations (51) into equations of equilibrium (45), (46)

allows to determine the unknown constants

This expression gives the quadratic equation for

Note that the simple link

The stresses are expressed through new potentials in such a way

*Way 3* [1, 16]. This way is proposed for equations of motion, but only for the isotropic theory of elasticity. It can be used for the static problems of transversely isotropic theory of elasticity. The initial equations here are the equations of motion (43) without inertial summands

The potentials are introduced like (51), but the representations are complicated by necessity of introducing two new unknown parameters:

A substitution of representations (57) into equations of motion (45) and (46) gives equations relative to the potentials. Eq. (45) gives two equations:

whereas Eq. (46) gives three equations:

The last two equations are identical. Eqs. (58) and (60) and (59) and (62) have to be identical. This means that the coefficients in these equations have to be identical. As a result, two equations can be obtained for the determination of unknown constants

The unknown potentials

Thus, three ways of introduction of potentials in the static problems of transversely isotropic theory of elasticity are shown. The different attempts to transfer these ways into the dynamic problems meet some troubles—the presence of inertial summands generates new additional conditions for the unknown constants in representations of potentials. Introducing the new constants does not help—the number of conditions is still more than the number of all constants.

### 3.4 Solving the problem on the propagation in the direction of vertical axis surface cylindrical wave for the case of transversal isotropy of medium

Consider now equations of motion (45) and (46) and introduce the potentials by the formula (57). A substitution of formula (57) into equations of motion gives five equations relative to the potentials. Eq. (57) gives two equations:

Eq. (46) gives three equations:

Two last equations are identical. Also the equations for potential

Note that characterization of an attenuation of wave depth down functions

As a result, two equations can be obtained that permit to determine the constants

Note that restriction on the kind of solution (it has to be a wave) allows to unite two different conditions into one—conditions for equaling coefficients in summands with the second derivative by time

A success in the determination of transformed potentials is accompanied by a complication of conditions which provide the wave attenuation. They have the form.

Let us recall that the similar conditions for the case of isotropic medium

If the conditions (83) are fulfilled, then the solution of wave equations for potentials can be written in the form.

With allowance for formulas (84), the representations of potentials becomes more definite

The formula (85) completes the first analytical part of solving the problem on cylindrical surface wave.

### 3.5 Boundary conditions: equations for the unknown wave number

This part of analysis can be treated as the second analytical part. The boundary conditions have the form identical for all kinds of symmetry of properties. That is, they have the form (17) or (21). The formulas for stresses depend already on the symmetry of medium. The expressions for stresses through the potential reflect the features of introducing the potentials. In this case, they have the form

Further, the representations (86) and (87) should be substituted into the boundary conditions, and the formulas on differentiation of Macdonald functions [3] should be taken into account:

Then the boundary conditions are transformed into the algebraic equations relative to quantities

When the determinant of linear homogeneous system of Eqs. (88) and (89) is equaled to zero, then the equations for the unknown wave number can be obtained:

Note that the sufficiently complex expression relative to the wave number is hidden coefficients

Note also that the simple and convenient condition from analysis of classical surface Rayleigh wave [6, 7, 8, 9, 10, 17], when the wave number depends only on ratio

## 4. Solving the problem on propagating in the direction of symmetry axis surface wave within the framework of linearized theory of elasticity with allowance for initial stresses

Note that analysis of cylindrical surface wave in isotropic medium was first carried out by Biot [1] in 1952 and the transversally isotropic medium with initial stresses was first carried out by Guz et al. in 1974 [18].

Let us show below an analysis of the problem in hand that is carried out in Subchapter “Longitudinal Waves” of Chapter 4 “Waves in Cylindrical Media” of volume 2 of edition [19]. Here, the cylinder of circular cross-section is considered, and the longitudinal wave is defined as the wave propagating in the direction of cylinder axis

The medium is assumed isotropic or transversally isotropic. The main relations for transversal isotropy are described by independent constant

Note that as shown in (92), eight constants are necessary in the linearized theory, but in the framework of linear theory, they have the form (33), and their number is five.

Further, the general solutions of basic equations in displacements are utilized. These equations have the form (3.174) [19]

where only eight independent constants (92) must be taken into account.

The corresponding equations of linear theory of elasticity for the case of transversally isotropic medium without of initial stresses are written above as Eqs. (45) and (46).

The general solutions for the case of axial symmetry are expressed through one potential in the form (4.13) [19]

Note that in Section 3 of this chapter, two potentials

The longitudinal harmonic wave is described analytically through the potential in the form (101) [19]

where the unknown amplitude

which further is written in the form

The unknown quantities

The solution (95) describes the surface wave, if amplitude

The shown part of analysis from introducing the potential by formula (94) to representation of solution by formula (99) inclusive can be compared with analogous part of analysis from Section 3 of this chapter (from introducing the potentials by formula (57) to the solution in the form of (85)). It is easy to see a difference in representations (99) and (85): formula (99) uses the Bessel functions and in particular the Macdonald function of zero index, whereas formula (85) uses (like the Biot’s solution (14)) the Macdonald functions (16) of the zero and first indexes.

The next part of analysis of cylindrical wave consists in substitution of solution into boundary conditions of the form (99) [19]

The case of oscillatory behavior of wave in the direction of radius is considered with pointing that the case of surface wave is the same type. A substitution of solution (99) into conditions (4.79) [19] gives the dependence of velocity of surface wave or its wave number on frequency—a dispersion equation in the form of determinant of the fourth order in the form (4.26) [19].

This finishes the analytical part of analysis shown in [19]. It corresponds to the part of Section 3.5 of this chapter, where the explicit form of dispersive equations is proposed in the form (90) that includes the Macdonald functions of the zero and first orders which represent some generalization of dispersion Eq. (27) obtained by Biot.

## 5. Conclusions

This chapter proposes three fragments of analytical analysis of the cylindrical surface wave propagating in the vertical direction of circular cylindrical cavity. The first fragment shows the analytical part of pioneer work of Biot. It represents the classicism of mathematical procedures and physical comments of Biot. Properly speaking, the clear and understandable Rayleigh’s scheme is saved, but it is complemented by some findings reflecting the features of cylindrical waves. Two next fragments show the more late development of the Biot’s problem. They are different by influence of the Biot’s procedure. The approach shown in Section 3 is more close to the Biot’s analytical scheme, whereas Section 4 proposes as an independent scheme that is more close to the Rayleigh scheme. Nevertheless, all fragments testify the mathematical complexity in solving the problem on the cylindrical surface waves. Thus, revisiting the old Biot’s problem shows that it still generates new scientific and practical problems.