PROPAGATION OF NON-STATIONARY ELASTIC WAVES IN A SPHERICAL BODY

АННОТАЦИЯ

Предложена методика проведения аналитического исследования собственных функций в рамках рассматриваемой физико-математической задачи. Определено, что данные функции не могут быть интерпретированы как векторы классического гильбертова пространства ввиду отсутствия их нормируемости. Этот эффект вызван выявленным экспоненциальным ростом амплитуды решений при неограниченном увеличении пространственной координаты. В связи с обнаруженной расходимостью обоснована необходимость выхода за рамки стандартного -подхода. В работе аргументировано использование аппарата обобщенных функций и специализированных методов теории рассеяния, что позволяет корректно описывать состояния системы, не принадлежащие дискретному спектру.

ABSTRACT

This paper proposes a methodology for the analytical investigation of eigenfunctions within the considered physical and mathematical problem. It is shown that these functions cannot be interpreted as vectors in a standard Hilbert space due to their non-normalizability. This phenomenon stems from the exponential growth of the solution amplitude at large spatial distances. Given this divergence, we justify the necessity of extending the analysis beyond the conventional $L^2$ framework. The study demonstrates that employing generalized functions (distributions) alongside specialized scattering theory methods enables a rigorous description of system states that lie outside the discrete spectrum.

Ключевые слова: сферическая оболочка, наполнитель, колебания, частотное уравнение, коэффициент демпфирования.

Keywords: spherical shell, filler, vibrations, frequency equation, damping coefficient.

Introduction. The scattering of a plane wave by a single spherical body is a frequent occurrence in many practical problems of geophysics and seismology [1, 2]. In exploration geophysics, spherical objects provide an acceptable approximation for real-world entities [3, 4]. Analytical models developed for a single sphere can serve as a foundation for constructing more complex solutions. In the petroleum industry, it is reasonable to suggest that if oil is trapped in cavities, seismic energy may be captured due to fluid resonance. However, observing such resonances is challenging because of the impedance contrast between the rock and the fluid [5, 6].

Exact solutions to wave propagation problems are highly relevant. Although analytical solutions exist only for certain types of obstacles (such as spheres, cylinders, or ellipsoids), the insights gained from them are of great significance [7, 8]. The diffraction of elastic waves in inhomogeneous media is inextricably linked to scattered waves; these represent classical fundamental problems in the dynamics of deformable solids, and their solution requires a sophisticated mathematical apparatus [9–11]. Furthermore, the method of adding fillers to materials is widely used to influence their mechanical and thermal characteristics [12, 13].

The problem of wave propagation in a spherical inhomogeneous medium was studied long ago in the process of solving numerous scientific and technical problems, particularly those related to the diffraction of electromagnetic [14], acoustic [15], and elastic [16] waves. This problem is typically considered in a stationary state, where the incident wave is assumed to be infinite in both space and time.

 and is treated as a harmonic wave of the form.

In this case, a number of difficulties arise related to the impossibility of treating the eigenfunctions of the problem under investigation as vectors in Hilbert space: they cannot be normalized due to their exponential growth as distance increases. This fact, well-known from scattering theory [17], arises from the following situation.

A wave propagating to infinity takes the form  where  are the complex eigenfrequencies. [18].

Thus,  and the amplitude of the scattered wave at a fixed point decreases over time due to radial losses. Conversely, the spatial distribution of amplitudes at any given moment grows exponentially with distance r, because the parts of the wave that have moved infinitely far away originated at much earlier times, when the vibration amplitude of the inhomogeneous region was infinitely large.

The infinite growth of eigenfunctions lacks a definitive physical meaning; according to established principles, no effect can occur at point r when . We replace this specific situation with another case related to a stationary state in infinite space, in which we encounter an "exponential catastrophe."

To overcome this, we must account for the fact that vibrations cannot exist for an infinitely long time. It is necessary to formulate and solve the problem of pulse diffraction in one clearly defined form or another.

Methodology

Problem Statement and Solution Method. Let the center of wave propagation be located at the origin of a spherical coordinate system , which coincides with the Cartesian coordinate system (x,y,z). Consider an incident plane wave. The equations of motion for the spherical body (k=2)  and the surrounding medium (k=1) take the following form:

                  (1)

Where- for the surrounding medium (k=1), For the spherical body (k=2) the Lamé parameters, - density, - the displacement vector.

At r=R the continuity of displacement and stress between the two bodies is satisfied:

. (2)

By applying the condition of wave attenuation at infinity, we have: 

da .

The initial conditions are given as follows:

                                  (3)

Let the longitudinal and transverse wave velocities be denoted as c_pk and c_sk (k=1,2) respectively. Assume that the wave is propagating in the positive direction along the - axis.

 the displacement vector

,               (4)

Where - the unit vector,  instantaneous time, H(x)- Heaviside function.

To solve the given problem, we employ the Fourier integral transform as follows:

 (5)

Where- the transformation paramete:

.

the propagation of longitudinal waves in the medium.

After applying the transform (5), the Lamé equation (1) takes the following form:

(6)

Applying the Fourier transform to the incident pulse, we obtain the following

.                                        (7)

Where. (8)

(4) The solution of the equation is presented in [7] and is given by

,                   (9)

 The scalar Helmholtz equation is satisfied, and its solution is expressed as:

 (10)

Where - Spherical Bessel function. For the exterior problem, we replace  with the spherical Hankel function of the second kind, which characterizes outgoing waves at infinity ,

satisfies the condition of boundedness at the origin. The incident plane wave can be expanded in terms of the regular vector eigenfunctions  of the vector Helmholtz equation: [19]:

                                (11)

Where  s and c The symbols denote the functions and  within the expressions of the eigenvectors. The vectors themselves are determined from the continuity conditions of the displacement vector .

,

The traction vectors, in turn, are determined by

at the interface between the inhomogeneous region and the surrounding medium. At r=R, the following relations hold:

                                           (12)

 By calculating the displacements and stresses from the potentials, substituting the resulting expressions into the boundary conditions (12), and utilizing the orthogonality of the spherical wave functions on the sphere's surface, we obtain a system of algebraic equations to determine the unknown coefficients. In the case of longitudinal wave pulse propagation, it follows from (11) that degeneracy occurs with respect to the parameter m (i.e., when m=0), and the general solution becomes independent of the azimuthal coordinate.

In the case of transverse wave propagation, we assume that the wave is polarized along the  direction. The displacements in the surrounding medium are expressed by the following formulas (since we are only concerned with the external medium, the index 2 is omitted):

 (13)

incidence of a longitudinal wave :

,

transverse wave incidence

.

The following shorthand notations are introduced in equation (13):

The formulas for the scattering coefficients characterizing the exterior diffracted field are expressed as follows:

                    (14)

where:  is the scattering coefficient of the transverse wave;

- is the longitudinal wave scattering coefficient, - Determinants,  Bessel functions,  Hankel functions. the following system of equations

                                    (15)

These equations determine the complex eigenfrequencies of spherical inhomogeneous spheroidal and torsional oscillations in an infinite elastic medium [20]. The time-domain solution is obtained by means of the inverse Fourier transform:

                         (16)

Where u(x) We determine them via Eq. (13).

We rewrite expression (16) in the wave interaction zone as follows. By neglecting terms of order and utilizing the asymptotic expansion of the Hankel function, we obtain the following

,

It follows that

 (17)

Where The functions take the following form

 (18)

longitudinal pulse propagation

transverse pulse propagation

To evaluate integral (17), we represent the incident pulse in the form of (4). As a probing signal, we employ the Berlage pulse, which provides a sufficiently accurate approximation of seismic variations:

Whereare the parameters defining the pulse. By utilizing (8), we derive the following:

.

Integrals (16) and (17) can be approximately calculated using a computer through direct numerical integration. To obtain the expressions for the original displacements and stresses, we utilize the residue theory. This method provides a physically accurate description of the process under consideration.  By replacing the integration along the real axis with integration along a closed contour consisting of the real axis and a semicircular arc in the complex half-plane, the integrals are reduced to the sum of residues at the poles of the integrand. Some of these poles are the roots of equations (15). This implies that the expansion is carried out in terms of functions whose arguments include complex natural frequencies—that is, the eigenfunctions of the spherical elastic inhomogeneous medium.

Expression (17) can be written in the following form:

. (19)

We choose the incident pulse in the form of its spectrum as   in  Consequently, the expression within the parentheses tends uniformly to zero.

Under the condition that , Jordan's lemma [21] holds, and the integration along the infinite semicircle can be neglected. The latter inequality reflects the principle of causality: the signal cannot reach the point r+R before the time moment . Thus, using the residue theory to calculate integral (19), we obtain the following

, (20)

Where-- are the poles of the functions. If the spherical inhomogeneity (medium) differs slightly from the surrounding medium, then the imaginary part of will be small, and the poles will be located close to the real axis.

In geophysical applications, cases frequently occur where the inclusion under consideration is sufficiently large in size  and does not differ significantly from the surrounding medium in its elastic-density properties (up to 20%–30% in terms of velocities and up to 3%–5% in terms of density).

In this case, the amplitude functions can be obtained using simple approximate formulas. As shown in [21], a weakly contrasting inhomogeneity is characterized by the following equalities:

,

We represent the resulting expression  as a sum of two terms:

.

The first term yields the well-known form of Fraunhofer diffraction, while the second is associated with rays transmitted through the inclusion.
Physically, this is fully justified because, due to the weak contrast of the inhomogeneity, rays undergoing reflections inside the sphere can be neglected. If the diffraction term

                                (21)

fully meets the requirements of geophysical accuracy [22], then the second term

, (22)

Where which is valid only in the small-angle approximation and has an approximate character. Using the weak-contrast approximation, we express the scattering coefficients as follows:

,                        (23)

Where

Substituting (23) into (22) and replacing the Legendre functions with their asymptotic expressions, we obtain the following [23]:

 (24)

The first term, denoted by unity in the parentheses, yields the form of Fraunhofer diffraction. To estimate the second term, we replace the sum with an integral expressed in the following form:

 (25)

where

We asymptotically calculate the integrals using the stationary phase method formulas [24]:

. (26)

First, by finding the point  near the stationary phase point, we obtain the condition which is  To obtain the scattered wave value, we utilize the method developed by V. A. Dubrovskiy and V. S. Marochnik [25]:

 (27)

Where

. (28)

Since the conditions of Jordan's lemma are satisfied for   it can be calculated using the residue theory. As a result, we obtain the following:

If   then we utilize the following integral form:[24, 25]:

Substituting the last formula into (26) and changing the order of integration, we calculate the inner integral using the residue theory:

Where. In the propagation of a longitudinal pulse

 (29)

These integrals are numerically calculated using the Romberg method.

The numerical results were obtained based on the MATLAB system. The roots of the transcendental equation are found using Muller's method.

Results and Discussion

The following parameter values were adopted for the calculations:

Figure 1 shows the results of the scattered radial displacement component  . calculated according to the developed methodologies for longitudinal wave pulse propagation. The comparison with the exact data from  for  shows a match with a difference of up to 9%.

Figure 1. Time dependence of the radial displacement component propagation under an incident longitudinal pulse

Based on the given parameter  it was determined that the approximate formulas can be utilized within the range of scattering angles .

Furthermore, it should be noted that for  the arrival time of the scattered wave is slightly "smeared." This occurs as a result of the Gibbs effect, and to eliminate it, in general, it is necessary to smooth the leading edge of the incident pulse.
The conducted numerical calculations showed that relatively simple approximate formulas can be fully utilized to solve the problem of non-stationary wave propagation in a weakly contrasting inhomogeneous medium.

Conclusion

A methodology has been developed for calculating wave propagation in a spherical body under incident longitudinal or transverse waves. Additionally, a methodology and an algorithm have been created for computing special Bessel and Hankel functions with complex arguments. The results obtained using the developed methodology were compared with well-known methods, showing good agreement.

It was determined that the proposed approximate formulas (by V.A. Dubrovskiy and V.S. Marochnik) can be utilized within the range of scattering angles.

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