Transient high-frequency spherical wave propagation in porous medium using fractional calculus technique

Abstract

Transient high-frequency spherical wave propagation in the porous medium is studied using the Biot-JKD theory. The porous media is considered to be a composed of deformable solid skeleton and viscous incompressible fluid inside the pores. In order to treat the fractional proportionality of the dynamic tortuosity to the frequency (or equivalently, to time) due to the viscous interaction between solid and fluid phases, the fractional calculus theory along with the Laplace and Fourier transforms are used to solve the coupled governing partial differential equations of the scaler and vector potential functions obtained from the Helmholtz’s decomposition in the Laplace domain. Both the longitudinal and transverse waves, additionally the reflection and transmission kernels are determined in detail at the Laplace domain. For the Laplace-to-time inversion transform, Durbin’s numerical formula is used and the independence of the results from the involved tuning and accuracy parameters is checked. The effects of the porosity, dynamic tortuosity, characteristics length, etc. on the reflected pressure and stress are investigated. The general pattern of the results is similar to our previous knowledge of wave propagation. Further works and experiments may be conducted in future works for various applications.

1 Introduction

Porous materials are commonly encountered in numerous engineering disciplines, such as geophysical [1], environmental engineering [2,3,4,5] earthquake [6], and oceanographic engineering [7], where soils, rocks, and gassy sediments are prevalent. In addition, porous metals have been successfully employed in the fabrication of acoustic liners for aero-engines [8]. The characterization of wave propagation in porous materials and the wave interaction with the boundary of such material enjoys great importance in various engineering problems such as architectural acoustics and machinery noise control [9], and medical diagnostics [10]. It is also very useful in interpreting the data of the nondestructive testing of porous materials, e.g., by applying ultrasound methods [11]. Due to the unusual nature of the medium in terms of the large contact area between the two phases, which is the primary characteristic of porous media, the diffusion, and transport of momentum during wave propagation in a porous medium are not the same as the classical single-phase medium and, as we will see, it has its unique characteristics [12,13,14,15,16].

In general, in all applications, an inverse and direct scattering strategy for analysis of the traveling pulses is conducted, and the characterization of the various property of the porous medium is specified [12, 17]. Due to the relative fluid–Structure Movement (FSM), the porous medium affects the transmitted signal and causes it to lose energy Several physical concepts and parameters are involved in wave propagation in the porous material problem, such as attenuation, porosity, characteristics length of pore fluid, dynamic tortuosity and permeability, etc., which must be considered and investigated. There are so many applications that engineers and researchers are encountered with a two-phase porous medium [12, 18]. The medical engineering branch is concerned with the characterization of this medium for the diagnosis of related diseases such as osteoporosis. One of the main and simplest ways of diagnosis is using acoustical wave propagation and analyzing the reflected and transmitted waves due to mathematical models. In the field of oil extraction and identification of environments with oil resources, engineers are dealt with porous environments containing two or more phases (Soil + Oil, Soil + Oil + Gas, etc.) [19, 20]. Due to this, oil companies are greatly interested in acoustic wave propagation and its properties for the characterization of natural porous media.

In addition, acoustic wave propagation through porous rocks in geophysics is very useful in determining the contained fluids and soil composition.[15, 16, 19]. Gas-Saturated porous materials such as fibrous materials or plastic foams are very attractive and widely used in noise suppression, passive control, coaxial cables, automotive and aeronautical structures, etc. [13, 21]. Biot’s theory of the porous media derives the equations of motions of each involved phase by modeling the inertial, potential, and phase coupling effects [14, 22, 23]. This theory predicts two compressional or longitudinal and one shear wave for an isotropic porous medium [1, 12]. This branch is not new, and many studies have been accomplished by scientists and researchers [24]. Various mathematical developments of different cases, including the coordinates, mechanical properties, different effects, etc., have been conducted in the literature [8]. The aim of the present paper is to provide deeper mathematical modeling to study the transient spherical high-frequency wave propagation in the porous medium based on the Biot-JKD theory by applying the fractional calculus and employing the Laplace and Fourier transforms [9, 14, 21]. The main difference of the present work relative to other similar ones is that here the displacement-based formulation of the Biot-JKD theory is directly used to obtain the dynamical response of high frequency wave propagation in the time domain (not the reduced or simplified equations). As the nearest example, Fellah et. at. [23] studied the transient spherical wave propagation in porous media using the fractional calculus and Green’s function to obtain the wave propagations characteristics in the fluid-saturated equivalent model, which assumes that the structure is motionless. Although their analytical results may be very useful, their assumption of the fixed structure may not be true in all cases, especially in cases where the scale of the domain is small.

This paper is organized as follows. In Sects. 2.1 and 2.2 physical modeling of the problem along with the used notations and main assumptions are presented. In Sect. 2.3 mathematical modeling and governing equations of the problem are brought and then using the Helmholtz decomposition and the properties of the Fourier transform they are reduced to the longitudinal and transversal components. In Sects. 2.4 and 2.5 based on the fractional calculus theory and also Laplace transform, the governing equations are solved for both longitudinal and transverse waves. Afterward, in subSect. 2.6 the acoustical boundary conditions and the kernel operators are applied to obtain the physical constants of the solution. In Sects. 2.7 and 2.8 Durbin’s discretized form for the numerical approximation of the Laplace inversion and also the incident signal used for numerical simulation are presented. Section 3 introduces the specifications of the system used for executing the codes. In Sect. 4, the numerical results are provided, and different effects are investigated.

2 Physical and mathematical modelling

In the present section, the configuration of the problem of the high-frequency spherical wave propagation in the porous medium is defined. Then the main conventions and assumptions are reviewed. Afterward, the partial differential governing equations of the high-frequency wave propagation through the spherical space for the porous medium is described. Next, using fractional calculus, Fourier, and Laplace transforms, the equations are solved for longitudinal and shear waves in the Laplace domain.

2.1 Problem definition

The schematic view of the hallow spherical space is illustrated in Fig. 1. The computational domain is consisting a boundary in $r={r}_{\mathrm{in}}$ where it has been assumed that it is the inner boundary of computational region (porous media) that separates the porous medium and the outer side of the computational domain, also, a boundary at $r={r}_{\mathrm{out}}$. The porous medium space is excited by a transient incident signal at the inner radius. The material model of this space is poroelastic media with viscous fluid as the second phase in the pore spaces of the solid skeleton. Because of the nature of spherical waves, here the spherical coordinate $\left(r,\vartheta ,\varphi \right)$ is used in which $r$, $\vartheta ,$ and $\phi$ are the radial, circumferential, and azimuthal directions, respectively. The $\left(x,y,z\right)$ is the components of the corresponding Cartesian coordinate.

Fig. 1

Schematic of the spherical geometry of the problem

2.2 Assumptions and conventions

In the proceeding paper, the state variables are the macroscopic displacements vectors of the solid skeleton, $\overleftarrow{\mathbf{u}}$, and pore fluid $\overleftarrow{\mathbf{U}}$. The medium is assumed as a continuum media composed of the deformable solid skeleton, $s$, in which the pores have been filled with the viscous fluid, $f$. The porous medium is considered to be homogeneous, linear, and isotropic. The deformation gradient, $\mathbf{F}$, is considered as $\mathbf{F}=\mathbf{I}+\nabla \mathbf{u}$, in which $\mathbf{I}$ is the second-order isotropic tensor with component ${\delta }_{ij}$ [25, 26]. The ${\delta }_{ij}$ is the Kronecker delta. The $\nabla$ is the gradient operator given by the following relationship in the spherical coordinate:

$$\nabla =\frac{\partial }{\partial r}{\widehat{e}}_{r}+\frac{1}{r}\frac{\partial }{\partial \vartheta }{\widehat{e}}_{\vartheta }+\frac{1}{r\mathrm{sin}\vartheta }\frac{\partial }{\partial \varphi }{\widehat{e}}_{\varphi }$$

(1)

${\varvec{r}}=\left({\varvec{r}},t\right)=\left(r,\vartheta ,\varphi ,t\right)$ represents the Eulerian position vector at time $t$ in the spherical coordinate. Also, $\mathbf{u}=\mathbf{r}-\mathbf{R}$, is the displacement vector in which $\mathbf{r}$ and ${\varvec{R}}$ are the initial and current positions, respectively. The Green–Lagrange strain tensor for infinitesimal deformation is $\varepsilon =\frac{1}{2}\left({\mathbf{u}}^{\mathbf{T}}+\mathbf{u}\right)$. Because of the incompressibility of the solid skeleton the volume dilatation, ${\varepsilon }_{ii}$ is match with the porosity, $\phi$. The pore spaces are assumed to be interconnected [26, 27].

2.3 Governing equations of the porous medium

The governing equations of wave propagation in the porous medium for solid skeleton displacement vector, $\overleftarrow{\mathbf{u}}=\left({u}_{r},{u}_{\vartheta },{u}_{\varphi }\right)$, and fluid displacement vector, $\overleftarrow{\mathbf{U}}=\left({U}_{r},{U}_{\vartheta },{U}_{\varphi }\right)$, based on the modified Biot’s theory (Biot’s-JKD) are given by the following [26, 27]:

$$\frac{{\partial }^{2}}{\partial {t}^{2}}\left[{\rho }_{11}\overleftarrow{\mathbf{u}}+{\rho }_{12}\overleftarrow{\mathbf{U}}\right]=\mathcal{P}\nabla \left(\nabla .\overleftarrow{\mathbf{u}}\right)+\mathcal{Q}\nabla \left(\nabla .\overleftarrow{\mathbf{U}}\right)-\mathcal{N}\nabla \times \left(\nabla \times \overleftarrow{\mathbf{u}}\right)$$

(2)

$$\frac{{\partial }^{2}}{\partial {t}^{2}}\left[{\rho }_{12}\overleftarrow{\mathbf{u}}+{\rho }_{22}\overleftarrow{\mathbf{U}}\right]=\mathcal{Q}\nabla \left(\nabla .\overleftarrow{\mathbf{u}}\right)+\mathcal{R}\nabla \left(\nabla .\overleftarrow{\mathbf{U}}\right)$$

(3)

where $\mathcal{N}$, is the shear modulus of the solid skeleton and the generalized elastic constants $\mathcal{P}$, $\mathcal{Q}$, and $\mathcal{R}$ are defined as follows:

$$ \begin{gathered} {\mathcal{P}} = \frac{{K_{s} \left[ {K_{b} \left[ {\left( {\phi - 1} \right)K_{f} + \phi K_{s} } \right] + \left( {\phi - 1} \right)^{2} K_{f} K_{s} } \right]}}{{K_{s} \left[ {\phi K_{s} - \left( {\phi - 1} \right)K_{f} } \right] - K_{b} K_{f} }} + \frac{{4{\mathcal{N}}}}{3}, \,\,{\mathcal{Q}} = \frac{{\phi K_{s} \left( { - \frac{{K_{b} }}{{K_{s} }} - \phi + 1} \right)}}{{ - \frac{{K_{b} }}{{K_{s} }} - \frac{{\phi K_{s} }}{{K_{f} }} - \phi + 1}} \hfill \\ {\mathcal{R}} = \frac{{\phi^{2} K_{s} }}{{ - \frac{{K_{b} }}{{K_{s} }} - \frac{{\phi K_{s} }}{{K_{f} }} - \phi + 1}} \hfill \\ \end{gathered} $$

(4)

The symbols $\nabla$, $\nabla .$ and $\nabla \times$ in Eqs. (2 and 3) are gradient, divergence, and curl operators, respectively. $\phi$, is the porosity. ${K}_{f}$, ${K}_{s}$ and ${K}_{b}$ are the bulk modulus of the fluid, solid, and porous skeleton, respectively. Also, we have:

$${K}_{\mathrm{s}}=\frac{{E}_{\mathrm{s}}}{3-6{v}_{\mathrm{s}}},\quad {K}_{\mathrm{b}}=\frac{{E}_{\mathrm{b}}}{3-6{v}_{\mathrm{b}}},\quad \mathcal{N}=\frac{{E}_{\mathrm{b}}}{2{v}_{\mathrm{b}}+2}$$

(5)

where ${E}_{\mathrm{s}}$,${v}_{\mathrm{s}}$, ${E}_{\mathrm{b}}$, and ${v}_{\mathrm{b}}$ are Young’s modulus and Poisson’s ratio of the solid and skeleton frame, respectively. The effective densities ${\rho }_{ij}$ ($i, j=1, 2$) are related to the fluid and solid skeleton densities, ${\rho }_{\mathrm{f}}$ and ${\rho }_{\mathrm{s}}$ respectively by the following relationships:

$${\rho }_{11}+{\rho }_{12}=\left(1-\phi \right){\rho }_{\mathrm{s}}, \quad{\rho }_{12}+{\rho }_{22}=\phi {\rho }_{\mathrm{f}}, \quad{\rho }_{12}=-\phi {\rho }_{\mathrm{f}}\left({\widetilde{\alpha }}_{\infty }-1\right)$$

(6)

${\rho }_{12}$ is called the mass coupling term. ${\widetilde{\alpha }}_{\infty }$ is the dynamic tortuosity. The tortuosity can be interpreted as a geometric parameter that describes the appearance of tortuous pores, besides inertial coupling between the fluid and the structure of the material, especially in the high-frequency regime. The lowest value tortuosity is ${\widetilde{\alpha }}_{\infty }=1$ in the case of the porous medium having straight pores, and its large values is ${\widetilde{\alpha }}_{\infty }=1.5, 2$ for the most resistive material. The permeability and the dynamic tortuosity of the porous medium can significantly affect the dynamical response of the wave propagation in the porous media materials. The formula developed for dynamic tortuosity in the high-frequency ranges is given by [26, 29]:

$${\widetilde{\alpha }}_{\infty }\left(\omega \right)={\overline{\alpha }}_{\infty }\left[1+\frac{2}{{\Lambda }_{f}}{\left(\frac{{\mu }_{\mathrm{f}}}{\mathcal{j}\omega {\rho }_{\mathrm{f}}}\right)}^{1/2}\right]$$

(7)

in which ${\mathcalligra{j}}^{2}=-1$. ${\mu }_{\mathrm{f}}$ is the fluid viscosity. ${\Lambda }_{\mathrm{f}}$ is viscous characteristics length that determines the viscosity effects in the pore spaces. $\omega$ is the oscillation frequency. As Eq. (7) shows for high-frequency wave motions, the tortuosity is frequency dependent.

Now based on the Helmholtz decomposition theory to derive the longitudinal and transverse part of the wave displacement can be written [30]:

$$\overleftarrow{\mathbf{u}}\left({\varvec{r}},t\right)=\nabla {{\varphi }}_{s}\left({\varvec{r}},t\right)+\nabla \times {{\varvec{\psi}}}_{s}\left({\varvec{r}},t\right),\quad\quad \overleftarrow{\mathbf{U}}\left({\varvec{r}},t\right)=\nabla {{\varphi }}_{f}\left({\varvec{r}},t\right)+\nabla \times {{\varvec{\psi}}}_{f}\left({\varvec{r}},t\right)$$

(8)

where $\varphi$ and ${\varvec{\psi}}\left({\varvec{r}},t\right)=\left({{\varvec{\psi}}}^{r},{{\varvec{\psi}}}^{\vartheta },{{\varvec{\psi}}}^{\varphi }\right)$ denote the longitudinal wave scaler potential and transversal wave vector potential. Due to the definitions, these potential functions satisfy the following conditions [30, 31]:

$$ \begin{gathered} \nabla .{\varvec{\psi}}_{f} = 0,\quad \nabla .{\varvec{\psi}}_{s} = 0 \\ \nabla \times \left( {\nabla \varphi_{s} } \right) = 0,\quad \nabla \times \left( {\nabla \varphi_{f} } \right) = 0 \\ \nabla .\left( {\nabla \times {\varvec{\psi}}_{s} } \right) = 0,\quad \nabla .\left( {\nabla \times {\varvec{\psi}}_{f} } \right) = 0 \\ \end{gathered} $$

(9)

Substituting Eqs. (8) in Eqs. (2 and 3) then employing Eq. (9) and some identities in vectorial mathematics the following equations can be obtained for the longitudinal and transversal components:

$$\frac{{\partial }^{2}}{\partial {t}^{2}}\left[{\rho }_{11}\left[{{\varphi }}_{s}\right]+{\rho }_{12}\left[{{\varphi }}_{f}\right]\right]={\nabla }^{2}\left[\mathcal{P}{{\varphi }}_{s}+\mathcal{Q}{{\varphi }}_{f}\right]$$

(10a)

$$\frac{{\partial }^{2}}{\partial {t}^{2}}\left[{\rho }_{12}\left[{{\varphi }}_{s}\right]+{\rho }_{22}\left[{{\varphi }}_{f}\right]\right]={\nabla }^{2}\left[\mathcal{Q}{{\varphi }}_{s}+\mathcal{R}{{\varphi }}_{f}\right]$$

(10b)

$$\frac{{\partial }^{2}}{\partial {t}^{2}}\left[\left[{\rho }_{11}+\frac{{\rho }_{12}}{{\rho }_{22}}\right]{{\varvec{\psi}}}_{s}\right]=-\mathcal{N}{\nabla }^{2}{{\varvec{\psi}}}_{s},\quad {{\varvec{\psi}}}_{f}=\frac{{\rho }_{12}}{{\rho }_{22}}{{\varvec{\psi}}}_{s}$$

(10c)

where ${\nabla }^{2}$, is Laplacian operator given by the following relationship [31]:

$${\nabla }^{2} =\frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}\frac{\partial }{\partial r}\right)+\frac{1}{{r}^{2}\mathrm{sin}\varphi }\frac{\partial }{\partial \varphi }\left(\mathrm{sin}\varphi \frac{\partial }{\partial \varphi }\right)+\frac{1}{{r}^{2}{\mathrm{sin}}^{2}\varphi }\frac{{\partial }^{2}}{\partial {\vartheta }^{2}}$$

(11)

Since we considered a linear homogeneous and isotropic medium the derivatives $\frac{\partial }{\partial \varphi }$ and $\frac{\partial }{\partial \vartheta }$ are zero. Indeed, the wave only propagates in the radial direction. So:

$${\nabla }^{2} =\frac{1}{{r}^{2}}\frac{\partial }{\partial r}\left({r}^{2}\frac{\partial }{\partial r}\right)=\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}$$

(12)

Taking the inverse Fourier transform from both sides of the Eqs. (10a)–(10c) and using the convolution integral definition, the following time-domain equations will be obtained [26]:

$${\pi }_{11}(t)*\left[\frac{{\partial }^{2}{\varphi }_{s}\left({\varvec{r}},t\right)}{\partial {t}^{2}}\right]+{\pi }_{12}(t)*\left[\frac{{\partial }^{2}{\varphi }_{f}\left({\varvec{r}},t\right)}{\partial {t}^{2}}\right]=\left[\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right]\left[P{\varphi }_{s}\left({\varvec{r}},t\right)+Q{\varphi }_{f}\left({\varvec{r}},t\right)\right]$$

(13a)

$${\pi }_{12}(t)*\left[\frac{{\partial }^{2}{\varphi }_{s}\left({\varvec{r}},t\right)}{\partial {t}^{2}}\right]+{\pi }_{22}(t)*\left[\frac{{\partial }^{2}{\varphi }_{f}\left({\varvec{r}},t\right)}{\partial {t}^{2}}\right]=\left[\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right]\left[Q{\varphi }_{s}\left({\varvec{r}},t\right)+R{\varphi }_{f}\left({\varvec{r}},t\right)\right]$$

(13b)

$${\pi }_{31}(t)*\left[\frac{{\partial }^{2}{{\varvec{\psi}}}_{s}\left({\varvec{r}},{\varvec{t}}\right)}{\partial {t}^{2}}\right]={\pi }_{12}(t)*\left[\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right]\left[{{\varvec{\psi}}}_{s}\left({\varvec{r}},{\varvec{t}}\right)\right]$$

(13c)

where $*$ stands for convolution integral by the following definition [31]:

$${\mathcal{F}}^{-1}\left\{F\left(\omega \right)G\left(\omega \right)\right\}=f\left(t\right)*g\left(t\right)={\int }_{-\infty }^{{\infty }}f\left(\tau \right)g\left(t-\tau \right)d\tau $$

(14)

also, ${\pi }_{ij}(t)$ are obtained as:

$$ \begin{aligned} \overline{\pi }_{11} \left( t \right) =\, & \left[ {\left( {1 - \phi } \right)\rho_{{\text{s}}} + \phi \rho_{{\text{f}}} \left( {at^{{\frac{ - 1 }{{ 2}}}} - 1} \right)} \right],\quad \overline{\pi }_{12} \left( t \right) = \left[ {\phi \rho_{{\text{f}}} \left( {at^{{\frac{ - 1 }{{ 2}}}} - 1} \right)} \right], \quad\overline{\pi }_{22} \left( t \right) = \left[ {\phi \rho_{{\text{f}}} at^{{\frac{ - 1 }{{ 2}}}} } \right] \\ \overline{\pi }_{31} \left( t \right) =\, & \left[ \overline{\alpha }_{\infty } \left[ {\phi \left( {1 - \phi } \right)\rho_{{\text{s}}} \rho_{{\text{f}}} - \phi^{2} \rho_{{\text{f}}}^{2} - \phi \rho_{{\text{f}}} } \right]\left[ {\delta \left( t \right) + \frac{{2 t^{{\frac{ - 1 }{{ 2}}}} }}{{\Lambda_{{\text{f}}} }}\left( {\frac{{\mu_{{\text{f}}} }}{{\pi \rho_{{\text{f}}} }}} \right)^{1/2} } \right]\right. \\& \left.+ \overline{\alpha }_{\infty }^{2} \phi^{2} \rho_{{\text{f}}}^{2} \left[ {\delta \left( t \right) + \frac{{4 t^{{\frac{ - 1 }{{ 2}}}} }}{{\Lambda_{{\text{f}}} }}\left( {\frac{{\mu_{{\text{f}}} }}{{\pi \rho_{{\text{f}}} }}} \right)^{1/2} + \frac{2}{{\Lambda_{{\text{f}}}^{2} }}\left( {\frac{{\mu_{{\text{f}}} }}{{\rho_{{\text{f}}} }}} \right){\text{sign}}\left( t \right) } \right] +\, \phi \rho_{f} \delta \left( t \right) \right] \\ \overline{\pi }_{32} \left( t \right) =\, & - \overline{\alpha }_{\infty } \phi \rho_{{\text{f}}} {\mathcal{N}}\left[ {\delta \left( t \right) + \frac{{2 t^{{\frac{ - 1 }{{ 2}}}} }}{{\Lambda_{{\text{f}}} }}\left( {\frac{{\mu_{{\text{f}}} }}{{\pi \rho_{{\text{f}}} }}} \right)^{1/2} } \right] \\ \end{aligned} $$

(15)

in which $\delta \left(t\right)$ and $\mathrm{sign}(t)$ Dirac-Delta and sign functions respectively. Also, $a$ is:

$$a(t)={\overline{\alpha }}_{\infty }\left[\delta \left(t\right)+\frac{2 }{{\Lambda }_{f}}{\left(\frac{{\mu }_{f}}{\pi {\rho }_{f}}\right)}^{1/2}\right]$$

(16)

2.4 Fractional derivative and fractional form of the equations

From the coefficients (15) and Eqs. (13) can be found that the convolution of the ${t}^{\frac{-1 }{ 2}}$ and a temporal function exists in the equations. Therefore, fractional calculus is the most suitable case to reform the equations in the fractional derivatives of order $\upsilon$. There is various definition for fractional derivative in the mathematics literature. Here, we use the Gorenflo and Mainardi definition given by [25, 32]:

$${D}^{\upsilon }\left\{f\left(t\right)\right\}=\frac{{d}^{\upsilon }f}{d{t}^{\upsilon }}=\frac{1}{\Gamma \left(m-\upsilon \right)}{\int }_{0}^{{t}}\frac{{f}^{\left(m\right)}\left(\tau \right)}{{\left(t-\tau \right)}^{\upsilon +1-m}}d\tau ;\quad \Gamma \left(\chi \right)={\int }_{0}^{{+\infty }}{\tau }^{\chi -1}{e}^{-\tau }d\tau $$

(17)

where ${f}^{\left(m\right)}\left(\tau \right)$ represents the ordinary derivative of the positive integer number $m$. Also, $\Gamma \left(\chi \right)$ is the Gamma function. In this paper, we use the case $m=0$ of the above definition which will be the following form [26, 32]:

$${D}^{\upsilon }\left\{f(t)\right\}=\frac{{d}^{\upsilon }f}{d{{\rm t}}^{\upsilon }}=\frac{1}{\Gamma \left(-\upsilon \right)}{\int }_{0}^{{t}}{\left(t-\tau \right)}^{-\upsilon -1}f\left(\tau \right)d\tau , \quad 0\le \upsilon <1 $$

(18)

Equation (18) shows that here the fractional derivative operates like the convolution integral. This definition enables us to use the Laplace transform of the fractional derivatives to treat the problem in the Laplace domain more simpler. It is worth mentioning that the Helmholtz decomposition states that any vector field, given its sufficient smoothness and rapid decay, can be expressed as the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. It's important to note that this decomposition is not universally unique and may not be applicable to all vector fields. The conditions of being "sufficiently smooth," "rapidly decaying," and subject to "suitable boundary conditions" are crucial for ensuring a unique decomposition. In the present study, the porous media under consideration is isotropic, and all the specified conditions are satisfied, including the fulfillment of well-defined physics, where solid and fluid displacements are smooth and rapidly decaying vector fields. Consequently, we can confidently employ the Helmholtz decomposition in the current work.

2.5 Solution: shear and longitudinal waves

2.5.1 Longitudinal waves

In the case of the longitudinal wave (due to Eqs. (13a) and (13b)), the contribution of the scalar displacement potentials can be obtained from the following system of fractional equations:

$$\left[\begin{array}{cc}{\overline{\pi }}_{11}& {\overline{\pi }}_{12}\\ & \\ {\overline{\pi }}_{12}& {\overline{\pi }}_{22}\end{array}\right]\frac{{\partial }^{2}}{\partial {t}^{2}}\left(\begin{array}{c}{\varphi }_{s}\left({\varvec{r}},t\right)\\ \\ {\varphi }_{f}\left({\varvec{r}},t\right)\end{array}\right)+\eta \left[\begin{array}{cc} 1& -1\\ & \\ -1& 1\end{array}\right]\frac{{\partial }^{\frac{3}{2}}}{\partial {t}^{\frac{3}{2}}}\left(\begin{array}{c}{\varphi }_{s}\left({\varvec{r}},t\right)\\ \\ {\varphi }_{f}\left({\varvec{r}},t\right)\end{array}\right)=\left[\begin{array}{cc}\mathcal{P}& \mathcal{Q}\\ & \\ \mathcal{Q}& \mathcal{R}\end{array}\right]{\nabla }^{2} \left(\begin{array}{c}{\varphi }_{s}\left({\varvec{r}},t\right)\\ \\ {\varphi }_{f}\left({\varvec{r}},t\right)\end{array}\right)$$

(19)

in which $\eta =\frac{2\phi {\rho }_{f}{\overline{\alpha }}_{\infty }}{{\Lambda }_{f}}\sqrt{\frac{{\mu }_{f}}{{\rho }_{f}}}$. To obtain the fast and slow longitudinal waves (${\varphi }_{1}\left({\varvec{r}},t\right)$, ${\varphi }_{2}\left({\varvec{r}},t\right)$) the eigenvalue analysis of the following system can be conducted:

$$\left[\begin{array}{cc}{\overline{\pi }}_{11}+\eta \frac{{\partial }^{\frac{3}{2}}}{\partial {t}^{\frac{3}{2}}}&\quad {\overline{\pi }}_{12}-\eta \frac{{\partial }^{\frac{3}{2}}}{\partial {t}^{\frac{3}{2}}}\\ & \\ {\overline{\pi }}_{12}-\eta \frac{{\partial }^{\frac{3}{2}}}{\partial {t}^{\frac{3}{2}}}&\quad {\overline{\pi }}_{22}+\eta \frac{{\partial }^{\frac{3}{2}}}{\partial {t}^{\frac{3}{2}}}\end{array}\right]\frac{{\partial }^{2}}{\partial {t}^{2}}\left(\begin{array}{c}{\varphi }_{s}\left({\varvec{r}},t\right)\\ \\ {\varphi }_{f}\left({\varvec{r}},t\right)\end{array}\right)-\left[\begin{array}{cc}\mathcal{P}& \mathcal{Q}\\ & \\ \mathcal{Q}& \mathcal{R}\end{array}\right]{\nabla }^{2} \left(\begin{array}{c}{\varphi }_{s}\left({\varvec{r}},t\right)\\ \\ {\varphi }_{f}\left({\varvec{r}},t\right)\end{array}\right)=0$$

(20)

By taking the Laplace transform from the above system, the fractional derivative term can be treated in the Laplace domain. We will have [14]:

$$\left[ \begin{array}{cc} \left({\pi }_{11}\overline{\mathcal{R} }-{\pi }_{12}\overline{\mathcal{Q} }\right){\mathcalligra{s}}^{2}+\eta {\mathcalligra{s}}^{\frac{3}{2}} \quad \left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)& \left({\pi }_{12} \quad \overline{\mathcal{R} }-{\pi }_{22}\overline{\mathcal{Q} }\right){\mathcalligra{s}}^{2}-\eta {\mathcalligra{s}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)\\ & \\ \left({\pi }_{12}\overline{\mathcal{P} }-{\pi }_{11}\overline{\mathcal{Q} }\right){\mathcalligra{s}}^{2}-\eta {\mathcalligra{s}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)&\quad \left({\pi }_{22}\overline{\mathcal{P} }-{\pi }_{12}\overline{\mathcal{Q} }\right){\mathcalligra{s}}^{2}+\eta {\mathcalligra{s}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)\end{array}\right]\left(\begin{array}{c}{\Phi }_{s}\left({\varvec{r}},\mathcalligra{s}\right)\\ \\ {\Phi }_{f}\left({\varvec{r}},\mathcalligra{s}\right)\end{array}\right)={\nabla }^{2} \left(\begin{array}{c}{\Phi }_{s}\left({\varvec{r}},\mathcalligra{s}\right)\\ \\ {\Phi }_{f}\left({\varvec{r}},\mathcalligra{s}\right)\end{array}\right)$$

(21)

in which, $\Phi \left({\varvec{r}},{\mathcalligra{s}}\right)$ is Laplace transform of $\varphi \left({\varvec{r}},t\right)$ as follows [31]:

$$ \Phi \left( {{\varvec{r}},{\mathcalligra{s}}} \right) = {\mathcal{L}}\left\{ {\varphi \left( {{\varvec{r}},t} \right)} \right\} = \mathop \int \nolimits_{0}^{ + \infty } e^{ - {\mathcalligra{s}}t} \varphi \left( {{\varvec{r}},t} \right)dt,{\text{ and }} \pi_{ij} = {\mathcal{L}}\left\{ {\overline{\pi }_{ij} } \right\} $$

(22)

also, the coefficients $\overline{\mathcal{P} }$, $\overline{\mathcal{Q} }$ and $\overline{\mathcal{R} }$ are:

$$ \left[ {\begin{array}{*{20}l} P \hfill &\quad { Q} \hfill \\ {Q } \hfill &\quad { R} \hfill \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}l} {\overline{R}} \hfill & { - \overline{Q}} \hfill \\ { - \overline{Q} } \hfill & { \overline{P}} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\frac{R}{{{\mathcal{P}\mathcal{R}} - {\mathcal{Q}}^{2} }}} \hfill & { \frac{{ - {\mathcal{Q}}}}{{{\mathcal{P}\mathcal{R}} - {\mathcal{Q}}^{2} }}} \hfill \\ {\frac{{ - {\mathcal{Q}}}}{{{\mathcal{P}\mathcal{R}} - {\mathcal{Q}}^{2} }} } \hfill & { \frac{{\mathcal{P}}}{{{\mathcal{P}\mathcal{R}} - {\mathcal{Q}}^{2} }}} \hfill \\ \end{array} } \right] $$

(23)

The system (21) can be solved by finding the corresponding eigenvalues of the following equivalent system:

$$\left[\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right] \left(\begin{array}{c}{\Phi }_{1}\left({\varvec{r}},{\mathcalligra{s}}\right)\\ \\ {\Phi }_{2}\left({\varvec{r}},{\mathcalligra{s}}\right)\end{array}\right)=\left[\begin{array}{cc}{\lambda }_{1}\left({\mathcalligra{s}}\right)& 0\\ & \\ 0& {\lambda }_{2}\left({\mathcalligra{s}}\right)\end{array}\right]\left(\begin{array}{c}{\Phi }_{1}\left({\varvec{r}},{\mathcalligra{s}}\right)\\ \\ {\Phi }_{1}\left({\varvec{r}},{\mathcalligra{s}}\right)\end{array}\right)$$

(24)

where ${\lambda }_{1}\left({\mathcalligra{s}}\right)$ and ${\lambda }_{2}\left({\mathcalligra{s}}\right)$ are the eigenvalues of the matrix $\mathcal{M}$ given below (Appendix A):

$$\mathcal{M}=\left[\begin{array}{cc}\left({\pi }_{11}\overline{\mathcal{R} }-{\pi }_{12}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}+\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)&\quad \left({\pi }_{12}\overline{\mathcal{R} }-{\pi }_{22}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}-\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)\\ & \\ \left({\pi }_{12}\overline{\mathcal{P} }-{\pi }_{11}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}-\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)&\quad \left({\pi }_{22}\overline{\mathcal{P} }-{\pi }_{12}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}+\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)\end{array}\right]$$

(25)

Equation (24) can be written as the following two spherical Bessel’s equations of order $0$ [31]:

$$\frac{{\partial }^{2}{\Phi }_{j}\left({\varvec{r}},{\mathcalligra{s}}\right)}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial {\Phi }_{j}\left({\varvec{r}},{\mathcalligra{s}}\right)}{\partial r}-{{\mathcalligra{k}}}_{j}^{2}\left({\mathcalligra{s}}\right){\Phi }_{j}\left({\varvec{r}},{\mathcalligra{s}}\right)=0;\quad {{\mathcalligra{k}}}_{j}^{2}\left({\mathcalligra{s}}\right)={\lambda }_{j}\left({\mathcalligra{s}}\right), j=\mathrm{1,2}$$

(26)

the general solution of Eq. (26) is [27]:

$${\Phi }_{j}\left({\varvec{r}},{\mathcalligra{s}}\right)={c}_{1j}\sqrt{\frac{\pi }{2{\mathcal{k}}_{j}r}}{\mathcal{I}}_{\frac{1}{2}}\left({{\mathcalligra{k}}}_{j}r\right)+{c}_{2j}\sqrt{\frac{2}{\pi {\mathcal{k}}_{j}r}}{\mathcal{K}}_{\frac{1}{2}}\left({{\mathcalligra{k}}}_{j}r\right),\quad j=\mathrm{1,2}$$

(27)

in which, ${\mathcal{I}}_{1/2} \left( {{\mathcalligra{k}}_{j} r} \right)$ and ${\mathcal{K}}_{1/2} \left( {{\mathcalligra{k}}_{j} r} \right)$ are the first and second kinds of the modified Bessel’s functions of order 1/2. Finally, the solution for longitudinal potential functions of the solid skeleton and fluid is obtained from the following relationships:

$$\left(\begin{array}{c}{\Phi }_{s}\left({\varvec{r}},{\mathcalligra{s}}\right)\\ \\ {\Phi }_{f}\left({\varvec{r}},{\mathcalligra{s}}\right)\end{array}\right)=\left[\begin{array}{cc}1& 1\\ & \\ {V}_{1}\left({\mathcalligra{s}}\right)& {V}_{2}\left({\mathcalligra{s}}\right)\end{array}\right] \left(\begin{array}{c}{\Phi }_{1}\left({\varvec{r}},{\mathcalligra{s}}\right)\\ \\ {\Phi }_{2}\left({\varvec{r}},{\mathcalligra{s}}\right)\end{array}\right)$$

(28)

where ${V}_{1}\left({\mathcalligra{s}}\right)$ and ${V}_{2}\left({\mathcalligra{s}}\right)$ are the corresponding eigenvectors of the matrix $\mathcal{M}$ (Appendix A).

2.5.2 Transverse (Shear) Waves

For the transverse wave propagation analysis, we take the Laplace transform of Eq. (13c), so, we will obtain:

$${\pi }_{31}\left({\mathcalligra{s}}\right){{\mathcalligra{s}}}^{2}{{\varvec{\Psi}}}_{s}\left({\varvec{r}},{\mathcalligra{s}}\right)={\pi }_{32}\left({\mathcalligra{s}}\right)\left[\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial }{\partial r}\right]\left[{{\varvec{\Psi}}}_{s}\left({\varvec{r}},{\mathcalligra{s}}\right)\right],\quad {{\varvec{\Psi}}}_{s}\left({\varvec{r}},{\mathcalligra{s}}\right)=\left({{\varvec{\Psi}}}_{s}^{r},{{\varvec{\Psi}}}_{s}^{\vartheta },{{\varvec{\Psi}}}_{s}^{\varphi }\right)$$

(29)

this equation is the spherical Bessel’s equation in three directions $r$, $\vartheta ,$ and $\varphi$. The solution of this equation for each direction is [31]:

$$ \begin{gathered} {{\varvec{\Psi}}}_{s}^{r} \left( {{\varvec{r}},{\mathcalligra{s}}} \right) = c_{31}^{r} \sqrt {\frac{\pi }{2{\mathcalligra{f}}r}} {\mathcal{I}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) + c_{41}^{r} \sqrt {\frac{2}{\pi {\mathcalligra{f}}r}} {\mathcal{K}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) \hfill \\ {{\varvec{\Psi}}}_{s}^{\vartheta } \left( {{\varvec{r}},{\mathcalligra{s}}} \right) = c_{31}^{\vartheta } \sqrt {\frac{\pi }{2{\mathcalligra{f}}r}} {\mathcal{I}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) + c_{41}^{\vartheta } \sqrt {\frac{2}{\pi {\mathcalligra{f}}r}} {\mathcal{K}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) \hfill \\ {{\varvec{\Psi}}}_{s}^{\phi } \left( {{\varvec{r}},{\mathcalligra{s}}} \right) = c_{31}^{\phi } \sqrt {\frac{\pi }{2{\mathcalligra{f}}r}} {\mathcal{I}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) + c_{41}^{\phi } \sqrt {\frac{2}{\pi {\mathcalligra{f}}r}} {\mathcal{K}}_{\frac{1}{2}} \left( {{\mathcalligra{f}}r} \right) \hfill \\ \end{gathered} $$

(30)

in which, ${\mathcalligra{f}}\left({\mathcalligra{s}}\right)=\frac{{\uppi }_{31}\left({\mathcalligra{s}}\right){{\mathcalligra{s}}}^{2}}{{\uppi }_{32}\left({\mathcalligra{s}}\right)}$. For the transverse potential function of the fluid, ${{\varvec{\Psi}}}_{f}\left({\varvec{r}},{\mathcalligra{s}}\right)$, can be written:

$${{\varvec{\Psi}}}_{f}\left({\varvec{r}},\mathcal{s}\right)={V}_{3}\left({\mathcalligra{s}}\right){{\varvec{\Psi}}}_{s}\left({\varvec{r}},{\mathcalligra{s}}\right)$$

(31)

${V}_{3}\left({\mathcalligra{s}}\right)$ is given in the Appendix A.

2.6 Acoustical boundary conditions–reflection and transmission coefficients

To determine the solution for wave displacements for solid skeleton and fluid the constants in Eqs. (27 and 30) must be obtained. In some previous studies, the following boundary conditions for fluid and solid skeleton displacements at $r={r}_{\mathrm{in}}$ and $r\to \infty$ (half-medium) has been used [24]:

$$\mathrm{BCs}: \left\{\begin{array}{l}r={r}_{\mathrm{in}}: \left\{\begin{array}{l}{\mathbf{u}}_{{\varvec{r}}}=\left(1-\phi \right){\overline{\mathbf{u}} }_{{\varvec{r}}}{\mathbf{e}}^{-j{{\varvec{l}}}_{{\varvec{p}}}{{\varvec{r}}}_{\mathbf{i}\mathbf{n}}}{\mathbf{e}}^{-j{{\varvec{\upomega}}}_{{\varvec{p}}}\mathbf{t}}\\ {\mathbf{u}}_{\boldsymbol{\vartheta }}=0 \\ {\mathbf{u}}_{{\varvec{\phi}}}=0 \end{array}\right.\\ \\ r\to \infty : \left\{\begin{array}{l}{\mathbf{u}}_{{\varvec{r}}}=0\\ {\mathbf{u}}_{\boldsymbol{\vartheta }}=0\\ {\mathbf{u}}_{{\varvec{\phi}}}=0\end{array}\right. \end{array}, \left\{\begin{array}{l}r={r}_{\mathrm{in}}: \left\{\begin{array}{l}{\mathbf{U}}_{{\varvec{r}}}=\phi {\overline{\mathbf{u}} }_{{\varvec{r}}}{\mathbf{e}}^{-j{{\varvec{l}}}_{{\varvec{p}}}{{\varvec{r}}}_{\mathbf{i}\mathbf{n}}}{\mathbf{e}}^{-j{{\varvec{\upomega}}}_{{\varvec{p}}}\mathbf{t}}\\ {\mathbf{U}}_{\boldsymbol{\vartheta }}=0 \\ {\mathbf{U}}_{{\varvec{\phi}}}=0 \end{array}\right.\\ \\ r\to \infty : \left\{\begin{array}{l}{\mathbf{U}}_{{\varvec{r}}}=0\\ {\mathbf{U}}_{\boldsymbol{\vartheta }}=0\\ {\mathbf{U}}_{{\varvec{\phi}}}=0\end{array}\right.\end{array}\right.\right.$$

(32)

in boundary conditions (32) it has been assumed that the incident wave causes a known displacement that distributes in the solid and fluid phases due to the porosity. Also, because of the nature of Bessel’s functions, the boundary conditions at $r\to \infty$ are satisfied automatically. Therefore, we still need two more conditions at $r={r}_{\mathrm{in}}$. Here, at first, we develop the solution for a finite spherical domain in which ${r}_{\mathrm{in}}\le r\le {r}_{\mathrm{out}}$ and then we use the acoustical boundary conditions. However, here we develop a more general process to obtain the coefficients. The present paper, the half-medium case is used for validation only and the results are presented for a finite domain. Note that ${r}_{in}$ and ${r}_{out}$ are the inner and outter diameter of the computational spherical domain as shown in Fig. 1.

When a normal sound wave impacts to the porous medium boundary, part of the wave is reflected back to the fluid and the other is transmitted through the porous media. Since only the normal incident wave impinges on the medium the shear is not involved. Accordingly, the pressure field for $r\le {r}_{\mathrm{in}}$ and $r\ge {r}_{\mathrm{out}}$ can be written as [14, 23]:

$${P}^{\mathrm{inr}}\left({\varvec{r}},t\right)={P}^{\mathrm{inc}}\left(t-\frac{r-{r}_{\mathrm{in}}}{{c}_{0}}\right)+{P}^{\mathrm{ref}}\left(t+\frac{r-{r}_{\mathrm{in}}}{{c}_{0}}\right);\quad r\le {r}_{\mathrm{in}}$$

(33a)

$${P}^{\mathrm{otr}}\left({\varvec{r}},t\right)={P}^{\mathrm{tr}}\left(t-\frac{r-{r}_{\mathrm{out}}}{{c}_{0}}\right); \quad r\ge {r}_{\mathrm{out}}$$

(33b)

where ${P}^{\mathrm{inc}}$, ${P}^{\mathrm{inr}}$, ${P}^{\mathrm{otr}}$, ${P}^{\mathrm{ref}}$ and ${P}^{\mathrm{tr}}$ are the incident, total inner, total outer, reflected, and transmitted wave pressures in the out of the porous domain, respectively [25]. In the present study because of the semi-infinity medium the transmitted wave passes through the media and finally, is damped. In order to relate the incident and scattered waves, reflection, $\mathfrak{R}\left(\tau \right)$, and transmission, $\mathcal{T}\left(\tau \right)$, ernel operators. The reflection and transmission pressures can be obtained by integrating the product of the incident wave and the kernel operators:

$$ P^{{{\text{ref}}}} \left( {r,t} \right) = \mathop \int \limits_{0}^{t} \Re \left( \tau \right)P^{{{\text{inc}}}} \left( {t - \tau + \frac{{r - r_{{{\text{in}}}} }}{{c_{0} }}} \right)d\tau = \Re \left( t \right)*P^{{{\text{inc}}}} \left( t \right)*\delta \left( {t + \frac{{r - r_{{{\text{in}}}} }}{{c_{0} }}} \right); \quad r \le r_{{{\text{in}}}} $$

(34a)

$$ \begin{aligned} P^{{{\text{ref}}}} \left( {r,t} \right) =\, & \mathop \int \limits_{0}^{t} {\mathcal{T}}\left( \tau \right)P^{{{\text{inc}}}} \left( {t - \tau - \frac{{r_{{{\text{out}}}} - r_{{{\text{in}}}} }}{c} - \frac{{r - \left( {r_{{{\text{out}}}} - r_{{{\text{in}}}} } \right)}}{{c_{0} }}} \right)d\tau \\ = \,& {\mathcal{T}}\left( t \right)*P^{{{\text{inc}}}} \left( t \right)*\delta \left( {t - \frac{{r - \left( {r_{{{\text{out}}}} - r_{in} } \right)}}{{c_{0} }}} \right); \quad r \ge r_{{{\text{out}}}} \\ \end{aligned} $$

(34b)

about the kernel operator, note that it only depends on the material properties of the porous medium. Besides, the lower limit of the integral Eq. (34a), $t=0,$, denotes that the incident wave hits the porous medium at zero time.

To obtain the kernels, $\mathfrak{R}\left(\tau \right)$ and $\mathcal{T}\left(\tau \right)$, first, we transform the domain into the Laplace domain then we apply the boundary conditions. Rewriting Eqs. (33a, 33b) due to the Eqs. (34a, 34b) and the convolution integral definition, the pressure fields are obtained as follows:

$${P}^{\mathrm{inr}}\left({\varvec{r}},t\right)=\left(\delta \left(t-\frac{r-{r}_{in}}{{c}_{0}}\right)+\mathfrak{R}\left(t\right)*\delta \left(t+\frac{r-{r}_{in}}{{c}_{0}}\right)\right)*{P}^{\mathrm{inc}}\left(t\right); \quad r\le {r}_{\mathrm{in}}$$

(35a)

$${P}^{\mathrm{otr}}\left({\varvec{r}},t\right)=\mathcal{T}\left(t\right)*\delta \left(t-\frac{r-\left({r}_{\mathrm{out}}-{r}_{\mathrm{in}}\right)}{{c}_{0}}\right)*{P}^{\mathrm{inc}}\left(t\right); \quad r\ge {r}_{\mathrm{out}}$$

(35b)

so, in the Laplace domain we can write:

$${\widetilde{P}}^{\mathrm{inr}}\left({\varvec{r}},{\mathcalligra{s}}\right)=\mathcal{L}\left\{{P}^{\mathrm{inr}}\left({\varvec{r}},t\right)\right\}=\left({e}^{-{\mathcalligra{s}}\frac{r-{r}_{in}}{{c}_{0}}}+\widetilde{\mathfrak{R}}\left({\mathcalligra{s}}\right).{e}^{{\mathcalligra{s}}\frac{r-{r}_{\mathrm{in}}}{{c}_{0}}}\right){\widetilde{P}}^{\mathrm{inc}}\left({\mathcalligra{s}}\right)$$

(36a)

$${\widetilde{P}}^{\mathrm{otr}}\left({\varvec{r}},{\mathcalligra{s}}\right)=\mathcal{L}\left\{{P}^{\mathrm{otr}}\left({\varvec{r}},t\right)\right\}=\widetilde{\mathcal{T}}\left({\mathcalligra{s}}\right)\left({e}^{-\mathcal{s}\frac{r-\left({r}_{\mathrm{out}}-{r}_{\mathrm{in}}\right)}{{c}_{0}}}\right){\widetilde{P}}^{\mathrm{inc}}\left({\mathcalligra{s}}\right)$$

(36b)

where ${\widetilde{P}}^{\mathrm{inc}}\left({\mathcalligra{s}}\right)$, $\widetilde{\mathfrak{R}}\left({\mathcalligra{s}}\right)$ and $\widetilde{\mathcal{T}}\left({\mathcalligra{s}}\right)$ are the Laplace transform of the ${P}^{\mathrm{inc}}\left(t\right)$, $\mathfrak{R}\left(t\right)$ and $\mathcal{T}\left(t\right)$, respectively. Here, part of the incident wave is reflected and the remaining part passes through the porous meduim and leads to deformation and hence stress in both solid and fluid phases. The stresses created in the medium is due to the incident signal (that acts the role of the external excitation) of the domain. Based on the fact that the normal and shear stresses are continuous at the boundaries the following boundary conditions in the Laplace domain can be written [23, 27]:

$$ \begin{aligned} 1)\, \sigma_{rr}^{s} \left( {r_{{{\text{in}}}}^{ + } ,{\mathcalligra{s}}} \right) = \,& - \left( {1 - \phi } \right)\tilde{P}^{inr} \left( {r_{{{\text{in}}}}^{ - } ,{\mathcalligra{s}}} \right) \\ 2)\, \sigma_{rr}^{f} \left( {r_{{{\text{in}}}}^{ + } ,{\mathcalligra{s}}} \right) =\, & - \phi \tilde{P}^{inr} \left( {r_{{{\text{in}}}}^{ - } ,{\mathcalligra{s}}} \right) \\ 3)\, \sigma_{rr}^{s} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right) =\, & - \left( {1 - \phi } \right)\tilde{P}^{otr} \left( {r_{{{\text{out}}}}^{ + } ,{\mathcalligra{s}}} \right) \\ 4)\, \sigma_{rr}^{f} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right) =\, & - \phi \tilde{P}^{otr} \left( {r_{{{\text{out}}}}^{ + } ,{\mathcalligra{s}}} \right) \\ \end{aligned} $$

(37)

now still two more equations are needed to have a determined system for obtaining all coefficients. On the other hand, the acoustic velocity field has the relationship with the solid and fluid velocities as [7, 25]:

$${\widetilde{V}}_{1}\left({r}_{\mathrm{in}}^{-},{\mathcalligra{s}}\right)=\left(1-\phi \right){\widetilde{V}}_{s}\left({r}_{\mathrm{in}}^{+},{\mathcalligra{s}}\right)+\phi {\widetilde{V}}_{f}\left({r}_{\mathrm{in}}^{+},{\mathcalligra{s}}\right)$$

(38a)

$${\widetilde{V}}_{2}\left({r}_{\mathrm{out}}^{+},{\mathcalligra{s}}\right)=\left(1-\phi \right){\widetilde{V}}_{s}\left({r}_{\mathrm{out}}^{-},{\mathcalligra{s}}\right)+\phi {\widetilde{V}}_{f}\left({r}_{\mathrm{out}}^{-},{\mathcalligra{s}}\right)$$

(38b)

where the ${\widetilde{V}}_{1}$ and ${\widetilde{V}}_{2}$ are obtained from the surrounding pressure field by using the Euler equation as [7, 25]:

$$\frac{\partial {\widetilde{P}}^{\mathrm{inr}}\left(r,{\mathcalligra{s}}\right)}{\partial r}={\widetilde{\rho }}_{f}{\mathcalligra{s}}{\widetilde{V}}_{1}\left(r,{\mathcalligra{s}}\right), \frac{\partial {\widetilde{P}}^{\mathrm{otr}}\left(r,{\mathcalligra{s}}\right)}{\partial r}={\widetilde{\rho }}_{f}{\mathcalligra{s}}{\widetilde{V}}_{2}\left(r,{\mathcalligra{s}}\right)$$

(39)

in which ${\widetilde{\rho }}_{f}$ is the density of the surrounding fluid. Additionally, the acoustic velocities of the solid and fluid in the porous medium can be obtained from the solid and fluid displacements, respectively. Note that here we only consider the longitudinal wave propagation, we will have [7]:

$$ \begin{aligned} {\mathbf{u}}_{{{\mathbf{Cmp}}.}} \left( {{\varvec{r}},t} \right) =\, & \nabla \varphi_{s} \left( {{\varvec{r}},t} \right) \to \tilde{V}_{s} \left( {r,{\mathcalligra{s}}} \right) = {\mathcalligra{s}}\frac{{\partial {\Phi }_{s} \left( {r,{\mathcalligra{s}}} \right)}}{\partial r} \\ {\mathbf{U}}_{{{\mathbf{Cmp}}.}} \left( {{\varvec{r}},t} \right) = \,& \nabla \varphi_{f} \left( {{\varvec{r}},t} \right) \to \tilde{V}_{f} \left( {r,{\mathcalligra{s}}} \right) = {\mathcalligra{s}}\frac{{\partial {\Phi }_{f} \left( {r,{\mathcalligra{s}}} \right)}}{\partial r} \\ \end{aligned} $$

(40)

The stress field equations for normal stresses ${\sigma }_{ij}^{s}$ and ${\sigma }_{ij}^{f}$ are given by [11, 13]:

$$ \begin{aligned} \sigma_{ij}^{s} =\, & \left( {\left( {{\mathcal{P}} - 2{\mathcal{N}}} \right)\nabla .u_{i,j} + {\mathcal{Q}}\nabla .U_{i,j} } \right)\delta_{ij} + {\mathcal{N}}\left( {u_{i,j} + u_{j,i} } \right) \\ \sigma_{ij}^{f} = \,& - \phi p_{f} \delta_{ij} = \left( {{\mathcal{R}}\nabla .U_{i,j} + {\mathcal{Q}}\nabla .u_{i,j} } \right)\delta_{ij} \\ \end{aligned} $$

(41)

where ${p}_{f}$, is the pore fluid pressure. At the end, all equations are summarized as:

$$ \begin{gathered} 1)\; {\mathcal{P}}\frac{{\partial^{2} \Phi_{s} \left( {r_{in}^{ + } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} + {\mathcal{Q}}\frac{{\partial^{2} \Phi_{f} \left( {r_{in}^{ + } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} = - \left( {1 - \phi } \right)\left( {1 + \tilde{\Re }\left( {{\mathcalligra{s}}} \right)} \right)\tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) \hfill \\ 2)\; {\mathcal{R}}\frac{{\partial^{2} \Phi_{f} \left( {r_{in}^{ + } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} + {\mathcal{Q}}\frac{{\partial^{2} \Phi_{s} \left( {r_{in}^{ + } ,s} \right)}}{{\partial r^{2} }} = - \phi \left( {1 + \tilde{\Re }\left( s \right)} \right)\tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) \hfill \\ 3)\; {\mathcal{P}}\frac{{\partial^{2} \Phi_{s} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} + {\mathcal{Q}}\frac{{\partial^{2} \Phi_{f} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} = - \left( {1 - \phi } \right){\tilde{\mathcal{T}}}\left( {{\mathcalligra{s}}} \right)e^{{ - s\frac{{r_{{{\text{in}}}} }}{{c_{0} }}}} \tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) \hfill \\ 4)\; {\mathcal{R}}\frac{{\partial^{2} \Phi_{f} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} + {\mathcal{Q}}\frac{{\partial^{2} \Phi_{s} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{{\partial r^{2} }} = - \phi {\tilde{\mathcal{T}}}\left( {{\mathcalligra{s}}} \right)e^{{ - s\frac{{r_{{{\text{in}}}} }}{{c_{0} }}}} \tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) \hfill \\ 5)\; \left( { - \frac{{\mathcalligra{s}}}{{c_{0} }} + \frac{{\mathcalligra{s}}}{{c_{0} }} \tilde{\Re }\left( {{\mathcalligra{s}}} \right)} \right)\tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) = \left( {1 - \phi } \right)\tilde{\rho }_{f} {\mathcalligra{s}}^{2} \frac{{\partial \Phi_{s} \left( {r_{{{\text{in}}}}^{ + } ,{\mathcalligra{s}}} \right)}}{\partial r} + \phi \tilde{\rho }_{f} {\mathcalligra{s}}^{2} \frac{{\partial \Phi_{f} \left( {r_{{{\text{in}}}}^{ + } ,{\mathcalligra{s}}} \right)}}{\partial r} \hfill \\ 6)\; - \frac{{\mathcalligra{s}}}{{c_{0} }} {\tilde{\mathcal{T}}}\left( {{\mathcalligra{s}}} \right)e^{{ - {\mathcalligra{s}}\frac{{r_{{{\text{in}}}} }}{{c_{0} }}}} \tilde{P}^{inc} \left( {{\mathcalligra{s}}} \right) = \left( {1 - \phi } \right)\tilde{\rho }_{f} {\mathcalligra{s}}^{2} \frac{{\partial \Phi_{s} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{\partial r} + \phi \tilde{\rho }_{f} {\mathcalligra{s}}^{2} \frac{{\partial \Phi_{f} \left( {r_{{{\text{out}}}}^{ - } ,{\mathcalligra{s}}} \right)}}{\partial r} \hfill \\ \end{gathered} $$

(42)

solving this system with respect to $\frac{{{\partial }^{2}\Phi }_{f}\left({r}_{\mathrm{in}, \mathrm{out}}^{-,+},\mathrm{{\mathcalligra{s}}}\right)}{\partial {r}^{2}}$, $\frac{{\partial\Phi }_{s,f}\left({r}_{\mathrm{in}, \mathrm{out}}^{-,+},\mathrm{{\mathcalligra{s}}}\right)}{\partial r}$, $\widetilde{\mathfrak{R}}\left(\mathrm{{\mathcalligra{s}}}\right)$ and $\widetilde{\mathcal{T}}\left(\mathrm{{\mathcalligra{s}}}\right)$ gives:

$$ \begin{aligned} \tilde{\Re }\left( {{\mathcalligra{s}}} \right) = \,& 1 + \gamma A + \theta B \\ {\tilde{\mathcal{T}}}\left( {{\mathcalligra{s}}} \right) = \,& - \gamma \mu C - \theta \mu D \\ \end{aligned} $$

(43)

$$\gamma =\frac{\left(1-\phi \right){c}_{0}{\widetilde{\rho }}_{f}\mathrm{{\mathcalligra{s}}}}{{\widetilde{P}}^{\mathrm{inc}}\left(\mathrm{{\mathcalligra{s}}}\right)}, \theta =\frac{\phi {c}_{0}{\widetilde{\rho }}_{f}\mathrm{{\mathcalligra{s}}}}{{\widetilde{P}}^{\mathrm{inc}}\left(\mathrm{{\mathcalligra{s}}}\right)}, \mu ={e}^{\mathrm{{\mathcalligra{s}}}\frac{{r}_{\mathrm{in}}}{{c}_{0}}}$$

(44)

where the coefficients $A$, $B$, $C,$ and $D$ are:

$$A=\frac{{\partial\Phi }_{s}\left({r}_{\mathrm{in}}^{+},{\mathcalligra{s}}\right)}{\partial r}, B=\frac{{\partial\Phi }_{f}\left({r}_{\mathrm{in}}^{+},{\mathcalligra{s}}\right)}{\partial r}, C=\frac{{\partial\Phi }_{s}\left({r}_{\mathrm{out}}^{-},{\mathcalligra{s}}\right)}{\partial r}, D=\frac{{\partial\Phi }_{f}\left({r}_{\mathrm{out}}^{-},{\mathcalligra{s}}\right)}{\partial r}$$

(45)

finally, the coefficients ${c}_{ij}$ ($i,j=\mathrm{1,2}$) can be obtained from the following algebraic linear system:

$$\left[\begin{array}{ccc}{\overline{\mathcal{C}} }_{11}^{1}& {\overline{\mathcal{C}} }_{21}^{1}& \begin{array}{cc}{\overline{\mathcal{C}} }_{12}^{1}& {\overline{\mathcal{C}} }_{22}^{1}\end{array}\\ {\overline{\mathcal{C}} }_{11}^{2}& {\overline{\mathcal{C}} }_{21}^{2}& \begin{array}{cc}{\overline{\mathcal{C}} }_{12}^{2}& {\overline{\mathcal{C}} }_{22}^{2}\end{array}\\ \begin{array}{c}{\overline{\mathcal{C}} }_{11}^{3}\\ {\overline{\mathcal{C}} }_{11}^{4}\end{array}& \begin{array}{c}{\overline{\mathcal{C}} }_{21}^{3}\\ {\overline{\mathcal{C}} }_{21}^{4}\end{array}& \begin{array}{cc}\begin{array}{c}{\overline{\mathcal{C}} }_{12}^{3}\\ {\overline{\mathcal{C}} }_{12}^{4}\end{array}& \begin{array}{c}{\overline{\mathcal{C}} }_{22}^{3}\\ {\overline{\mathcal{C}} }_{22}^{4}\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}{c}_{11}\left(\mathrm{{\mathcalligra{s}}}\right)\\ {c}_{21}\left(\mathrm{{\mathcalligra{s}}}\right)\\ \begin{array}{c}{c}_{12}\left(\mathrm{{\mathcalligra{s}}}\right)\\ {c}_{22}\left(\mathrm{{\mathcalligra{s}}}\right)\end{array}\end{array}\right]=\left[\begin{array}{c}{e}_{1}\\ {e}_{2}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\right]$$

(46)

in which the parameters ${\overline{\mathcal{C}} }_{ij}^{k}$ ($i,j=\mathrm{1,2}, k=1,..,4$), ${e}_{1}$ and ${e}_{2}$ have been given in the Appendix B. Therefore, replacing these coefficients into Eqs. (45) due the Eq. (28) then replacing the outcome into Eq. (43), $\widetilde{\mathfrak{R}}\left({\mathcalligra{s}}\right)$ and $\widetilde{\mathcal{T}}\left({\mathcalligra{s}}\right)$ are obtained.

2.7 Temporal domain

To obtain the dynamical response of the displacement and stress components, and also the acoustic pressure in the time domain the Laplace inversion must be taken from the above Laplace domain based relationships. The Laplace inversion is defined by the following complex form integral [31, 33]:

$$\mathcal{H}\left(r,\vartheta ,\varphi ,t\right)=\frac{1}{2\pi j}{\int }_{\mathcalligra{o}-j\infty }^{{\mathcalligra{o}+j\infty }}\overline{\mathcal{H} }\left(r,\vartheta ,\varphi ,{\mathcalligra{s}}\right)d{\mathcalligra{s}}$$

(47)

where $\mathcal{o}$ is an arbitrary real number greater than all the real parts of the singularities of the $\overline{\mathcal{H} }\left(r,\vartheta ,\varphi ,{\mathcalligra{s}}\right)$. In the present work, Durbin’s discretized form for the numerical approximation of the Laplace inversion in the interval [0, 2 ${T}_{0}$] is used which its relationship is given below [33]:

$$\mathcal{H}\left(t\right)\cong \frac{2{e}^{\mathcal{o}t}}{{T}_{0}}\left[\frac{1}{2}\mathrm{Re}\left[\overline{\mathcal{H} }\left(\mathcalligra{o}\right)\right]+\sum_{k=1}^{\widehat{N}}\left[\mathrm{Re}\left[\overline{\mathcal{H} }\left(\mathcalligra{o}+jk\frac{2\pi }{{T}_{0}}\right)\right]\mathrm{cos}\left(\frac{2\pi kt}{{T}_{0}}\right)-\mathrm{Im}\left[\overline{\mathcal{H} }\left(\mathcal{o}+jk\frac{2\pi }{{T}_{0}}\right)\right]\mathrm{sin}\left(\frac{2\pi kt}{{T}_{0}}\right)\right]\right]$$

(48)

In which $\widehat{N}$ is the number of samples (or the truncation number). It is suggested that the value for $\mathcal{o}{T}_{0}$ is between 5 and 10 with $\widehat{N}$ in the interval 50 to 5000. The results are valid for ${T}_{0}\ge 2{t}_{\mathrm{max}}$ where ${t}_{\mathrm{max}}$ is the maximum time value that the equations are solved [33].

2.8 The Incident Signal

The incident signal is considered as the following function in the temporal domain:

$${P}^{\mathrm{inc}}\left(t\right)=\left\{\begin{array}{cc}\mathfrak{f}\left(t\right)={\overline{P} }_{\mathrm{m}}^{\mathrm{i}}{e}^{\xi \left[t-{t}_{1}\right]} \mathrm{sin}\left[{\omega }_{s}\left[t-{t}_{1}\right]\right] &\quad 0\le t<{t}_{1}\\ \\ \mathfrak{g}\left(t\right)={\overline{P} }_{\mathrm{m}}^{\mathrm{i}}{e}^{-\xi \left[t-{t}_{1}\right]}\mathrm{sin}\left[{\omega }_{s}\left[t-{t}_{1}\right]\right]&\quad {t}_{1}\le t\end{array}\right.$$

(49)

where the values of the coefficients have been given in Table 1.

Table 1 Coefficients for functions $\mathfrak{f}\left(t\right)$ and $\mathfrak{g}\left(t\right).$

In fact, in the present study the incident signal is a combinitation two part as showd by Eq. (49). Indeed, one is amplitude growth untile time ${t}_{1}$ and another amplitude decay from time ${t}_{1}$. Both parts have a single growth and damped frequency ${\omega }_{s}$, respectively.

Since we solved the governing equations in the Laplace domain, we take a Laplace transform of the incident wave signal given by the following relationship:

$$ \begin{aligned} \tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right) = \,& {\mathcal{L}}\left\{ {P^{{{\text{inc}}}} \left( t \right)} \right\} = \mathop \int \limits_{0}^{{t_{1} }} {\mathfrak{f}}\left( t \right)e^{ - {\mathcalligra{s}}t} dt + \mathop \int \limits_{{t_{1} }}^{\infty } {\mathfrak{g}}\left( t \right)e^{ - {\mathcalligra{s}}t} dt \\ =\, & \overline{P}_{{\text{m}}}^{{\text{i}}} e^{{ - {\mathcalligra{s}}t_{1} }} \left[ {\frac{{\left( {e^{{\left( {{\mathcalligra{s}} - \xi } \right)t_{1} }} \left( {\omega_{s} \cos \omega_{s} t_{1} - \left( {\xi - {\mathcalligra{s}}} \right)\sin \omega_{s} t_{1} } \right) - \omega_{s} } \right)}}{{s^{2} - 2s\xi + \omega_{s}^{2} + \xi^{2} }} + \frac{{\omega_{s} }}{{s^{2} + 2s\xi + \omega_{s}^{2} + \xi^{2} }}} \right] \\ \end{aligned} $$

(50)

Figure 2 illustrates the transient acoustical incident signal used in this study.

Fig. 2

The transient acoustical incident signal used for numerical simulation

3 Computational utilized system

The wave propagation governing equations of the porous medium was solved through coding in the MATLAB environment. The computational process was performed by an Intel(R) Xeon(R) with a Core i5 CPU at 2.53 GHz. The code and numerical procedure were optimized as much as possible to implement more efficiently and faster.

4 Results and discussion

In the present section, the results of solving a spherical wave propagation problem in the porous medium are brought. The detailed effects of the different physical parameters on the dynamical response of the problem are investigated. The mechanical properties of the material of the solid skeleton and pore fluid are summarized in Table 2. Indeed, here, all the properties considered are related to bone as a palpable example of the two-phase porous medium. In the present paper, the purpose is to determine the effect of the different physical parameters and properties on the presented mathematical modeling of spherical wave propagation for porous media applications based on the Biot-JKD equations (mainly ultrasonic waves). All the results have been obtained using the data provided in Table 2 unless mentioned in the manuscript.

Table 2 Mechanical Properties of Porous Medium

4.1 Convergence and validation

Investigation of the independency of physical response from the temporal and the spatial meshgrid is necessary when using numerical methods. To this end, we need to show independence from the time step and also $\widehat{N}$ for a specific $\mathcal{o}{T}_{0}$. Figure 3 shows the maximum stress, $\mathrm{max}\left\{{\sigma }_{rr}^{s}\left(\frac{{r}_{\mathrm{out}}+{r}_{\mathrm{in}}}{2},t\right)\right\}$, in the middle of the domain. As can be seen, mesh independence has been achieved. Besides, the convergence with respect to $\widehat{N}$ as shown in Table 3.

Fig. 3

Convergence with respect to time step

Table 3 Convergence with respect to parameter $\widehat{N}$

In order to verify the current results obtained from the conducted mathematical modeling, an infinite-porous medium (finite inner radius and infinite outer radius) is considered the same as the work done by Ozyazicioglu [28]. All the material and mechanical properties are set similarly to that work. Besides, in our mathematical outcomes to model the infinity-media the outer radius is approached to infinity (${r}_{\mathrm{out}}\to \infty$). The incident signal is step (Heaviside) pulse. Table 4 summarizes the comparison between pore pressure and solid displacement for $r=15\mathrm{ m}$ at time $t=0.00463\mathrm{ s}$ and $t=0.00488\mathrm{ s}$, respectively. As can be seen, there is an acceptable agreement between the results that confirms the present mathematical outcomes.

Table 4 Validation of the present reults

4.2 Porosity, tortuosity and characteristics length effects

The dynamical response of the porous medium to the ultrasonic incident signal for different has been shown in Fig. 4. Decreasing the porosity leads to a decrease in the proportion of the fluid which leads to the medium being more resistive, stiffer, and absorbent. Figure 4a confirms this reality, as can be seen, maximum stress (generally amplitude of the oscillations solid part) increases when the porosity decreases. Also, Fig. 4b shows that the reflected pressure oscillation amplitude varies directly with the porosity. It should be mentioned that the wavefront velocity remains constant and it does not dependent on the porosity. Figure 4 shows that although the changes in the reflected pressure in considerable (about 30% for 50% changes in porosity), the solid stress is more sensitive to porosity relative to the reflected pressure.

Fig. 4

Dynamical Response of the porous medium to incident signal a solid stress at $r={r}_{\mathrm{in}}+0.1h$ and reflected pressure at $r=\frac{{r}_{\mathrm{in}}}{2}$ for different porosities

The tortuosity effect has been plotted in Fig. 5. From Fig. 5a we can see that by increasing the tortuosity, the transmission amplitude of the solid stress increase. However, due to the time lag (here about 1 $\mathrm{ms}$) between the two obtained signals, the transmission wave velocity decreases. The dynamic tortuosity is very important and meaningful in the dynamics of the porous medium, which describes the inertial effects between the solid and fluid parts of porous material since it provide information on the geometry of the pores and therefore affects the relative displacements and hence created stresses, directly. In fact, the geometry of the pores directly affects the fast and slow waves and their velocities. The opposite to the solid stress, the reflected pressure does not changes significantly with the tortuosity (about 0.5% in amplitude and less than 0.1 $\mathrm{\mu s}$ time lag). Of course, these values are sensible, and they can be detected with measurement instruments.

Fig. 5

Dynamical Response of the porous medium to incident signal a solid stress at $r={r}_{\mathrm{in}}+0.9h$ and reflected pressure at $r=\frac{{r}_{\mathrm{in}}}{2}$ for different tortuosity

Figure 6 illustrates the viscous characteristics length effects on the solid stress variations. The viscous characteristic length is a function of the pores' geometrical size. This parameter considers the small pores in which the viscous effects are important. As it is seen, the solid stress amplitude considerably decreases for higher values of the viscous characteristics length. The interpretation of this phenomenon is because of the smaller viscous characteristic lengths show higher viscosity effects, and therefore, the momentum is transported more effectively. From the results shown above we can conclude that the reflection (and transmission pressure) changes conversely relative to the solid stress. Therefore, for a more absorbent medium (lower porosity and higher viscous characteristics length) the transmission pressure decreases with decreasing the viscous characteristics length.

Fig. 6

Variation of the solid stress with time for different values of the viscous characteristics length

Figure 7 shows the sensitivity of the solid stress to the bulk modulus of the pore fluid. As it is expected, higher values of the bulk modulus of the fluid (stiffer fluid and so stiffer medium) lead to the lower amplitude of the transmitted wave in terms of the solid stress.

Fig. 7

Variation of the solid stress with time for different values of the bulk modulus of the pore fluid

It should be mentioned that the investigation of some parameters can be performed together to have a more realistic result. For instance, changes in density and viscosity of the fluid (also bulk modulus) can be studied for different pore fluids. In Fig. 8 the created solid stress for SAE 30 Oil (${K}_{\mathrm{f}}=1.5\,\mathrm{ GPa}$, ${\mu }_{f}=1.53\,\mathrm{ Pa s}$ and ${\rho }_{\mathrm{f}}=886\,\mathrm{ kg }{\mathrm{m}}^{-3}$) and water a pore fluid is showed. As can be seen for a more viscous fluid like SAE 30 Oil and lower bulk modulus relative to water, the solid stress created by an identical incident signal is more. In future studies, with respect the developed mathematical modelling effects of different pore fluids and parameters can be investigated to conclude more practical results.

Fig. 8

Variation of the solid stress with time for water and SAE 30 Oil as the pore fluid

4.3 Outlook and future works

The focus of the present paper was based on modeling and obtaining the dynamical response of the high-frequency spherical wave propagation based on the Biot-JKD formulation. In Future works, we can investigate several effects and situations in various applications. For example, wave propagation through different materials and also study the effect of the transverse wave propagations on the longitudinal component. It is essential to note that when dealing with various fluid types, encompassing both lower and higher viscosity fluids, the outcomes can be significantly affected by fluid viscosity. Consequently, the specific findings of this study may not be universally applicable to all pore fluid types.

5 Conclusion

In the present work, transient high-frequency spherical wave propagation through a saturated porous media was investigated for the case of the deformable solid skeleton and incompressible but viscous fluid inside the pores. Because of the nature of the viscous interaction between the fluid and solid phases, the damping behavior (dynamic tortuosity) has a fractional proportionality to the frequency. The coupled governing partial differential equations based on the Biot-JKD model were used. By employing Helmholtz’s decomposition, the displacement-based formulation was reduced to the scaler and vector potential functions. Afterward, the potential-based equations using the advantages of the fractional calculus along with the Laplace transform for treatment of the fractional order of the time after the frequency-to-time transformation using the Fourier transform were solved for both longitudinal and transversal wave components. Because of the nature of the spherical wave propagation, Bessel’s functions appeared in the solutions. At the end of the solution part, a numerical technique was introduced to inverse-transform of the Laplace domain for the obtained solutions. The solution was validated for discretization independence. Then the effects of the different physical parameters such as porosity, dynamic tortuosity, characteristics length, etc. were investigated on the wave propagation. The trend results were exactly showed that the effects of these parameters are the same as our previous knowledge, as it is expected. The achievements of the present work can be very helpful for the characterization and analysis of acoustical wave propagation in porous mediums in different applications. In the same direction, future works may investigate the coupled effects of longitudinal and transverse wave propagation and also employing the present results for practical examples.

Appendices

Appendix 1

The matrix $\mathcal{M}$ is:

$$\mathcal{M}=\left[\begin{array}{cc}\left({\pi }_{11}\overline{\mathcal{R} }-{\pi }_{12}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}+\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)&\quad \left({\pi }_{12}\overline{\mathcal{R} }-{\pi }_{22}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}-\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)\\ & \\ \left({\pi }_{12}\overline{\mathcal{P} }-{\pi }_{11}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}-\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)&\quad \left({\pi }_{22}\overline{\mathcal{P} }-{\pi }_{12}\overline{\mathcal{Q} }\right){{\mathcalligra{s}}}^{2}+\eta {{\mathcalligra{s}}}^{\frac{3}{2}}\left(\overline{\mathcal{P} }+\overline{\mathcal{Q} }\right)\end{array}\right]$$

(A1)

the eigenvalues of this matrix are simply calculated by the following relationship:

$${\lambda }_{\mathrm{1,2}}\left({\mathcalligra{s}}\right)=\frac{\mathrm{Tr}\left(\mathcal{M}\right)\pm \sqrt{{\left[\mathrm{Tr}\left(\mathcal{M}\right)\right]}^{2}-4\mathrm{Det}\left(\mathcal{M}\right)}}{2}$$

(A2)

in which $\mathrm{Tr}\left(\mathcal{M}\right)$ and $\mathrm{Det}\left(\mathcal{M}\right)$ represent the trace and determinant of matrix $\mathcal{M}$, respectively which they are given by:

$$ \begin{aligned} Tr\left( {\mathcal{M}} \right) =\, & \left( {{\overline{\mathcal{R}}}\pi_{11} - 2{\overline{\mathcal{Q}}}\pi_{12} + {\overline{\mathcal{P}}}\pi_{22} } \right){\mathcalligra{s}}^{2} + \eta {\mathcalligra{s}}^{\frac{3}{2}} \left( {{\overline{\mathcal{R}}} + 2{\overline{\mathcal{Q}}} + {\overline{\mathcal{P}}}} \right) \\ Det\left( {\mathcal{M}} \right) =\, & \left( {{\overline{\mathcal{P}}\overline{\mathcal{R}}} - {\overline{\mathcal{Q}}}^{2} } \right)\left[ {\left( {\pi_{11} \pi_{22} - \pi_{12}^{2} } \right){\mathcalligra{s}}^{4} + \eta {\mathcalligra{s}}^{\frac{7}{2}} \left( {\pi_{11} + \pi_{22} - 2\pi_{12} } \right)} \right] \\ \end{aligned} $$

(A3)

finally, the expression for the eigenvectors are as follows:

$${V}_{\mathrm{1,2}}\left(\mathrm{{\mathcalligra{s}}}\right)=\frac{{\lambda }_{\mathrm{1,2}}\left(\mathrm{{\mathcalligra{s}}}\right)-\left[\left(\overline{\mathcal{R}}{\pi }_{11}-\overline{\mathcal{Q}}{\pi }_{12}\right){\mathrm{{\mathcalligra{s}}}}^{2}+\eta {\mathrm{{\mathcalligra{s}}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)\right]}{\left(\overline{\mathcal{R}}{\pi }_{12}-\overline{\mathcal{Q}}{\pi }_{22}\right){\mathrm{{\mathcalligra{s}}}}^{2}-\eta {\mathrm{{\mathcalligra{s}}}}^{\frac{3}{2}}\left(\overline{\mathcal{R} }+\overline{\mathcal{Q} }\right)}$$

(A4)

Appendix 2

The coefficients ${\overline{\mathcal{C}} }_{ij}^{k}$ ($i,j=\mathrm{1,2}, k=1,..,4$), ${e}_{1}$ and ${e}_{2}$ are given by:

$$ \begin{aligned} {\bar{\mathcal{C}}}_{{11}}^{1} =\, & {\mathcal{C}}_{{11}}^{1} + \left( {1 - \phi } \right)\left[ {\gamma + \theta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{21}}^{1} =\, & {\mathcal{C}}_{{21}}^{1} - \left( {1 - \phi } \right)\left[ {\gamma + \theta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{12}}^{1} =\, & {\mathcal{C}}_{{12}}^{1} + \left( {1 - \phi } \right)\left[ {\gamma + \theta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{22}}^{1} =\, & {\mathcal{C}}_{{22}}^{1} - \left( {1 - \phi } \right)\left[ {\gamma + \theta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} v}}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{11}}^{2} & = {\mathcal{C}}_{{11}}^{2} + \phi \left[ {\gamma + \theta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{21}}^{2} =\, & {\mathcal{C}}_{{21}}^{2} - \phi \left[ {\gamma + \theta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{12}}^{2} =\, & {\mathcal{C}}_{{12}}^{2} + \phi \left[ {\gamma + \theta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{22}}^{2} =\, & {\mathcal{C}}_{{22}}^{2} - \phi \left[ {\gamma + \theta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{in}}}}^{ - } } \right) \\ {\bar{\mathcal{C}}}_{{11}}^{3} =\, & {\mathcal{C}}_{{11}}^{3} + \left( {1 - \phi } \right)\left[ {\alpha + \beta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r_{{{\text{out}}}} }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{21}}^{3} =\, & {\mathcal{C}}_{{21}}^{3} - \left( {1 - \phi } \right)\left[ {\alpha + \beta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r_{{out}} }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{12}}^{3} =\, & {\mathcal{C}}_{{12}}^{3} + \left( {1 - \phi } \right)\left[ {\alpha + \beta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r_{{{\text{out}}}} }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{22}}^{3} =\, & {\mathcal{C}}_{{22}}^{3} - \left( {1 - \phi } \right)\left[ {\alpha + \beta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r_{{{\text{out}}}} }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{11}}^{4} =\, & {\mathcal{C}}_{{11}}^{4} + \phi \left[ {\alpha + \beta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r_{{{\text{out}}}} }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {k_{1} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{21}}^{4} =\, & {\mathcal{C}}_{{21}}^{4} - \phi \left[ {\alpha + \beta V_{1} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r_{{{\text{out}}}} }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {k_{1} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{12}}^{4} =\, & {\mathcal{C}}_{{12}}^{4} + \phi \left[ {\alpha + \beta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r_{{{\text{out}}}} }}} {\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r_{{{\text{out}}}} } \right) \\ {\bar{\mathcal{C}}}_{{22}}^{4} =\, & {\mathcal{C}}_{{22}}^{4} - \phi \left[ {\alpha + \beta V_{2} } \right].\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right)\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r_{{{\text{out}}}} }}} {\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{s}}_{2} r_{{{\text{out}}}} } \right) \\ e_{1} =\, & - 2\left( {1 - \phi } \right)\tilde{P}^{{{\text{inc}}}} \left( {{\mathcalligra{s}}} \right),~~~~e_{2} = - 2\phi \tilde{P}^{{{\text{inc}}}} \left( k \right) \\ ~\alpha =\, & - \gamma \mu ,~~~~~\beta = - \theta \mu \\ \end{aligned} $$

(A5)

where the coefficients ${\mathcal{C}}_{ij}^{k}$ can be calculated by the following relationships:

$$\begin{aligned} {\mathcal{C}}_{{11}}^{1} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{P}} + V_{1} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{21}}^{1} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{P}} + V_{1} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{12}}^{1} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{P}} + V_{2} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{22}}^{1} &= \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{P}} + V_{2} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{11}}^{2} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{Q}} + V_{1} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{21}}^{2} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{Q}} + V_{1} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{12}}^{2} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{Q}} + V_{2} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{22}}^{2} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{Q}} + V_{2} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{in}}}}^{ + } }} \\ {\mathcal{C}}_{{11}}^{3} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{P}} + V_{1} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{21}}^{3} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{P}} + V_{1} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{12}}^{3} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{P}} + V_{2} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{22}}^{3} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{P}} + V_{2} {\mathcal{Q}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{11}}^{4} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{Q}} + V_{1} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{21}}^{4} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{1} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{1} r} \right)} \right]\left( {{\mathcal{Q}} + V_{1} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{12}}^{4} =\, & \left. {\left[ {\sqrt {\frac{\pi }{{2{\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{I}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) + \frac{1}{r}{\mathcal{I}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{Q}} + V_{2} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ {\mathcal{C}}_{{22}}^{4} =\, & \left. {\left[ {\sqrt {\frac{2}{{\pi {\mathcalligra{k}}_{2} r}}} \left[ {{\mathcal{K}}_{{\frac{5}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right) - \frac{1}{r}{\mathcal{K}}_{{\frac{3}{2}}} \left( {{\mathcalligra{k}}_{2} r} \right)} \right]\left( {{\mathcal{Q}} + V_{2} {\mathcal{R}}} \right)} \right]} \right|_{{r = r_{{{\text{out}}}}^{ - } }} \\ \end{aligned}$$

(A6)