# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 2 Model and Solution
### Wave Field Analysis and Solutions  
Assuming a plane harmonic P\({}_{1}\) or SV wave with circular frequency \(\omega\) propagates in the positive direction of the x-axis, which may be expressed in terms of the Bessel-Fourier series:  
\[\varphi^{i}=\varphi_{0}\sum\limits_{n=0}^{\infty}\varepsilon_{n}i^{n}J_{n}(k _{0}r)\cos n\theta e^{-i\omega t} \tag{27}\]  
\[\Psi^{i}=\Psi_{0}\sum\limits_{n=0}^{\infty}\varepsilon_{n}i^{n}J_{n}(k_{0}r) \cos n\theta e^{-i\omega t} \tag{28}\]  
where \(k_{0}=\omega/c_{p}\) (for P\({}_{1}\) wave incidence) or \(k_{0}=\omega/c_{s}\) (for SV wave incidence) represents the wave number of the incident wave, \(\varphi_{0}(\Psi_{0})\) represents the incident wave amplitude, \(J_{n}(. )\) is the Bessel function of the first kind of order \(n\), \(\varepsilon_{n}=1\) if \(n=0\) and \(\varepsilon_{n}=2\) if \(n\geq 1\), and \(i=\sqrt{-1}\). When the incident plane wave is on the cavity surface, waveform conversion will occur, and the cavity surface will produce scattered waves. The total wave field in an unsaturated porous medium consists of incident waves and scattered waves. The potential of the total wave field in an unsaturated porous medium can be expressed as follows [18]. For the P\({}_{1}\) wave incidence, the total scalar potentials of each phase can be written as:  
\[\left\{ \begin{array}{l}\varphi^{S}(r,\theta)=\sum\limits_{j=1}^{3}\sum \limits_{n=0}^{\infty}A_{n,j}K_{n}\left(k_{j}r\right)\cos n\theta+\varphi_{0} \sum\limits_{n=0}^{\infty}\varepsilon_{n}i^{n}J_{n}(k_{0}r)\cos n\theta\\ \varphi^{W}=\sum\limits_{j=1}^{3}\delta_{j}^{N}\sum\limits_{n=0}^{\infty}A_{ n,j}K_{n}\left(k_{j}r\right)\cos n\theta+\delta_{1}^{N}\varphi_{0}\sum \limits_{n=0}^{\infty}\varepsilon_{n}i^{n}J_{n}(k_{0}r)\cos n\theta\\ \varphi^{N}=\sum\limits_{j=1}^{3}\delta_{j}^{N}\sum\limits_{n=0}^{\infty}A_{ n,j}K_{n}\left(k_{j}r\right)\cos n\theta+\delta_{1}^{N}\varphi_{0}\sum \limits_{n=0}^{\infty}\varepsilon_{n}i^{n}J_{n}(k_{0}r)\cos n\theta\\ \end{array} \right.\] (29a) The total vector potential functions can be written as: \[\left\{ \begin{array}{l}\Psi^{S}=\sum\limits_{n=0}^{\infty}B_{n}K_{n}(k_{4}r )\sin n\theta\\ \Psi^{W}=\zeta^{W}\Psi^{S}\\ \Psi^{W}=\zeta^{W}\Psi^{S}\\ \end{array} \right.\] (29b) For the SV wave incidence, the total scalar potentials of each phase can be written as: \[\left\{ \begin{array}{l}\varphi^{S}(r,\theta)=\sum\limits_{j=1}^{3}\sum \limits_{n=0}^{\infty}A_{n,j}K_{n}\left(k_{j}r\right)\cos n\theta\\ \varphi^{W}=\sum\limits_{j=1}^{3}\delta_{j}^{N}\sum\limits_{n=0}^{\infty}A_{ n,j}K_{n}\left(k_{j}r\right)\cos n\theta\\ \varphi^{N}=\sum\limits_{j=1}^{3}\delta_{j}^{N}\sum\limits_{n=0}^{\infty}A_{ n,j}K_{n}\left(k_{j}r\right)\cos n\theta\\ \end{array} \right. \tag{29b}\]The total vector potential functions can be written as:  
\[\left\{ \begin{array}{c}\Psi^{\text{S}}=\sum\limits_{n=0}^{\infty}B_{n}K_{n}(k_{ 4}r)\sin n\theta+\Psi_{0}\sum\limits_{n=0}^{\infty}\varepsilon_{n}t^{n}J_{n}(k _{0}r)\text{cos }n\theta\\ \Psi^{\text{W}}=\zeta^{\text{W}}\Psi^{\text{S}}\\ \Psi^{\text{W}}=\zeta^{\text{W}}\Psi^{\text{S}}\end{array} \right. \tag{30b}\]  
where \(A_{n,j}(j=1,2,3)\) and \(B_{n}\) refer to the unknown amplitudes of a P wave and an S wave, which should be determined by the boundary conditions. Assuming the permeable boundary conditions of the cavity and the continuity conditions of the displacement between the fluid and the solid skeleton, the boundary conditions at the surface of the cavity (\(r=a\)) can be expressed as:  
\[\sigma_{rr}^{\text{S}}=\sigma_{rr}^{\text{W}}=\sigma_{rr}^{\text{N}}=\sigma_{ r\theta}^{\text{S}}=0(\text{at }r=a) \tag{31}\]  
Substituting Equation (5) into the constitutive relations of Equation (4) in cylindrical coordinates [1], the stress and displacement components in Equation (31) can be expressed by the potentials of the total wave field.