## 2 The Model of Incident Elastic Wave Scattering by a Two-Ring Borehole  
A scattering model of an incident plane elastic P or S wave by a two-ring borehole in an infinite space is given to investigate theoretical frequency response and the corresponding bandwidth for borehole tensor strainmeters, as shown in Figure 1, where the two rings represent the instrument casing and the expansive grout, respectively. The inner radius and outer radius of the stainless steel instrument casing and the radius of the borehole are \(a\), \(b\), and \(c\), respectively. The harmonic plane elastic P or S wave propagating in the plane perpendicular to the axis of the borehole is incident at the incident angle \(\alpha\) with respect to the \(x\) direction. Because of the existence of the borehole, grout and instrument casing, two kinds of scattering waves or standing waves (P and S waves) are generated from layer interfaces. The displacement \(\mathbf{u}\) induced by total wave fields around the borehole, which include the incident and scattering wave fields, can be represented in terms of displacement potential \(\phi\) of P wave and displacement potential \(\psi\) of S wave [28]:  
\[\mathbf{u}=\
abla\phi+\
abla\times(\mathbf{e_{z}}\psi) \tag{1}\]  
where \(\mathbf{e_{z}}\) is a unit vector along the borehole axis. Figure 1: The scattering model of the incident elastic plane wave by a two-ring borehole. The incident P and S waves in the rock can be expressed in the form of displacement potential as:  
\[ \begin{array}{ll}\varphi_{p}^{(i)}&=\varphi_{0}e^{i(k_{11}r\cos(\theta-\alpha) -\omega t)}\\ &=\varphi_{0}\sum\limits_{n=-\infty}^{\infty}i^{n}J_{n}(k_{11}r)e^{in(\theta- \alpha)}e^{-i\omega t}\end{array} \Bigg{\}},\text{ for the incident P wave} \tag{2}\]  
\[ \begin{array}{ll}\psi_{S}^{(i)}&=\psi_{0}e^{i(k_{21}r\cos(\theta-\alpha)- \omega t)}\\ &=\psi_{0}\sum\limits_{n=-\infty}^{\infty}i^{n}J_{n}(k_{21}r)e^{in(\theta- \alpha)}e^{-i\omega t}\end{array} \Bigg{\}},\text{ for the incident S wave} \tag{3}\]  
where \(k_{11}=\omega/c_{1p}\) and \(k_{21}=\omega/c_{1s}\) are the wave number of the incident P and S waves, respectively. \(c_{1p}=\sqrt{(\lambda_{1}+2\mu_{1})/\rho_{1}}\) and \(c_{1s}=\sqrt{\mu_{1}/\rho_{1}}\) are P and S wave velocities in the surrounding rock, respectively. \(\rho_{1}\) is the mass density of the surrounding rock. \(\varphi_{0}\) and \(\psi_{0}\) are the amplitudes of the incident P and S wave displacement potentials, respectively. \(\omega\) is the circular frequency of the incident wave. \(J_{n}\) denotes Bessel function of the first kind of order \(n\). If no borehole, the incident wave-induced strains at the borehole center, along the propagation direction of incident P/S waves, are expressed as:  
\[ \begin{array}{ll}\varepsilon_{x^{\prime}}^{(i)}=-\varphi_{0}k_{11}^{2}e^{-i \omega t}\\ \varepsilon_{y^{\prime}}^{(i)}=\varepsilon_{x^{\prime}y^{\prime}}^{(i)}=0 \end{array} \Bigg{\}},\text{ for the incident P wave} \tag{4}\]  
\[ \begin{array}{ll}\varepsilon_{x^{\prime}}^{(i)}=\varepsilon_{y^{\prime}}^{( i)}=0\\ \varepsilon_{x^{\prime}y^{\prime}}^{(i)}=\frac{1}{2}\psi_{0}k_{21}^{2}e^{-i \omega t}\end{array} \Bigg{\}},\text{ for the incident S wave} \tag{5}\]  
The scattering wave fields in the rock, grout layer and instrument casing layer can be denoted by wave functions of Bessel function or Hankel functions, with the unknown constants as following [28]:  
\[ \begin{array}{ll}\varphi_{1}^{(s)}=\sum\limits_{n=-\infty}^{\infty}A_{n}^{ 1}H_{n}(k_{11}r)e^{in(\theta-\alpha)}e^{-i\omega t}\\ \psi_{1}^{(s)}=\sum\limits_{n=-\infty}^{\infty}C_{n}^{1}H_{n}(k_{21}r)e^{in( \theta-\alpha)}e^{-i\omega t}\end{array} \Bigg{\}},\text{ for the surrounding rock} \tag{6}\]  
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