## 2 Theoretical Analysis of Shock-Wave Propagation
### Analysis Model of Shock-Wave Propagation  
For wave-impedance gradient materials with the same structure, the shock wave generated by the impact point is simplified as a one-dimensional stress wave, and only its propagation in the thickness direction of the target plate is considered, as shown in Figure 1. Within a time interval \(\Delta T\) when the stress wave is transmitted from the generation end to the distal end, the stress in the distal layer can be obtained according to the reflection coefficient and transmission coefficient \(\sigma_{n}\). From Equation (1) to (2), the following can be deduced:  
\[\left\{ \begin{array}{c}\sigma_{2}=T_{1}\sigma_{1}\\ \sigma_{3}=T_{2}\sigma_{2}\\ \sigma_{4}=T_{3}\sigma_{3}\\ \cdots\cdots\cdots\\ \sigma_{n-1}^{{}^{\prime}}=F_{n-1}\sigma_{n-1}\end{array} \right. \tag{5}\]  
where \(n=2,\ 3,\ 4,\cdots;\sigma_{1},\sigma_{2},\sigma_{3},\cdots,\sigma_{n}\) are the incident wave intensities at the interface of each layer; \(\sigma_{1}^{{}^{\prime}},\ \sigma_{2}^{{}^{\prime}},\ \sigma_{3}^{{}^{\prime}} \cdots,\ \sigma_{n-1}^{{}^{\prime}}\) are the reflected wave intensities at the interface of each layer; and \(T_{1},\ T_{2},\ T_{3},\ \cdots,\ T_{n-1}\) and \(F_{1},\ F_{2},\ F_{3},\ \cdots,\ F_{n-1}\) are the inter-layer projection coefficient and the reflection coefficient, respectively. Iterate Equation (5) to obtain:  
\[\left\{ \begin{array}{c}\sigma_{n}=T_{n-1}T_{n-2}\cdots T_{2}T_{1}\sigma_{1}= \sigma_{1}2^{n-1}\prod\limits_{i=1}^{n-1}\left(\frac{Z_{i+1}}{Z_{i+1}+Z_{i}} \right)\\ \\ \sigma_{n-1}^{{}^{\prime}}=F_{n-1}T_{n-2}T_{n-3}\cdots T_{2}T_{1}\sigma_{1}= \sigma_{1}\frac{Z_{n}-Z_{n-1}}{Z_{n}+Z_{n-1}}\prod\limits_{i=1}^{n-1}\left( \frac{Z_{i+1}}{Z_{i+1}+Z_{i}}\right)\end{array} \right. \tag{6}\]  
Figure 1: Reflection and transmission of shock wave in the wave-impedance gradient material. Let \(T^{\prime}\) be the total transmission coefficient when the incident shock wave passes through the wave-impedance gradient-attenuation region. \[T^{\prime}=T_{n-1}T_{n-2}\cdots T_{2}T_{1}=2^{n-1}\prod_{i=1}^{n-1}\biggl{(} \frac{Z_{i+1}}{Z_{i+1}+Z_{i}}\biggr{)} \tag{7}\]  
The wave impedance and the total gradient attenuation of the surface material of the wave-impedance gradient material are kept constant as \(Z_{1}\), and \(\Delta Z\), respectively. For the wave-impedance gradient material with N layers, it can be assumed that the wave impedance of each layer of material changes as follows:  
\[\frac{\Delta Z}{n-1} \tag{8}\]  
The wave impedance of the \(i\)-th layer material is:  
\[Z_{i}=Z_{1}-(i-1)\frac{\Delta Z}{n-1},\ (2\leq i\leq n) \tag{9}\]  
When the wave-impedance gradient material is two layers:  
\[T^{\prime}=T_{1}=\frac{2(Z_{1}-\Delta Z)}{2Z_{1}-\Delta Z} \tag{10}\]  
When the wave-impedance gradient material is three layers:  
\[T^{\prime}=T_{2}T_{1}=\left(\frac{2Z_{1}-\Delta Z}{2Z_{1}-\frac{\Delta Z}{2}} \right)\left(\frac{2Z_{1}-2\Delta Z}{2Z_{1}-\frac{3\Delta Z}{2}}\right) \tag{11}\]  
When the wave-impedance gradient material is four layers:  
\[T^{\prime}=T_{3}T_{2}T_{1}=\left(\frac{2Z_{1}-\frac{2\Delta Z}{3}}{2Z_{1}- \frac{\Delta Z}{3}}\right)\left(\frac{2Z_{1}-\frac{4\Delta Z}{3}}{2Z_{1}- \Delta Z}\right)\left(\frac{2Z_{1}-2\Delta Z}{2Z_{1}-\frac{5\Delta Z}{3}}\right) \tag{12}\]  
When the wave-impedance gradient material is \(n\) layers:  
\[T^{\prime}=T_{n-1}T_{n-2}\cdots T_{2}T_{1}=\left(\frac{2Z_{1}-2\frac{\Delta Z} {n-1}}{2Z_{1}-\frac{\Delta Z}{n-1}}\right)\left(\frac{2Z_{1}-2\frac{2\Delta Z }{n-1}}{2Z_{1}-\frac{3\Delta Z}{n-1}}\right)\cdots\left(\frac{2Z_{1}-2\frac{( n-1)\Delta Z}{n-1}}{2Z_{1}-\frac{(2n-3)\Delta Z}{n-1}}\right) \tag{13}\]  
Assuming that the surface material of the wave-impedance gradient material is titanium alloy and the back material is nylon, the constant \(Z_{1}\) and \(\Delta Z\) are determined. The total transmission coefficient of Ti-Ny is 0.206, that of Ti-Al-Ny is 0.236, and that of Ti-Al-Mg-Ny is 0.273.