# A Numerical Study on Computational Time Reversal for Structural Health Monitoring
## 2 Time Reversal Approaches for Structures
### Time Reversal Refocusing as Outcome of Reciprocity
#### 2.2.1 Switching Time Reversal  
We choose the \((v,w)\) state to be the \((v(x,t),u(x,T-t))\). In this case, which is sometimes called a switching time reversal approach [31], the reciprocal statement would be:  
\[\int\limits_{0}^{L}u(x,t^{*})*u(x,t)\,dx-\int\limits_{0}^{L}q(x, t)*v(x,t)\,dx=\] \[EI\left[\frac{\partial^{3}u(x,t)}{\partial x^{3}}*v(x,t)-\frac{ \partial^{2}u(x,t)}{\partial x^{2}}*\frac{\partial v(x,t)}{\partial x}+\frac {\partial u(x,t)}{\partial x}*\frac{\partial^{2}v(x,t)}{\partial x^{2}}-u(x,t )*\frac{\partial^{3}v(x,t)}{\partial x^{3}}\right]_{0}^{L}\] \[+\rho A\int\limits_{0}^{L}\dot{v}_{0}(x)u(x,t)+v_{0}(x)\dot{u}(x, t)-\dot{u}_{0}(x)v(x,t)-u_{0}(x)\dot{v}(x,t)\,dx. \tag{10}\]  
It can be shown that if the forward state is such that produces wave propagation motion originated from a limited range on the spatial \(x\) axis of the beam, then a wave refocusing on the same spatial range is expected to take place at the end of the reversed motion in time.  
In the case of switching time reversal approaches, based on the reciprocal statement of Equation (10), we expected to observe wave refocusing on the part of the initial disturbance which produced the forward wave propagation motion. Such an approach could lead to interesting time reversal computational tools [32,33]. A usual choice of such a disturbance would be a concentrated impulsive load on point \(x_{s}\) given by \(q(x,t)\)=\(\delta(x-x_{s})\delta(t)\) resulting in motion \(u(x,t)\). In other words, we choose as forward state the one corresponding to the fundamental solution for specific boundary conditions. For the current theoretical approach, this is a good choice, which however could lead to numerical problems and therefore we approximate it with some smoother equivalent. Using the method of eigenmodes, the fundamental solution, also known as [PERSON]'s function, is given as:  
\[u(x,t)=\rho A\sum_{n=1}^{\infty}\frac{\phi_{n}(x)\phi_{n}(x_{s})}{M_{n}\omega_ {n}}\sin\omega_{n}t \tag{11}\]  
As regards the backward state, it is selected to be the motion \(v(x,t)\) produced by a load distribution equal with the reversed in time forward displacement field, that is \(w(x,t)=u(x,t^{*})\), where \(t^{*}\)=\(T\)=\(t\).  
After substitution into the reciprocal statement of Equation (10) and some algebraic manipulation we obtain:  
\[v(x_{s},t)=(\rho A)^{2}\sum_{n=1}^{\infty}\frac{\phi_{n}^{2}(x_{s})}{M_{n} \omega_{n}^{2}}(\sin\omega_{n}(T-t)*sin\omega_{n}t) \tag{12}\]  
calculating the time convolution integral we may further write as  
\[v(x_{s},t)=-(\rho A)^{2}\sum_{n=1}^{\infty}\frac{\phi_{n}^{2}(x_{s})}{M_{n} \omega_{n}^{2}}\frac{\sin\omega_{n}(t+T)+\sin\omega_{n}(t-T)-2\omega_{n}t\cos \omega_{n}(t-T)}{4\omega_{n}} \tag{13}\]  
which having its maximum value for time \(t\)=\(T\):  
\[v(x_{s},T)=-(\rho A)^{2}\sum_{n=1}^{\infty}\frac{\phi_{n}^{2}(x_{s})}{M_{n} \omega_{n}^{2}}\frac{\sin 2\omega_{n}T-2\omega_{n}T}{4\omega_{n}}, \tag{14}\]  
it indicates a refocus on \(x_{s}\) for the time \(T\).