## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.8 The Pulse Repetition Rate  
Another factor in obtaining a better smoothness of the cut walls is that the filaments in each of their arrays, despite their possible mutual influence on each other due to the temperature fields and gradients created by them, must, nevertheless, be straight. Otherwise, when the filaments are curved, after the material separation passing through these filaments, it is unlikely to expect flat smooth sidewalls of separation--most likely they will be rough. From this point of view, it looks reasonable and useful to minimise the influence of neighbour filaments in the array on each other, i.e., to minimise the influence of the temperature gradient and heating from the previous filament at the location and at the time of occurrence of the next new filament in time and space. We will show in this section that in the framework of the thermal or thermal deformation mechanism when \(R=R_{T}\) or \(R=R_{TD}\) and the impact zone is formed due to the energy transfer in the material by thermal diffusion, this requires the choice of the PRR\(f\) in some specific ranges. From the above expression (32) for the spatio-temporal temperature change in the material due to the energy release of the filament, we estimate the material heating from the previous filament at the location and at the time of occurrence of the next new filament in time and space, i.e., at time \(t=t_{rep}\) (\(t\) = 0 is taken as the time of occurrence of the previous filament) and at a distance \(r=s_{1}\) from this previous filament, as shown in Figure 5a:  
\[T\big{(}r=s_{1},\ t=t_{rep}\big{)}-T_{i}=\frac{Q}{\pi C\rho H\big{(}4\chi t_{ rep}\big{)}}\exp\biggl{(}-\frac{{s_{1}}^{2}}{4\chi t_{rep}}\biggr{)}. \tag{49}\]  
Considering \(s_{1}\) to be fixed, we search for such \(t_{rep}\) at which this heating (49) is small, i.e., for example, is no more than 10% of the characteristic heating value (\(T_{mod}-T_{i}\)) corresponding to the threshold point \(T_{mod}\) in the considered case. In the limiting case, we use for estimates not an inequality, but a strict equality:  
\[T\big{(}r=s_{1},\ t=t_{rep}\big{)}-T_{i}=0.1(T_{mod}-T_{i}). \tag{50}\]  
To find solutions of this equation for \(t_{rep}\), a new dimensionless variable \(a\) is introduced, such that:  
\[t_{rep}={s_{1}}^{2}/(4a\chi),\,\text{i.e.,}\,a(4\chi t_{rep})={s_{1}}^{2}. \tag{51}\]  
Substituting (51) into (49), then into the condition (50) and using also the expression (36) for \(T_{mod}\) results in a transcendental equation for \(a\): \(a\text{e}^{-a}=(0.1/\text{e}){(s_{1}/R)}^{2}\). For each given ratio (\(s_{1}\)/\(R\)), the solution of this equation has two roots \(a_{1}\) and \(a_{2}\), which can be easily found numerically. They are shown in Table 3. According to (51), \(a_{1}\) and \(a_{2}\) correspond to two solutions for \(t_{rep}\):  
\[t_{rep1}={s_{1}}^{2}/(4a_{1}\chi),\,t_{rep2}={s_{1}}^{2}/(4a_{2}\chi), \tag{52}\]  
and, respectively, two solutions for the PRR\(f=1/t_{rep}\):  
\[f_{1}=4a_{1}\chi/s_{1}^{2},\ f_{2}=4a_{2}\chi/s_{1}^{2}. \tag{53}\]Thus, the condition \(T\big{(}r=s_{1},\ t=t_{rep}\big{)}-T_{i}\leq 0.1(T_{mod}-T_{i})\) is fulfilled in two ranges of pulse repetition frequency: in a so-called \"high-frequency\" range:  
\[f\geq f_{1}, \tag{54}\]  
where \(f_{1}\) is the lower limit of this high-frequency range, and in the so-called \"low-frequency\" range:  
\[f\leq f_{2}, \tag{55}\]  
where \(f_{2}\) is the upper limit of the low-frequency range, with \(f_{2}\leq f_{1}\).