## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.9 \(T_{mod}\) and \(s_{1}\) at High PRR  
Having in hands expressions (52) for \(t_{np}\) in two practically interesting ranges of pulse repetition frequencies (54) and (55), we can now estimate the modification radius \(R\) taking into account a finite \(t_{np}\), when the resulting heating from two neighbour filaments becomes important for reaching the critical for separation temperature at the point \(r=R\), as shown in Figure 5. If we assume that the pulse Y\({}_{1}\) falls on the material at time \(t=0\) and creates a filament at a point \(x=0\), then the next pulse Y\({}_{2}\) creates its filament, respectively, at time \(t=t_{rp}\) (\(t_{rp}\) is the pulse repetition time) and at a point \(x=s_{1}\). Here \(s_{1}\) is, as previously, the step of the filaments following in the material, and the \(x\)-axis is directed along the irradiated surface of the material in the direction of the beam movement relative to it--see Figure 4a. We estimate the contributions of these pulses to the heating of the point \(x=R\), taking into account the obtained above finite time delay \(t_{rp}\) between them. We take for estimates according to (22) \(s_{1}=s_{1 max}=2R\), i.e., \(R=0.5s_{1}\). Thus, the location of this point is at the same distance \(R=0.5s_{1}\) from both these filaments. As can be seen from Figure 5b, the filament created by the pulse Y\({}_{1}\) causes at this point \(x=R\) the maximum heating, denoted as \(\Delta T_{1}\), at the time, which according to (34) is:  
\[t_{max1}=\ \frac{s_{1}^{2}}{16\chi}=0.0625\frac{s_{1}^{2}}{\chi}. \tag{58}\]  
The second filament created by the pulse Y\({}_{2}\) falling on the material with a delay of \(t_{rp}\) to the first pulse causes the maximum heating at the same point \(R\), as shown by curve 2, of the same value \(\Delta T_{1}\) (since this point is located at the same distance \(R\) from this second filament), but, accordingly, at the time delayed by \(t_{rp}\):  
\[t_{max2}=\ t_{max1}+t_{rep}. \tag{59}\]  
The resulting heating under the effect of these two neighbour filaments is represented by curve 3 (Figure 5b). It is seen that this curve has a maximum value of:  
\[\Delta T=\Delta T_{1}+\eta\Delta T_{1}=(1+\eta)\Delta T_{1} \tag{60}\]  
at the time:  
\[t_{max}\approx t_{max2}=t_{max1}+t_{rep}. \tag{61}\]  
Here \(\eta\Delta T_{1}\) is the value of the heating from the first pulse at the stage of the cooling down at the indicated time \(t=t_{max2}\). As can be seen from Equation (60) and Figure 5b, taking into account the action of two neighbour filaments gives rise to an increase in the maximum heating by \((1+\eta)\) times compared to the above-considered heat diffusion heating from only one filament. To estimate the coefficient of heat preservation \(\eta\) for the above practically interesting ranges of pulse repetition frequency \(f\geq f_{1}\) and \(f\leq f_{2}\), a relation obvious from Figure 5b is used:  
\[\eta=\ \frac{T\big{(}r=0.5s_{1},\ \ t=t_{max1}+t_{rep}\big{)}-T_{i}}{\Delta T _{1}}, \tag{62}\]  
in which the temperatures in the numerator and the denominator are calculated by Equation (32). Using (52), we substitute into (32) and (62) the parameter \(t_{rp}\), whose value according to Table 3 at \(s_{1}=2R\) is \(t_{rp1}=s_{1}^{2}/(12.1\chi)=0.0826s_{1}^{2}/\chi\), as well as \(t_{max1}\) from (58) and the expression(35) for \(\Delta T_{1}\). It gives: \(\eta(t_{rp1})=0.76\). Thus, for \(f=f_{1}\) one has \(\eta=0.76\). For \(f>>f_{1}\), when \(t_{rp}<<t_{rp1}\), i.e., the delay between pulses is practically absent, we have obviously \(\eta\approx 1\).