## 1 Introduction  
Abrasive water jet (AWJ) technology is an advanced, unconventional machining tool widely used for processing various materials, especially those that are challenging to cut by traditional methods [1, 2, 3]. AWJ is distinctive in its ability to create precise, cold cutting without causing thermal deformation of the materials [4], a crucial advantage when working with sensitive or multi-layered materials [5]. By combining a high-pressure water jet with abrasive particles, AWJ can effectively separate inhomogeneous materials with minimal waste, reducing both processing costs and environmental impact. Among typical materials used as abrasives are aluminium oxide, silicon carbide, garnet, diamond, glass beads, and cubic boron nitride [6]. These properties give AWJ technology an increasingly valuable alternative for applications requiring high precision, low thermal impact, and clean cuts, such as woodworking and wood composites [7, 8]. Pioneers in this area, including [PERSON] [9], [PERSON] et al. [10], [PERSON] and [PERSON] [11] and [PERSON] et al. [12], laid the foundational knowledge for AWJ technology. Their research enabled subsequent researchers, such as [PERSON] [13], [PERSON] [14], [PERSON] [15], and [PERSON] [16], to expand the understanding of the AWJ machining process, focusing on both the basic principles [17] and the practical optimization of cutting parameters across various industries. This technology is well established in fields ranging from precision component manufacturing and advanced functional materials to intelligent automotive engineering, aviation technology, renewable energy sources, and high-end medical device development [18]. In these areas, AWJ has proven to be a critical technological need, including optimized cutting parameters to increase the efficiency and quality of machining, dynamic simulations to predict material behavior under a water jet, and real-time process monitoring [19]. This technology is now essential not only in traditional industries but also in cutting-edge, high-precision applications [20]. AWJ cutting is a complex multi-parameter process in which various input factors, such as water pressure, traverse speed, abrasive concentration, and other factors, interact with each other [21]. Each input setting plays a crucial role in achieving precise, clean, and efficient cuts and in optimizing desired product properties, such as surface smoothness, dimensional accuracy, and minimal defects [22]. Among these factors, the lagging of the jet in the cutting plane is particularly significant, resulting from complex interactions between the stream of abrasive water and the material, which can influence geometry and final properties of the cut. As with other high-energy jet technologies, AWJ leaves surface traces known as grooves, which can impact the surface quality and dimensional accuracy of the final products [1, 23]. The cutting process is initiated slightly below the surface of the material and gradually deepens, resulting in a machined surface that consists of a smooth zone followed by a grooved zone [16]. The traces reveal the trajectory of the water jet as it moves through the material, whilst the lag of the water jet also visibly affects the uniformity and quality of the cut [24]. The mass flow rate of the abrasive material has a fundamental influence on the efficiency of the AWJ cutting. As the abrasive particle content in the liquid stream increases, the depth of cut also increases [25]. This phenomenon is due to the increased kinetic energy (\(\mathrm{E_{a}}\)) of the abrasive particles, a key factor in the cutting process. The kinetic energy is defined as follows:  
\[\mathrm{E_{a}}=\frac{1}{2}\mathrm{m_{a}.\mathrm{v_{a}}}^{2}\left[\
icefrac{{1} }{{\mathrm{s}}}\right], \tag{1}\]  
where the following is true:  
\(\mathrm{m_{a}}\)--abrasive mass flow rate of the abrasive particles [kg/s];  
\(\mathrm{V_{a}}\)--speed of abrasive material exiting the focusing tube [kg/s].