## 2 Characteristics of the Shock Wave and Its Sources
The described collision is inelastic; therefore, based on the conservation of momentum, we can determine the initial velocity of an isolated system consisting of a projectile and a transmitter:  
\[v_{bt}(p)=\frac{m_{b}\;\sqrt{\frac{2\;\;d_{t}\;p\;S_{b}}{m_{b}}}}{m_{b}+m_{t}} \;\;\;\;\;\;\left[\frac{\mathrm{m}}{\mathrm{s}}\right] \tag{11}\]  
where:  
\(m_{t}\)**--mass of the transmitter (a steel 15F type transmitter was used during the experiments). Thus, the kinetic energy accumulated in the transmitter after the collision is given by:  
\[E(p)=p\frac{m_{b}\;d_{t}\;S_{b}}{m_{b}+m_{t}}\;\;\;\;\;\;\;\left[J\right] \tag{12}\]  
Formula (12) shows that the energy accumulated in the transmitter after the collision depends directly proportional to the pressure of air forced into the tube (a parameter set by the user). Figure 4a shows a model of a typical transmitter. It is held in the applicator holder by a torus-shaped elastomer. It makes the movement of the projectile and transmitter system a delayed movement that can be modeled with a progressive spring system (Figure 4b). The elasticity coefficient depends on the hardness of the elastomer (expressed in Shore A degrees). When the system stops, it returns to its initial position (elastomer decompression), and the projectile is ejected [22]. In practical solutions, the elastomer, during compression, reduces its thickness (the path of the transmitter) by a maximum of 1 mm [1,23]. The authors in formula (12) omitted the component responsible for the momentum of the transmitter in the numerator, as the transmitter was at the initial state before the first hit. Nevertheless, as it turned out in the experiments, this component was not zero. We suspect that this explains the hypothesis that increasing the applicator frequency corresponds to a decrease in the energy flux density of the shock wave. It can be described according to the following:  
\[v_{bt}=\frac{m_{b}v_{b}+g(f)}{m_{b}+m_{t}}\;\mathrm{where}\;\exists_{\theta}\; \;k(f)=\varrho(g(f)) \tag{13}\]  
In other words, the projectile collides with a transmitter that has not returned to rest after the previous cycle (\(g\
eq 0\)). Moreover, as the experiments show, the energy flux density decreases with increasing frequency, which would mean that the return of the transmitter to the rest state is a vibrating motion (damped), and the g component depends on the current amplitude of these vibrations. To put it simply, the higher the frequency, the earlier in the vibration cycle of the transmitter the collision with the projectile takes place, i.e., with the greater the amplitude the transmitter counteracts the collision with the projectile. Electromagnetic generators have a simpler design, and they transform the electrical energy supplied to the device into kinetic energy of the piston (accelerated in the magnetic field of the electromagnet). Figure 5 shows an example of an electromagnetic solution. A ferromagnetic piston set in motion and accelerated in a magnetic field, after crossing the length of the solenoid, hits the transmitter. In practical solutions, a spring is also used to counteract the force of pulling the piston into the solenoid--thus solving the problem of the free return to the initial position of the piston after the disappearance of the magnetic field (coil voltage). Nevertheless, this elastic force is ignored in the following considerations. Secondly, it is quite important in the context of the applicator construction to ensure good cooling of the device, something that was not required in the pneumatic version. It is important not only to ventiate the solenoid itself, but also to prevent the piston from heating up because the increase in temperature deteriorates the ferromagnetic properties of the piston and thus reduces the force with which it is pulled into the solenoid. The magnetic permeability of the piston decreases as it approaches the Curie temperature. The parameters controlling the solenoid work in this case are the current \(i\).