## 1 Introduction  
Hot dry rock and coalbed methane are very important clean energy sources with abundant resource reserves and huge mining potential, which can effectively support China's \"carbon peak\" and \"carbon neutral\" programs [1]. In the process of underground resource extraction, rock fragmentation, especially hard rock fragmentation, is required. Compared to traditional mechanical and explosive rock fragmentation, water jet rock fragmentation technology has advantages such as non-contact, precision, and low cost [2, 3, 4, 5, 6, 7, 8, 9, 10]. The crushing of hard rock often requires extremely high jet pressure, which also means that larger pump sets are not suitable for practical on-site applications. Pulse water jet technology has received widespread attention for its ability to enhance the water hammer effect generated by the jet's impact on rocks. Because of their unique water hammer effect, pulsed water jets (PWJs) can periodically apply surge pressure waves when they erode the rock mass [11, 12, 13]. Currently, this is considered a good way to break rock in underground space excavation engineering and is particularly suitable for breaking brittle rock masses. When a PWJ impacts a rock mass, the tensile stress formed on the rock mass surface is utilized to break it through tensile failure, and the tensile strength of a rock mass is significantly less than its compressive strength. The PWJ can effectively utilize the physical and mechanical properties of the rock and improve crushing efficiency [14, 15]. Furthermore, water jet technology helps reduce the dust produced in the process of rock breaking, and the rock-breaking device does not need to contact the rock mass directly, thus avoiding wear on the device and making it suitable for breaking and tunneling different strata [16; 17; 18]. The PWJ is superior to the CWJ in many aspects because of its unique erosion ability. Many researchers have changed the focus of their research from simply increasing the pressure of the water jet to improving its striking ability, thus creating the PWJ. The existing PWJ technology can be divided into two types: associated pulsed water jet (APWJ) and modulated pulsed water jet (MPWJ). In the APWJ, the free water mass is interfered with by an external force before the hydrostatic pressure in the pipe is converted into jet pressure by the transducer element (orifice or nozzle). The water jet ejected by the transducer has the characteristics of pulse, pulsation, intermittence, and oscillation. Water cannons and other devices [19; 20] belong to the APWJ type. In the MPWJ, the free water mass is converted into jet flow pressure through transducer elements (such as or nozzles) by hydrostatic pressure. After the formation of a CWJ, the original fluid characteristics are changed by external disturbances or mechanical interference, resulting in characteristics such as jet pulse and intermission. The disc-truncated pulsed water jet [21], internal excitation pulsed water jet [22], self-excited oscillating water jet [23], and acoustic wave-modulated pulsed jet [24] belong to the MPWJ type. Current research has been mainly focused on the erosion mechanism and performance enhancement of the MPWJ on the rock. The longitudinal pulsation generated by ultrasonic interference in the cavity can effectively improve the rock-breaking ability of the water jet. Under the same hydraulic parameters, the rock-breaking effect is substantially better than that of the CWJ. [PERSON] et al. [25] found that the depths of trace formed by 20 Hz and 40 Hz PWJs are 11 times and 15 times greater than those of a CWJ, respectively. Moreover, they compared the erosion of granite by a sonic-modulated pulsating water jet and a CWJ, and the groove formed by erosion was analyzed by profile scanning and electron microscopy. The results show that the CWJ erosion is mainly composed of microcracks, fractures, and lamellar grain removal, whereas that of the pulsating water jet is mainly composed of a deep depression, a meteorite crater, and wavy deformation.