## 1 Introduction  
Cavitation is a process of bubble oscillation and rupture in the medium under ultrasonic irradiation. Bubbles can either oscillate stably, that is, stable cavitation, or expand gradually and finally collapse, that is, transient cavitation [1]. Generally speaking, cavitation refers to transient cavitation, and the generation of transient cavitation is affected by many factors, including ultrasonic frequency, bubble size, and characteristics of the surrounding medium [2]. It is widely used in the treatment of tumor and thrombus in the medical field to produce high temperature and pressure [3; 4; 5]. Compared with traditional single-frequency ultrasound, multifrequency ultrasound has unique advantages in enhancing the cavitation effect, such as reducing cavitation threshold and improving reaction efficiency [6; 7; 8]. In order to study the cavitation effect of multifrequency ultrasound, many experiments have been carried out. [PERSON] et al. [6] and [PERSON] et al. [9] found that when the dual-frequency high-intensity focused ultrasound (HIFU) transducer with different frequencies was used to ablate chicken, compared with the single-frequency HIFU alone, the dual-frequency irradiation could obtain faster temperature rise. [PERSON] et al. [10] used transducers with ultrasonic frequencies of 1563 MHz and 1573 MHz to irradiate the target porcine liver tissue at the same time, and carried out dual-frequency tissue ablation experiments. The results showed that under the same power condition, dual-frequency HIFU could produce greater lesions than traditional single-frequency HIFU, which may be the result of dual-frequency ultrasonic cavitation enhancement. When [PERSON] et al. [11] superposed two sine wave signals with ultrasonic frequencies of 531 kHz and 565 kHz for irradiation, compared with the single-frequency stimulation of 552 kHz, the transient cavitation threshold was reduced by nearly 30%. [PERSON] et al. [12] tested the thrombol efficiency of dual-frequency excitation composed of 535 kHz and 565 kHz on the blood clot in vitro and compared with the thrombol efficiency of single-frequency excitation of 550 kHz with the same power. The dual-frequency excitation doubled the thrombol efficiency and dissolved a 0.8 cm\({}^{3}\) blood clot in 5 min. [PERSON] et al. [13] used 1.45 MHz and 1.5 MHz dual-frequency ultrasound to achieve the same ablation efficiency under the same acoustic power and excitation conditions and found that, compared with single-frequency ablation, dual-frequency ultrasound could save about 30% power. [PERSON] et al. [14] superposed 0.8 MHz and 1.6 MHz ultrasound to irradiate biological tissues, which confirmed that dual-frequency HIFU irradiation was more effective in enhancing tissue coagulative necrosis volume. [PERSON] et al. [15] found that, compared with dual-frequency HIFU irradiation, triple-frequency (28 kHz, 1 MHz, and 1.87 MHz) irradiation could further improve cavitation efficiency. In the study of temperature rise of tissue ablation, under the same irradiation power and time, [PERSON] et al. [16] used any two frequencies of 0.95 MHz, 1.5 MHz, and 3.3 MHz for dual-frequency combined irradiation. Compared with the combination of triple-frequencies of 0.95 MHz, 1.5 MHz, and 3.3 MHz, under the same excitation conditions, the temperature rise of the triple-frequency combination irradiation was higher than that of the dual-frequency combination. Many experimental results show that the cavitation effect can be significantly enhanced by the combination of multifrequency ultrasound irradiation, but there are few theoretical studies on this problem. On the one hand, although some unique characteristics have been found in some preliminary studies of cavitation, the mechanism of multifrequency combination to enhance bubble cavitation has not been fully understood, and on the other hand, the combination of multifrequency ultrasound has an impact on the transient cavitation threshold in the medium.