## 1 Introduction
Firstly, the Kramers-Kronig dispersion relations indicate a power law between the attenuation and the working frequency, i.e., \(\alpha\propto\omega^{y}\), \(y=1-2\). For most tissues, the attenuation factor \(y\) sits in the range from 1 to 1.7 [32; 33]. However, in the KZK model, the propagation loss caused by viscosity and thermal conduction is considered to be proportional to the square of the working frequency (\(y=2\)), which is actually only valid for fresh water [32; 33]. Furthermore, \(y=2\) for fresh water causes a third derivative term to appear in the KZK equation. The third derivative as well as the derivative of acceleration have not yet been well clarified in the physical overview. In the context of [PERSON]'s law of motion, acceleration is directly affected by the force, and the derivative of acceleration has no obvious physical meaning. Secondly, sound velocity, which is treated as a constant in the KZK equation, usually changes with frequency in biological tissues. In HIFU fields, propagation nonlinearity could be non-negligible due to the high acoustic pressure, and the orders of harmonic waves could be rather high in many cases [5; 6; 7; 9; 29]. Therefore, the descriptions of both attenuation and sound velocity should be modified when biological tissues are present on the wave path. To overcome these shortcomings in the existing models, efforts need to be made to account for the case of a non-integer power index in the Kramers-Kronig relationship. In the frequency domain, with the help of the classic Laplace Transform, the acoustic field could be easily accessible, but only valid for linear models. The Rayleigh model proposed by [PERSON] et al. in 1995 was also not applicable in cases using wide-bandwidth acoustic pulses [34]. The fractional order derivative method described by [PERSON] and [PERSON] [35] was believed to be an effective approach to solve this problem. However, the complexity in mathematics has made numerical analysis hard to achieve, hence restrained its application in developing HIFU theories. After that, [PERSON] et al. utilized a convolution integral and further developed the theory of the fractional order derivative, in which Fourier Transform was adopted to define a derivative of non-integer order, while the basic principles of the Kramers-Kronig relationship was still preserved [36; 37; 38]. Some other calculation algorithms have also been proposed. [PERSON] et al. developed a popular, open source nonlinear simulation tool named the k-wave toolbox to simulate nonlinear wave propagation [39]. [PERSON] et al. proposed time-fractional acoustic wave equations [40]. Inspired by these algorithms, simulation of the KZK equation has made definite progress in the past few years. In this paper, a fractional order derivative was introduced to modify the KZK model to address the power-law relationship (non-integer power index), and a frequency-dependent sound velocity was employed to account for the dispersive behaviors of media. Numerical and experimental verifications were carried out to demonstrate the different behaviors between the modified model and the original KZK model. Results showed that the obtained modified KZK (mKZK) equation is in better agreement with the experimental results. The findings in this paper will promote not only the prediction and design of focused sound fields and acoustic transducers, but also the therapeutic applications of ultrasonic treatments.