## 4 The Disturbance Propagation Model Based on the Aw-Rascle Model in Crowd Flow
### Disturbance Propagation Model Based on the Aw-Rascle Model  
The propagation of disturbances in a passenger crowd is assumed to follow the mass conservation law of the macroscopic hydrodynamic model [12], that is  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{h})}{\partial x}+\frac {\partial(\rho v_{l})}{\partial y}=0 \tag{19}\]  
where \(v_{h}\) is the velocity in the horizontal direction and \(v_{l}\) is the velocity in the lateral direction. In addition, the crowd to the rear can adjust their movement by directly observing the abnormal motion of the luggage-laden passenger, while the crowd in front cannot. This reflects the anisotropy of disturbance propagation in crowds. Thus, to describe the process of disturbance propagation, the Aw-Rascle model [40,44], a typical anisotropic traffic flow model, is employed. The classic Aw-Rascle model can be expressed by Equations (20) and (21):  
\[\frac{\partial(v_{h}+P_{h})}{\partial t}+v_{h}\frac{\partial(v_{h}+P_{h})}{ \partial x}=s_{1}=\frac{1}{\tau}(V_{eh}-v_{h}) \tag{20}\]  
\[\frac{\partial(v_{l}+P_{l})}{\partial t}+u\frac{\partial(v_{l}+P_{l})}{ \partial y}=s_{2}=\frac{1}{\tau}(V_{el}-v_{l}) \tag{21}\]  
where \(\tau\) is the relaxation time, \(V_{eh}\) and \(V_{el}\) are the equilibrium velocities in the horizontal and lateral direction, respectively, \(s_{1}\) and \(s_{2}\) are the relaxation terms, and \(P_{h}\) and \(P_{l}\) refer to the horizontal and lateral pressure components, respectively. We replace the pressure term of the Aw-Rascle model with the pressure term of crowd interior disturbance in Equation (18), and then we obtain  
\[\frac{\partial(v_{h}+P_{gh})}{\partial t}+v_{h}\frac{\partial(v_{h}+P_{gh})}{ \partial x}=\frac{1}{\tau}(V_{eh}-v_{h}) \tag{22}\]  
\[\frac{\partial(v_{l}+P_{gl})}{\partial t}+u\frac{\partial(v_{l}+P_{gl})}{ \partial y}=\frac{1}{\tau}(V_{el}-v_{l}) \tag{23}\]  
In the propagation process of a disturbance, motion damping exists. Due to a damping effect, energy is absorbed by passengers in the process of disturbance propagation until all the energy is dissipated. This phenomenon is called the damping wave elimination effect of crowds. The damping wave elimination of a crowd is carried out in the direction of the velocity of the disturbance.  
To simulate the damping effect of crowds, it is reasonable to add the damping wave elimination terms to construct a disturbance propagation model in (22) and (23), as follows:  
\[Sx=-\rho u_{0}uf(x) \tag{24}\]\[Sy=-\rho\mu_{0}vf(x) \tag{25}\]  
where \(\rho\) is the crowd density and \(\mu_{0}\mu\) and \(\mu_{0}v\) reflect the loss of disturbance during propagation. The damping distribution function is approximated by a linear damping function  
\[f(x)=\frac{x-x_{s}}{x_{e}-x_{s}} \tag{26}\]  
where \(x_{s},x_{e}\) are the horizontal coordinates of the start and end of the wave elimination zone, respectively.  
Multiplying both sides of Equation (22) by density and adding the damping wave elimination term, we obtain:  
\[\rho\frac{\partial(v_{h}+P_{\xi h})}{\partial t}+\rho v_{h}\frac{\partial(v_{ h}+P_{\xi h})}{\partial x}-\rho\mu_{0}\mu f(x)=\rho\frac{1}{\tau}(V_{eh}-v_{h}) \tag{27}\]  
For the vertical direction, similarly, we obtain:  
\[\rho\frac{\partial(v_{l}+P_{\xi l})}{\partial t}+\rho u\frac{\partial(v_{l}+P _{\xi l})}{\partial y}-\rho\mu_{0}vf(x)=\rho\frac{1}{\tau}(V_{el}-v_{l}) \tag{28}\]  
Equations (27) and (23) are the dynamic model for the disturbance propagation of a luggage-laden passenger. Note that the AR traffic flow model was originally designed for the analysis of car movement. When adopting the AR traffic flow into pedestrian flow modeling, several premises should be ensured: (1) mass conservation; (2) continuous flow; and (3) the velocity of pedestrian media is regarded as the equilibrium velocity of continuum pedestrian flow and is the reflection of the desired velocity of several pedestrians.