## 3 Quasi-Steady Assumption-Based Communication Port Impedance  
The pressure difference across a communication port can be decomposed into two parts: resistive losses and inductive losses, which can be characterized by resistance and inertance. Impedance is a complex ratio between the Fourier transform of pressure difference and the Fourier transform of flow rate. This consideration assumes linearity for small perturbations around the operating point and corresponds to the two terms within the equation:  
\[Z=R_{\rm LA}+j\omega L_{\rm LA} \tag{1}\]  
where \(Z\) is the impedance of the communication port, \(R_{\rm LA}\) is its linear flow resistance, and \(L_{\rm LA}\) is its linear flow inertance. Theoretically, resistance and inertance are parameters related to fluid physical properties and flow patterns. Therefore, it is not entirely accurate to solely correlate resistance and inertance with the mass flow rate in the communication port. However, considering that pogo frequencies are typically below 50 Hz, we can assume that resistive losses and inductive losses are functions of the mass flow rate within the communication port, based on the quasi-steady assumption. Thus, we can analyze the impedance of the communication port under a series of steady states. It is known that the flow resistance of an orifice exhibits a non-linear dependency on the flow rate [15]. The relationship between flow resistance and orifice geometry can be expressed as follows:  
\[R_{\rm QA}=\frac{\Delta P}{\dot{m}_{\rm A}^{2}}=\frac{1}{2\rho C_{\rm D}^{2}A_ {\rm A}^{2}} \tag{2}\]  
where \(R_{\rm QA}\) is the non-linear flow resistance of communication port, \(\Delta\)P is the instantaneous pressure drop across resistances, \(\dot{m}_{\rm A}\) is the instantaneous mass flow rate through a resistance, \(\rho\) is the density of the fluid, \(C_{\rm D}\) is the orifice discharge coefficient, and \(A_{\rm A}\) is the total flow area of all the communication ports. In the case of small system perturbations around steady-state flow conditions, \(R_{\rm LA}\) is approximated through perturbation linearization of Equation (2) with respect to steady-state flow conditions. In the context of pulsating flow with zero mean flow weight, as previously mentioned, [PERSON] devised an energy equivalence method that ultimately leads to the following relationship:  
\[R_{\rm LA,2}=\frac{8\widetilde{M}_{\rm A}}{3\pi}R_{\rm QA}=\frac{4}{3\pi} \frac{\widetilde{M}_{\rm A}}{\rho C_{\rm D,0}^{2}A_{\rm A}^{2}} \tag{3}\]  
where \(\widetilde{M}_{\rm A}\) is the amplitude of the oscillatory mass flow rate. However, despite the similarities in geometry between the communication port and an orifice, their flow coefficients exhibit distinct differences. [PERSON] [12] proposed a theory that describes their relationship as follows:  
\[\frac{C_{\rm D,0}}{C_{\rm D}}=k\left|\frac{V}{V_{\rm P}}+a\right|+b \tag{4}\]  
where \(C_{\rm D,0}\) is the orifice discharge coefficient under zero main pipe flow velocity, \(V\) and \(V_{\rm P}\) represent surface average velocity in the supply pipeline and the communication port, and \(k\), \(a\), \(b\) are coefficients to be determined through the experiment. By combining Equations (2) and (4) and conducting further deduction, it becomes possible to quantify the resistance of the communication port based on its geometry for small system perturbations around steady-state flow conditions. For oscillatory flow, the linear flow resistance is defined as the ratio of pressure difference to flow rate after Fouriertransformation.