## 2 Materials and Methods
### Signal Processing Based on Laplace Techniques  
Transforming functions from the time domain into the Laplace domain is a well-established technique in fields dealing with differential equations [24; 25; 26; 27]. When this transformation is applied, all functions that depend on \(t\) are expressed in terms of \(s\), where \(s\) is the Laplace variable. One key advantage of this approach is that time derivatives become simple multiplications by \(s\), simplifying the differential equations. In the case of a differential equation that depends on one spatial operator and one temporal operator,  
Figure 1: Laminar velocity profiles for various Womersley numbers ranging from 1.5 to 30 with a phase angle of 90\({}^{\circ}\) (right side) and laminar velocity profiles for various phase angles ranging from 90\({}^{\circ}\) to 190\({}^{\circ}\) with a Womersley number of 30.  
the partial differential equation is reduced to an ordinary differential equation. Moreover, the Laplace transformation enables the analysis of arbitrary pressure signals, which is impossible with the Fourier method as it restricts solutions to harmonic, oscillating signals. This work aims to transform the system equations derived in the Laplace domain back into the time domain, ultimately obtaining an accurate equation that links the flow to two pressure signals in the time domain. One needs to perform an inverse Laplace transformation to revert from the Laplace domain to the time domain. The function must meet the conditions for an ILT, notably that the function in the Laplace domain must converge to zero as \(s\to\infty\). The ILT can be computed using a correspondence table after performing partial fraction decomposition for many Laplace domain functions. For example, [PERSON] et al. calculated the hydrodynamic force and torque on a sphere using ILT techniques aided by integral tables [25]. However, in cases where partial fraction decomposition is not feasible, and no correspondence table exists, a general formula is needed. This is provided by the Bromwich contour integral (Equation (2)), which can be used to obtain the ILT of most functions [28]. In this transformation, \(f(t)\) and \(F^{*}(s)\) represent the same function in the time and Laplace domains, respectively, with \({}^{*}\) indicating the function in the Laplace domain. The poles of the function \(F^{*}(s)\) are denoted as \(s=s_{j}\). The Bromwich contour integral does not need to be directly computed in many cases. It can be evaluated indirectly using [PERSON]'s contour integral, the Residue theorem from complex analysis, or numerical methods. For instance, [PERSON] et al. used a numerical approach to determine the ILT and obtain a solution for three-dimensional time-dependent circulation in shallow lakes [24]. [PERSON] conducted further work for instantaneous volumetric flow rates in circular pipes [26, 27]. He utilized the fast inverse Laplace transformation (FILT) to obtain a representation of the physical signal. The FILT inherently contains a truncation error and limits the validity range of the determined result to instantaneous volumetric flow rates. The Residue theorem's advantage over numerical methods is that it provides an exact formula, offering more profound insight into the investigated system. Additionally, it extends the range of validity compared to numerically derived solutions and avoids concerns about numerical instabilities. The integral over the complex plane is replaced by the sum of the residues of the function at its poles. Residues are specific coefficients found in the Laurent series expansion of a function around a pole [29]. It is worth noting that the Residue theorem is especially useful for dealing with functions that have complex poles. The following theorems and formulae (Equations (2)-(4)) regarding the fundamentals of the residue theorem and Laplace transformation can be found in the work by [PERSON] [29].