This paper investigates the numerical instabilities encountered in the simulation of fluid-structure interaction (FSI) problems using loosely coupled time advancing schemes. It shows that certain combinations of physical parameters can lead to numerical instabilities in loosely coupled schemes, while strongly coupled schemes may require more computational effort to solve. The study focuses on a simplified model representing the interaction between a potential fluid and a linear elastic thin tube. This model reproduces propagation phenomena and accounts for the added-mass effect of the fluid on the structure, which is known to be a source of numerical difficulties. The mathematical analysis reveals that the stability of the solution depends on the structure density and the geometry of the domain. The results are validated through numerical experiments and are shown to agree with observations in more complex FSI problems. The paper also discusses the use of different explicit and implicit schemes for time marching in FSI problems and proposes mathematical criteria for stability and convergence. The results are further extended using scaling arguments to a wider range of situations. The study concludes that the added-mass effect is a key factor in the numerical stability of partitioned algorithms for FSI problems.This paper investigates the numerical instabilities encountered in the simulation of fluid-structure interaction (FSI) problems using loosely coupled time advancing schemes. It shows that certain combinations of physical parameters can lead to numerical instabilities in loosely coupled schemes, while strongly coupled schemes may require more computational effort to solve. The study focuses on a simplified model representing the interaction between a potential fluid and a linear elastic thin tube. This model reproduces propagation phenomena and accounts for the added-mass effect of the fluid on the structure, which is known to be a source of numerical difficulties. The mathematical analysis reveals that the stability of the solution depends on the structure density and the geometry of the domain. The results are validated through numerical experiments and are shown to agree with observations in more complex FSI problems. The paper also discusses the use of different explicit and implicit schemes for time marching in FSI problems and proposes mathematical criteria for stability and convergence. The results are further extended using scaling arguments to a wider range of situations. The study concludes that the added-mass effect is a key factor in the numerical stability of partitioned algorithms for FSI problems.