Nodes and energy flow in the transverse oscillations of resonantly end-driven stretched strings with viscous damping

Abstract

Abstract Most of the works that analyze the forced oscillations of strings assume that, although they can support a tension T, they are perfectly elastic/flexible entities that do not have losses. In this work we will show that the incorporation of losses, even maintaining the analysis in a linear regime, allows us to clarify some details of the behavior that is observed in the forced oscillations of a real string. By introducing a dissipative force into the analysis, it is possible to predict not only the resonance frequencies but also how the oscillation amplitudes depend on the strength of the driving force. We can also analyze the variation of the amplitude of the antinodes when the excitation frequency approaches the resonance frequency. This amplitude variation has a behavior very similar to that of the resonance amplitude of a spring-mass oscillator. It is generally recognized that the nodes of a string, in forced oscillation, are not points of absolute rest because the energy of vibration must be transmitted through these points. However, it is difficult to find works that analyze the peculiar properties of this small movement. In this work we calculate the amplitudes of the nodes and deduce the peculiar properties of their movement. We also calculate the time averaged energy that flows along the string, from its excitation point to its fixed end.