Research and Mathematical Modeling of The Dependence of The Frequency of Transverse Vibrations of a Variable-Length String on The Amplitude

When studying string vibrations, the linear wave equation is most often used. This does not take into account the dependence of the string tension force on the amplitude. For metals, even with a slight elongation, the tension force will increase significantly, which will affect the speed of propagation of transverse waves and the frequency of vibrations. The speed of transverse vibrations is limited by the tensile strength of the material. Thus, for high-strength silicon-chromium-manganese steel, the ratio of the speed of longitudinal waves to the maximum speed of transverse waves is approximately equal to 10. For other materials, this ratio is much larger, so we will assume that longitudinal deformations at each moment of time have time to be evenly distributed along the length of the string, because during one transverse vibration a large number of longitudinal vibrations occur.