In this paper, we first present an overview of Da'lembert's solution and impedance boundary conditions, and then outline the excitation boundary conditions, including stick-slip vibration of a string and an excitation model of an expansion part of an acoustic self-excited phenomenon with a heat source. After that, we summarize the reflection law of a rigid wall from the viewpoint of the law of conservation of energy, and then introduce the process of examining the reflection law of the end when the string length changes.
Utsuno et al. (2025) studied this question.