## 1 Introduction  
Rotating machinery is extensively used in today's industries such as power plants, automobiles etc. and a crack in the rotating shaft can cause a costly failure, even a significant accident [1]. The detection of cracks in structures is a very important issue and more effective detection methods allowing assessment of the location and depth of the cracks are continuously investigated [2; 3; 4]. The vibration of the shaft caused by the rotation or the activation from the transducers can stimulate the elastic waves, and the propagation characteristics of the which are affected by the cracks, thus, the research into the propagation characteristics can help to detect the cracks. The study of the cracks and the propagation characteristics of the elastic waves has drawn wide attention [5; 6; 7; 8]. Recently, wave-based damage-detection techniques have been employed prevalently in the health-monitoring of structural components, such as in 1-D structure [9], in a plate [10], in metamaterial rods [11], in a square pile [12], or in a non-uniform cross-section pile [13] etc. The damage in the structures can affect the propagation characteristics of the elastic waves, thus, the damage can be detected by studying the change of the propagation characteristics. In another aspect, the wave-basedtechniques can offer advantages in terms of robustness and higher sensitivity, which make these techniques increasingly popular [14; 15]. The transfer matrix is often used to indicate the propagation process of the waves when studying the crack with the elastic wave method [16], and the propagation characteristics of the waves can be obtained by calculating the eigenvalues of the matrix [17]. The transverse cracks are often modeled as local springs in the transfer matrix [2; 16; 18]. The stiffness coefficient of the local spring is determined by the flexibility of the crack tip, i.e., the crack modes, which have been studied widely [19; 20; 21]. Usually, the transverse cracks on the rotating shaft possess different crack modes due to the different external loads, and there are three typical modes: the opening mode (Mode I), the edge-sliding mode (Mode II) and the tearing mode (Mode III) [22]. In a real practice, the crack mode can be a single one or a mixed one [23; 24; 25], and mode conversion may happen due to the damage and the boundary [26]. When a crack appears, the crack-tip plastic zone will affect the propagation characteristics, for example, the propagating speed of the waves may change [5] and the reflected and transmitted energy is related to the size of crack size or the magnitude of the loads [6], and the energy of the wave may dissipate due to the internal wave component reflections when the waves pass through a crack [27]. Recently, the authors of this paper also studied the propagation characteristics of the elastic waves in a non-uniform shaft by using the transfer matrix method with only considering Mode I [28; 29]. Mode II and Mode III do exist in a real application, and the crack modes will affect the local flexibility of the transverse crack which is modeled as a local spring, thus, the characteristics of the transfer matrix for the elastic waves will be affected too. The recent studies of the authors have researched the influence of the geometry of the shaft as well as the rotating speed to the propagation characteristics of the elastic waves. It is of great importance to study the propagation characteristics with different crack modes as well as the crack locations and depths. The influence of three typical crack modes, the location and depth of the transverse crack as well as the rotating speed of the non-uniform shaft to the propagation characteristics of the elastic wave is studied in this paper. The local stiffness coefficients are deduced firstly in Section 2, under different external loads and only considering the individual mode for simplicity. In Section 3, the transfer matrix equation is built, and the most suitable mode is then chosen by using the numerical and experimental methods.