## 1 Introduction
The periodicity of the passive structure with the geometrical and the material discontinuities influences the wave propagation predicted by the Brillouin theory. As a theory assumes the discontinuities and resonances among periodic structure, it can be applied to wave motion found in nature such as optical metamaterials, the wave mechanics of the spinning electrons, and the mechanical metamaterials. As a result, research on the shape of the periodic structure has been advanced. In the case of a general periodic structure used as a mechanical filter, it is classified into passive and active periodic structure. Existing passive periodic structures exhibit intrinsic dynamic properties that act as mechanical  
Figure 1: A representation of the periodic structure cell (**a**) Cylindrical metamaterials and (**b**) cone shape metamaterials. Figure 2: The general concept of periodic structures. (**a**) Generally known periodic structure and (**b**) Use various shapes as a periodic structure. filters for wave propagation. The harmonic analysis solution has been proposed with rectangular elastic rods of arbitrary aspect ratio [16]. The research on the longitudinal wave propagation of exponentially tapered bars was studied from theoretical and experimental point of view [17]. Additionally, the study of longitudinal wave control was advanced using the periodic structure with a conical shape of a different size and array [18]. It is known that changes in the cone's cross section cause wave dispersion, that conical rods are commonly used as high-pass filters, and that blocked frequencies vary with cone angle [19]. In the case of longitudinal waves of different cross-sections, inhomogeneities are characterized by density and elastic modulus [20]. To enhance the performance of the passive metamaterials, it is possible to add active materials. As for active periodic structures, the structure with active control was studied with the focus on the adjustment of spectral widths and positions of structural vibration and stop band [21]. The longitudinal wave transmission can be controlled by shape memory inserts which are periodically arranged adjustable properties according to load and active periodic structures that cause impedance mismatches [22]. Metamaterials based on liquid-crystal layers were studied in [23; 24]. Some sensitive mechanical device, audio, and electrical devices require vibration isolators to prevent vibration transmission among their components. To prevent vibration transmission, the isolation of components may be the best, but it loses the mechanical supporting function. The cylindrical shape metamaterials were commonly used, and the conical spikes were also used. The conical spikes were used to isolate environmental vibration from mechanical and electronic devices. It was reported that the single cone as an isolator and that the magnitude of the transmitted acceleration can be further reduced compared with a cylindrical shape isolator [18]. In this paper, we investigate the wave motions in cylindrical metamaterials, conical metamaterials, and the combined metamaterials. [PERSON] investigated the wave motion in conical periodic structure and [PERSON] investigated that the wave dispersion can be induced by varying the cross section [18; 19]. The cutoff frequency is dependent on the profiles of the conical shape. However, there is no research regarding the longitudinal wave motion of the combined type of cylindrical and conical metamaterials. In addition, this research studies the wave motions of the curved rod with the cylindrical and the conical metamaterials to extend the usage of the metamaterials. This research conducts the theoretical and experimental studies for the mechanical metamaterials for longitudinal wave in which the displacements show the same direction or the opposite direction to the propagation direction of wave. This research is interested in the metamaterials controlling the longitudinal wave.