Phononic crystals and phononic metamaterials are popular structures for manipulating acoustic waves with artificially arranged units that have different elastic constants. These structures are also used in acousto-optic coupling and optomechanical structures. In such research, a 1-D nanobeam containing a cavity region sandwiched by two mirror regions is one of the most common designs. However, searching bandgaps for suitable operation modes and the need for the mirror region are limitations in the device design. Therefore, we introduce the slow sound mode as the operating acoustic mode and use an acoustic potential well to further trap the phonons in the cavity. Three types of structures are introduced to investigate the effect of the potential well. The products of the mode frequencies and the quality factors of the modes are used to demonstrate the performance of the structures. The displacement field and the strain field show the concentrated slow sound modes of the potential wells and produce high quality factors.