## 1 Introduction  
The classical elastic wave theory posits that the stress at a given point is solely determined by the strain at that point. However, the constitutive equation that describes the mechanical behavior of materials does not incorporate the characteristic length of microscopic particles and thus fails to reflect the size effect exhibited by materials. In fact, in the propagation problem of elastic waves, when the incident wavelength of elastic waves approaches the characteristic length of the material microstructure, the microstructural effect of the material greatly increases. To consider the microstructural effect of materials, nonlocal theory [1], micropolar and micromorphic theory [2; 3], coupled stress elasticity theory [4], and various other generalized continuum media theories [5; 6] have been successively proposed. In 1964, [PERSON] [7] proposed the strain gradient theory, which is a linear elasticity theory that considers microstructural effects. The strain gradient theory is a generalized continuum theory developed to explain the size effect phenomenon of materials at the micro and nano scales. According to this theory, the stress at a point is related not only to the strain at that point but also to the strain gradients of various orders at that point; which characterizes such materials as higher-order materials. Strain gradient theory introduces a strain gradient into the constitutive equation to reflect the microstructural effect, enabling it to capture the size effect of materials. Additionally, the strain gradient theory posits that macroscopic and microscopic motions occur concurrently, with the coupling of these macroscopic and microscopic vibrations leading to twelve types of waveforms; of which eight constitute dispersive waves. In the traditional strain gradient elasticity theory established by [PERSON], the additional deformation, which includes the second-order deformation gradient, comprises 18 independent components: eight in the antisymmetric part and 10 in the symmetric part. Even for isotropic materials, the second-order deformation gradient corresponds to seven linear elastic constants, including two Lame coefficients and five non-classical constants related to the microstructure of the material. The control equations and boundary conditions derived from the principle of virtual work also contain five additional constants, making the structure still quite complex. Therefore, [PERSON] [8] first proposed the \"dipole gradient elasticity\" theory for studying crack problems in microstructured solids. In this theory, because the strain energy depends on the strain gradient, the characteristic length of the microparticles in the material is included, allowing the size effect to be reflected in the constitutive equations. [PERSON] et al. [9] studied the propagation of Rayleigh waves on a solid surface by considering the dipole gradient elasticity properties. Their research showed that Rayleigh waves are dispersive, consistent with experimental observations, thereby correcting the assertion of classical elastic theory that Rayleigh waves are non-dispersive at any frequency. In 2013, [PERSON] and [PERSON] [10] studied the reflection of elastic waves in an elastic half-space based on [PERSON]'s theory, discovering four wave modes, two body waves, P waves, and SV waves, as well as two surface waves, SP waves and SS waves in dipolar gradient solids. Building on the dipole strain gradient theory, [PERSON] and [PERSON] [11] investigated the reflection and transmission problems at the interfaces between two different dipole strain gradient elastic solids in 2015. Subsequently, [PERSON] and [PERSON] [12; 13; 14; 15] applied the dipole strain gradient elasticity theory to study the propagation of elastic waves in sandwich and infinite periodic structures. In 2021, [PERSON] and [PERSON] et al. [16] utilized a nonlocal strain gradient model and fractional derivatives to examine the propagation of bending waves in nanobeams, whereas [PERSON] and [PERSON] et al.