The homogenized medium of the gyro lattice with the mechanical quantum Hall effect, here denoted as the gyroscopic elastic medium, is governed by a new set of elastodynamic equations with the chiral density. This paper presents a generic theory based on the Stroh formalism to disclose the non-reciprocal Rayleigh waves supported by the gyroscopic medium. The Stroh formulation is investigated in the general case of anisotropic elasticity, and after introducing the chiral density, it is extended to yield new forms of eigenvalue equations, which are distinct for surface waves traveling in opposite directions. Numerical examples are performed in the isotropic elasticity scheme, illustrating the non-reciprocal Rayleigh waves as well as the evolution of the particle motion trajectory and polarization at shallow depths as the gyro coupling magnitude increases. The developed theory confirms the prediction for the classical Rayleigh wave that is reciprocal when the gyro coupling term vanishes. Finally, a general effective-medium method that relies on the surface response to external excitation is proposed for the estimation of the overall properties of gyroscopic composite materials in the long-wavelength regime. Consistency in the wave response of gyroscopic composites with periodic inclusions and their effectively homogeneous materials is demonstrated for a large range of gyro coupling constants.

具有机械量子霍尔效应的陀螺晶格的均质介质（这里称为陀螺弹性介质）由一组具有手性密度的弹性动力学方程组控制。本文提出了一种基于Stroh形式主义的通用理论，以揭示陀螺介质支持的不可逆瑞利波。在各向异性弹性的一般情况下研究Stroh公式，并在引入手性密度后将其扩展以生成新形式的特征值方程，这对于沿相反方向传播的表面波是不同的。在各向同性弹性方案中进行了数值示例，图中显示了不可逆的瑞利波，以及随着陀螺耦合量的增加，粒子运动轨迹和极化在浅深度的演变。发达的理论证实了经典的瑞利波的预测，当陀螺仪耦合项消失时，瑞利波是相反的。最后，提出了一种通用的有效介质方法，该方法依赖于对外部激发的表面响应，来估计长波长范围内陀螺复合材料的整体性能。对于大范围的陀螺耦合常数，具有周期性夹杂物的陀螺复合材料及其有效均质材料的波响应具有一致性。发达的理论证实了对经典瑞利波的预测，当陀螺仪耦合项消失时，该瑞利波是相反的。最后，提出了一种通用的有效介质方法，该方法依赖于对外部激发的表面响应，来估计长波长范围内陀螺复合材料的整体性能。对于大范围的陀螺耦合常数，具有周期性夹杂物的陀螺复合材料及其有效均质材料的波响应具有一致性。发达的理论证实了对经典瑞利波的预测，当陀螺仪耦合项消失时，该瑞利波是相反的。最后，提出了一种通用的有效介质方法，该方法依靠表面对外部激发的响应来估计陀螺复合材料在长波长范围内的总体性能。对于大范围的陀螺耦合常数，具有周期性夹杂物的陀螺复合材料及其有效均质材料的波响应具有一致性。