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**Equations for high-frequency, long-wavelength vibrations of an elastic layer on an acoustic half-space.**
*(English.
Russian original)*
Zbl 0729.73096

Sov. Phys., Dokl. 34, No. 12, 1106-1108 (1989); translation from Dokl. Akad. Nauk SSSR 309, No. 5, 1077-1081 (1989).

It is important to develop asymptotic approaches to the determination of the long-wavelength integrals of high-frequency dynamical problems. In the present article we discuss this problem in application to the vibrations of an elastic layer in contact with an acoustic half-space. We propose approximate two-dimensional equations for determining the long- wavelength solutions. For their derivation we use the asymptotic method for integration of the elasticity equations in thin domains, extending it to the high-frequency range. In terms of this method, we give a rigorous determination of the high-frequency, long-wavelength solutions, indicate the limits of validity, and estimate the error of the resulting equations.

### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

35C20 | Asymptotic expansions of solutions to PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

74J10 | Bulk waves in solid mechanics |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |