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

Kaplunov, Yu. D.

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

Keywords:

long-wavelength integrals; high-frequency dynamical problems; approximate two-dimensional equations