ENERGY PROPAGATION OF THE RAYLEIGH WAVES IN 3-DIMENSIONAL HALF SPACE

MISHIO KAWASHITA AND WAKAKO KAWASHITA(NÉE DAN)^{†} FACULTY OF EDUCATION, IBARAKI UNIVERSITY, † INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA

We study the energy behaviour of the Rayleigh surface waves in a homogeneous isotropic 3-dimensional half space with the Neumann boundary condition. It is known that the Rayleigh waves with finite energy are asymptotically concentrated along the boundary. Our aim of this study is to clarify the domain where the energy of the Rayleigh waves concentrate and to obtain the decay order of the energy distributed in the complement of the domain.

First, we estimate local energy of the Rayleigh component in the flat domains G_{L} = {x ∈ R_{+}^{3}; x_{3} > L} and obtain an estimate of energy in G_{L} with respect to L but not time t. This estimate explains that energy of the Rayleigh waves concentrates in the complement of the region G_{L} in some sense. On the other hand, we can not expect decay properties of energy in G_{L} as t → ∞. Indeed, we obtain an example pointing out this fact, which means the energy of the Rayleigh waves may spread to G_{L} generally. Guillot [1] obtained energy of the Rayleigh component in non-flat domain C_{ϵ} = {(x′, x_{3}) ∈ R_{+}^{3}; x_{3} > ϵ |x′| } tends to 0 as t → ∞. Hence, our results show different feature of the energy behaviour of the Rayleigh waves.

Second, to make the energy behaviour more clear, we also obtain decay order of energy with respect to t in flat and annular domains. The decay property of the Rayleigh component in [1] is showed by constructing asymptotic wave functions introduced by Wilcox [2], however, it seems to be hard to obtain decay estimates by the approach using the asymptotic wave functions. In stead of that, we rewrite the Rayleigh component by a kernel representation, which plays an important role in our approach. As a result, we conclude that the energy of the Rayleigh waves concentrates in the domain {(x′, x_{3}) ∈ R_{+}^{3}; ||x′| − c_{R}t| < L′, 0 < x_{3} < L}, by obtaining various estimates in the complement of the domain above, where c_{R} is a velocity of the Rayleigh waves.

References

[1] J. C. Guillot, “Existence and uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space”, Math. Meth. in the Appl. Sci. 8 (1986), 289–310

[2] C. H. Wilcox, “Scattering Theory for the d’Alembert Equation in Exterior Domains”, Lect. Notes in Math., no. 442, Springer, Berlin (1975)