Internal rectification for elastic surface waves

Abstract

We prove that fast oscillatory elastic surface waves can produce nontrivial internal nonoscillatory displacements. We consider elastic surface waves of the form, in y > 0 : U ε ( t , x , y ) ∼ ∑ k = 2 ∞ ε k U k ( t , x , y , x − c t ε , y ε ) , with profiles U k ( t , x , y , Y , θ ) = U ̲ k ( t , x , y ) + U k ⁎ ( t , x , θ , Y ) , where U k ⁎ is periodic in θ and exponentially decaying to 0 in Y. We prove that, in general, the corrector U 3 is not purely localized near the boundary, that is U ̲ 3 does not vanish. U 3 depends on the slow variable y and does not decay to 0 when Y tends to +∞, even if the source terms are exponentially decaying to 0.