Rayleigh waves guided by topography

Abstract

It would not be far-fetched to say that the work of Lord Rayleigh on surface guided waves has had fundamental and far-reaching effects upon modern life and many things we take for granted today, stretching from mobile phones through to the study of earthquakes. Many of these things take advantage of surface waves that are topographically guided thereby allowing energy to be carried in specific directions along some topography of the surface. Much of the emphasis has so far been placed on devices which are essentially thin rectangular ridge-like defects upon the surface. Indeed, experiments with these so-called SAW devices have demonstrated the presence of trapped waves. It remains to provide any convincing theory that captures this trapping effect for surface waves—this is our aim. We seek to resolve the issue of whether topography can support a trapped wave, whose energy is localized to within some vicinity of the topography, and to explain physically how trapping occurs. The trapping is first addressed by developing an asymptotic scheme that exploits a small parameter associated with the surface topography. We then provide numerical evidence to support results obtained from the asymptotic scheme; however, no rigorous proof of existence is presented.

Existence and uniqueness of Rayleigh waves with normal impedance boundary conditions and formula for the wave velocity

“…Proof First, from Proposition 3 (i,ii), it follows: z (1) = −1, z (2) = 1 are two different (real) roots of equation P(z) = 0. Suppose there exists a Rayleigh wave with the corresponding velocity x (3) ⇒ x (3) is a (real) root of Eq.…”

“…Rayleigh surface waves, discovered by Rayleigh [1] in 1885, attracted continued attention of researchers due to their wide range of applications in various engineering fields and their effect on many aspects of modern life [2]. There have been a large number of investigations of Rayleigh surface waves propagating in various elastic half-spaces and most of them assumed that the elastic half-spaces are free from traction, i.e., on their surface, the stresses vanish (see Eq.…”

“…( 14) and 0 < x (1) , x (2) < 1, according to Remark 1. From this fact, it follows: z (1) = (1 + x (1) )/2x (1) and z (2) = (1 + x (2) )/2x (2) are two different (real) roots of equation F(z) = 0 and |z (k) | > 1 (k = 1, 2), according to Remark 2. By the statement (26) of Proposition 2, we conclude that z (1) and z (2) are two different (real) roots of equation P(z) = 0 and |z (k) | > 1 (k = 1, 2).…”

“…Proof Suppose there exist two different Rayleigh waves with the corresponding velocities x (1) and x (2) (x (1) = x (2) ). Then x (1) and x (2) are two different (real) roots of Eq. ( 14) and 0 < x (1) , x (2) < 1, according to Remark 1.…”

“…Then x (1) and x (2) are two different (real) roots of Eq. ( 14) and 0 < x (1) , x (2) < 1, according to Remark 1. From this fact, it follows: z (1) = (1 + x (1) )/2x (1) and z (2) = (1 + x (2) )/2x (2) are two different (real) roots of equation F(z) = 0 and |z (k) | > 1 (k = 1, 2), according to Remark 2.…”