The paper is concerned with development of the asymptotic formulation for surface wave field induced by vertical surface stress under the effect of gravity in the short-wave region. The approach relies on the methodology of hyperbolic-elliptic models for the Rayleigh wave and results in a regularly perturbed hyperbolic equation on the surface acting as a boundary condition for the elliptic equation governing decay over the interior. A special value of the Poisson’s ratio v = 0.25 is pointed out, at which the effect of gravity disappears at leading order.
FotiSLaiCGRixGJ, et al. Surface wave methods for near-surface site characterization. Boca Raton, FL: CRC Press, 2014.
2.
KrylovVV (ed.). Noise and vibration from high-speed trains. London: Thomas Telford, 2001.
3.
GuptaMKhanMAButolaR, et al. Advances in applications of Non-Destructive Testing (NDT): a review. Adv Mater Process Technol. Epub ahead of print 4April2021. DOI: 10.1080/2374068X.2021.1909332.
4.
PalermoAKrödelSMarzaniA, et al. Engineered metabarrier as shield from seismic surface waves. Sci Rep2016; 6(1): 1–10.
5.
ColquittDColombiACrasterR, et al. Seismic metasurfaces: sub-wavelength resonators and Rayleigh wave interaction. J Mech Phys Solid2017; 99: 379–393.
6.
WoottonPKaplunovJColquittD.An asymptotic hyperbolic-elliptic model for flexural-seismic metasurfaces. Proc R Soc A2020; 475(2227): 20190079.
7.
Lord RayleighDCL. On waves propagated along the plane surface of an elastic solid. Proc London Math Soc1885; 1(1): 4–11.
8.
ChadwickPSmithGD.Foundations of the theory of surface waves in anisotropic elastic materials. Adv Appl Mech1977; 17: 303–376.
9.
FuYBMielkeA.A new identity for the surface–impedance matrix and its application to the determination of surface-wave speeds. Proc R Soc London A: Math Phys Eng Sci2002; 458(2026): 2523–2543.
10.
DestradeM.Seismic Rayleigh waves on an exponentially graded, orthotropic half-space. Proc R Soc London A: Math Phys Eng Sci2007; 463(2078): 495–502.
11.
BromwichTIA.On the influence of gravity on elastic waves, and, in particular on the vibrations of an elastic globe. Proc London Math Soc1898; 1(1): 98–165.
12.
LoveAEH.Some problems of geodynamics. Cambridge: Cambridge University Press, 1911.
13.
BiotMA.Mechanics of incremental deformations. New York: John Wiley & Sons, 1965.
14.
DeSNSen-GuptaPR.Influence of gravity on wave propagation in an elastic layer. J Acous Soc Am1974; 55(5): 919–921.
15.
Abd-AllaAMAhmedSM.Stoneley and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity. Appl Math Comput2003; 135(1): 187–200.
16.
VinhPCSerianiG.Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity. Wave Motion2009; 46(7): 427–434.
17.
TingTCT.Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity. Wave Motion2011; 48(4): 335–344.
18.
AlamPKunduSGuptaS.Effect of magneto-elasticity, hydrostatic stress and gravity on Rayleigh waves in a hydrostatic stressed magneto-elastic crystalline medium over a gravitating half-space with sliding contact. Mech Res Commun2018; 89: 11–17.
19.
SharmaMD.Propagation of Rayleigh waves at the boundary of an orthotropic elastic solid: influence of initial stress and gravity. J Vib Control2020; 26(21–22): 2070–2080.
20.
KaplunovJPrikazchikovDA.Asymptotic theory for Rayleigh and Rayleigh-type waves. Adv Appl Mech2017; 50: 1–106.
21.
ChadwickP.Surface and interfacial waves of arbitrary form in isotropic elastic media. J Elast1976; 6(1): 73–80.
22.
KiselevAPParkerDF.Omni-directional Rayleigh, Stoneley and Scholte waves with general time dependence. Proc R Soc London A: Math Phys Eng Sci2010; 466(2120): 2241–2258.
23.
NobiliAPrikazchikovD.Explicit formulation for the Rayleigh wave field induced by surface stresses in an orthorhombic half-plane. Eur J Mech-A/Solids2018; 70: 86–94.
24.
FuYKaplunovJPrikazchikovD.Reduced model for the surface dynamics of a generally anisotropic elastic half-space. Proc R Soc London A: Math Phys Eng Sci2020; 476(2234): 20190590.
25.
KhajiyevaLPrikazchikovDPrikazchikovaL.Hyperbolic-elliptic model for surface wave in a pre-stressed incompressible elastic half-space. Mech Res Commun2018; 92: 49–53.
26.
ErbaşBKaplunovJNoldeE, et al. Composite wave models for elastic plates. Proc R Soc London A: Math Phys Eng Sci2018; 474(2214): 20180103.
27.
WoottonPKaplunovJPrikazchikovD.A second-order asymptotic model for Rayleigh waves on a linearly elastic half plane. IMA J Appl Math2020; 85(1): 113–131.
28.
KaplunovJPrikazchikovDSultanovaL.Rayleigh-type waves on a coated elastic half-space with a clamped surface. Philos Trans R Soc A2019; 377(2156): 20190111.
29.
BratovVKaplunovJLapatsinS, et al. Elastodynamics of a coated half-space under a sliding contact. Math Mech Solid2022.
30.
DaiH-HKaplunovJPrikazchikovDA.A long-wave model for the surface elastic wave in a coated half-space. Proc R Soc London A: Math Phys Eng Sci2010; 466(2122): 3097–3116.
31.
AchenbachJD.Wave propagation in elastic solids. Amsterdam: Elsevier, 1973.
32.
MalischewskyPG. Comment to “A new formula for the velocity of Rayleigh waves” by D. Nkemzi [Wave Motion 26 (1997) 199-205]. Wave Motion2000; 31(1): 93–96.
33.
EgeNErbasBKaplunovJ, et al. Approximate analysis of surface wave-structure interaction. J Mech Mater Struct2018; 13(3): 297–309.
34.
BratovVAIlyashenkoAVKuznetsovSV, et al. Homogeneous horizontal and vertical seismic barriers: mathematical foundations and dimensional analysis. Mat Phys Mech2020; 44(1): 61–65.
35.
TovstikPY.The vibrations and stability of a prestressed plate on an elastic foundation. J Appl Math Mech2009; 73(1): 77–87.
36.
BratovV.Numerical simulations of dynamic fracture. Crack propagation and fracture of initially intact media. Mater Phys Mech2021; 47(3): 455–474.
