This paper aims to make some comparative studies between heterogeneous and homogeneous layers for nonlinear shear horizontal (SH) waves in terms of the heterogeneous and nonlinear effects. Therefore, with this aim, two layers are defined as follows: on the one hand, one layer consists of hyperelastic, isotropic, heterogeneous, and generalized neo-Hookean materials; on the other hand, another layer is made up of hyperelastic, isotropic, homogeneous, and generalized neo-Hookean materials. Moreover, it is assumed that upper boundaries are stress-free and lower boundaries are rigidly fixed. The method of multiple scales is used in both analyses, in addition to using the known solutions of the nonlinear Schrödinger (NLS) equation, called bright and dark solitary wave solutions; these comparisons are made, numerically, and then all results are given for the lowest branch of both dispersion relations, graphically. Moreover, these comparisons are observed both on a large scale and on a small scale, not only in terms of the bright and dark solitary wave solutions but also in terms of the heterogeneous and nonlinear effects.
AchenbachJD.Wave propagation in elastic solids. Amsterdam: North-Holland, 1973.
2.
EringenACSuhubiES.Elastodynamics, vol. 2. New York: Academic Press, 1975.
3.
EwingWMJardetskyWSPressF.Elastic waves in layered media. New York: McGraw-Hill, 1957.
4.
FarnellGW. Types and properties of surface waves. In: OlinerAA (ed.) Acoustic surface waves. Berlin: Springer, 1978, 13–60.
5.
GraffKF.Wave motion in elastic solids. New York: Dover, 1975.
6.
AblowitzMJClarksonPA.Solitons, nonlinear evolution equations and inverse scattering. Cambridge: Cambridge University Press, 1991.
7.
DoddRKMorrisHCEilbeckJC, et al. Solitons and nonlinear wave equations. London: Academic Press, 1982.
8.
JeffreyAKawaharaT.Asymptotic methods in nonlinear wave theory. Boston: Pitman, 1981.
9.
MauginGA. Physical and mathematical models of nonlinear waves in solids. In: JeffreyAEngelbrechtJ (eds.) Nonlinear waves in solids. New York: Springer, 1994, 109–233.
10.
NorrisA. Finite amplitude waves in solids. In: HamiltonMFBlackstockDT (eds.) Nonlinear acoustics. San Diego: Academic Press, 1998, 263–277.
11.
ParkerDF. Nonlinear surface acoustic waves and waves on stratified media. In: JeffreyAEngelbrechtJ (eds.) Nonlinear waves in solids. New York: Springer, 1994, 289–347.
12.
PorubovAV.Amplification of nonlinear strain waves in solids. Singapore: World Scientific, 2003.
13.
SamsonovAM. Nonlinear strain waves in elastic waveguides. In: JeffreyAEngelbrechtJ (eds.) Nonlinear waves in solids. New York: Springer, 1994, 349–382.
14.
WhithamGB.Linear and nonlinear waves. New York: John Wiley and Sons, 1974.
15.
AhmetolanSTeymurM.Nonlinear modulation of SH waves in a two-layered plate and formation of surface SH waves. Int J Non Linear Mech2003; 38: 1237–1250.
16.
AhmetolanSTeymurM.Nonlinear modulation of SH waves in an incompressible hyperelastic plate. Z Angew Math Phy2007; 58: 457–474.
17.
DemirkuşDTeymurM. Shear horizontal waves in a nonlinear elastic layer overlying a rigid substratum. Hacet J Math Stat2017; 46(5): 801–815.
18.
DemirkuşD. Nonlinear bright solitary SH waves in a hyperbolically heterogeneous layer. Int J Non Linear Mech2018; 102: 53–61.
19.
DemirkuşD.Nonlinear dark solitary SH waves in a heterogeneous layer. TWMS J Appl Eng Math2019. DOI: 10.26837/jaem.627563.
20.
DemirkuşD.Antisymmetric dark solitary SH waves in a nonlinear heterogeneous plate. Z Angew Math Phys2019; 70(6): 173.
21.
DemirkuşD. A comparison between heterogeneous and homogeneous layers for bright solitary shear horizontal waves in terms of heterogeneous effect. In: AltenbachHEremeyevVAPavlovI, et al. (eds.) Nonlinear wave dynamics of materials and structures (Advanced Structured Materials, vol. 122). Cham: Springer, 2020.
22.
DestradeMGorielyMASaccomandiG.Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations. Proc R Soc London, Ser A2011; 467: 1823–1834.
23.
DestradeMSaccomandiG.Solitary and compactlike shear waves in the bulk of solids. Phys Rev E2006; 72: 065604(R).
24.
FuY.On the propagation of nonlinear traveling waves in an incompressible elastic plate. Wave Motion1994; 19(3): 271–292.
25.
MayerAPParkerDFMaradudinAA.On the role of second-order nonlinearity in the evolution of shear-horizontal guided acoustic waves. Phys Lett A1992; 164(2): 171–176.
26.
PorubovAVSamsonovAM.Long nonlinear strain waves in layered elastic half-space. Int J Non Linear Mech1995; 30(6): 861–877.
27.
TeymurM.Nonlinear modulation of Love waves in a compressible hyperelastic layered half space. Int J Eng Sci1998; 26: 907–927.
28.
BhattacharyaSN.Exact solutions of SH wave equation for inhomogeneous media. Bull Seismol Soc Am1970; 60: 1847–1859.
29.
Danishevs’kyyWKaplunovJDRogersonGA.Anti-plane shear waves in a fibre-reinforced composite with a nonlinear imperfect interface. Int J Non Linear Mech2015; 76: 223–232.
30.
HudsonJA.Love waves in a heterogeneous medium. Geophys J Int1962; 6(2): 131–147.
31.
SahuSASarojPKDewanganN.SH-waves in viscoelastic heterogeneous layer over half-space with self-weight. Arch Appl Mech2014; 84: 235–245.
32.
VardoulakisIGeorgiadisHG.SH surface waves in a homogeneous gradient-elastic half-space with surface energy. J Elast1997; 47: 147–165.
33.
MauginGA.Material inhomogeneities in elasticity. London: Chapman and Hall, 1993.
34.
PeregrineDH.Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J1983; 25: 16–43.
35.
ZakharovVEShabatAB.Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. J Exp Theor Phys1972; 34: 62–69.
36.
ZakharovVEShabatAB.Interaction between solitons in a stable medium. Sov Phys JETP1973; 37: 823–828.
37.
BhattacharyaSN.Love wave dispersion: A comparison of results for a semi-infinite medium with inhomogeneous layers and for its approximation by homogeneous layers. Pure Appl Geophys1976; 114: 1021–1029.
38.
CrasterRJosephLKaplunovJ.Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media. Wave Motion2014; 51(4): 581–588.
