Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.6 Solutions to dynamic problems for isotropic linear elastic solids

 
Dynamic problems are even more difficult to solve than static problems.   Nevertheless, analytical solutions have been determined for a wide range of important problems.  There is not space here to do justice to the subject, but a few solutions will be listed to give a sense of the general features of solutions to dynamic problems.

 

5.6.1 Love potentials for dynamic solutions for isotropic solids

 

In this section we outline a general potential representation for 3D dynamic linear elasticity problems.  The technique is similar to the 3D  Papkovich-Neuber representation for static solutions outlined in Section 5.5.

 

Assume that

 The solid has Young’s modulus E, mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@ .

 Define longitudinal and shear wave speeds (see Sect 4.4.5)

c L = E(1ν) ρ 0 (1+ν)(12ν) c s = E 2(1+ν) ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaamitaaqabaGccqGH9aqpdaGcaaqaamaalaaabaGaamyraiaa cIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaaqaaiabeg8aYnaaBaaale aacaaIWaaabeaakiaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaiaa cIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMcaaaaaleqaaOGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqaai aadohaaeqaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaadweaaeaacaaI YaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaeqyWdi3aaSbaaS qaaiaaicdaaeqaaaaaaeqaaaaa@6C37@

 Body forces are neglected (a rather convoluted procedure exists for problems involving body force)

 The solid is assumed to be at rest for t<0

 Part of the boundary S 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGymaaqabaaaaa@3227@  is subjected to time dependent prescribed displacements u i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaqhaaWcbaGaamyAaaqaaiaacQ caaaaaaa@332B@

 A second part of the boundary S 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaaGOmaaqabaaaaa@3228@  is subjected to prescribed tractions t i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaqhaaWcbaGaamyAaaqaaiaacQ caaaaaaa@332A@

 

The procedure can be summarized as follows:

1.      Find a vector function Ψ i ( x 1 , x 2 , x 3 ,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki aacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWg aaWcbaGaaGOmaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIZaaabe aakiaacYcacaWG0bGaaiykaaaa@3D4A@   and a scalar function ϕ( x 1 , x 2 , x 3 ,t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjaacIcacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGG SaGaamiEamaaBaaaleaacaaIZaaabeaakiaacYcacaWG0bGaaiykaa aa@3C5E@  which satisfy

Ψ i x i =0 2 Ψ i x j x j = 1 c s 2 2 Ψ i t 2 2 ϕ x k x k = 1 c L 2 2 ϕ t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeuiQdK1aaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaa beaaaaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8+aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccqqHOoqwdaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWa aSbaaSqaaiaadQgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQb aabeaaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGJbWaa0baaSqa aiaadohaaeaacaaIYaaaaaaakmaalaaabaGaeyOaIy7aaWbaaSqabe aacaaIYaaaaOGaeuiQdK1aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOa IyRaamiDamaaCaaaleqabaGaaGOmaaaaaaGccaaMbSUaaGzaRpaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqy1dygabaGaeyOa IyRaamiEamaaBaaaleaacaWGRbaabeaakiabgkGi2kaadIhadaWgaa WcbaGaam4AaaqabaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaam4y amaaDaaaleaacaWGmbaabaGaaGOmaaaaaaGcdaWcaaqaaiabgkGi2o aaCaaaleqabaGaaGOmaaaakiabew9aMbqaaiabgkGi2kaadshadaah aaWcbeqaaiaaikdaaaaaaaaa@847F@

as well as boundary conditions

ϕ x i + ijk Ψ k x j = u i * on  S 1 2 ϕ x i x j n j + 1 2 ( ilk 2 Ψ k x l x j + jlk 2 Ψ k x l x i ) n j + ν n i 12ν 2 ϕ x k x k = (1+ν) E t i * on  S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacqGHciITcqaHvpGzae aacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabgUcaRiab gIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOWaaSaaaeaacq GHciITcqqHOoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG 4bWaaSbaaSqaaiaadQgaaeqaaaaakiabg2da9iaadwhadaqhaaWcba GaamyAaaqaaiaacQcaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caqGVbGaaeOBaiaabccacaWGtbWaaSbaaSqa aiaaigdaaeqaaaGcbaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaik daaaGccqaHvpGzaeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqa aOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccaWGUbWaaS baaSqaaiaadQgaaeqaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOm aaaadaqadaqaaiabgIGiopaaBaaaleaacaWGPbGaamiBaiaadUgaae qaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqqHOoqw daWgaaWcbaGaam4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaai aadYgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGc cqGHRaWkcqGHiiIZdaWgaaWcbaGaamOAaiaadYgacaWGRbaabeaakm aalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeuiQdK1aaSba aSqaaiaadUgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGSb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaaGccaGL OaGaayzkaaGaamOBamaaBaaaleaacaWGQbaabeaakiabgUcaRmaala aabaGaeqyVd4MaamOBamaaBaaaleaacaWGPbaabeaaaOqaaiaaigda cqGHsislcaaIYaGaeqyVd4gaamaalaaabaGaeyOaIy7aaWbaaSqabe aacaaIYaaaaOGaeqy1dygabaGaeyOaIyRaamiEamaaBaaaleaacaWG RbaabeaakiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaOGaey ypa0ZaaSaaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaeaa caWGfbaaaiaadshadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeOBaiaabcca caWGtbWaaSbaaSqaaiaaikdaaeqaaaaaaa@7434@

and initial conditions Ψ i =ϕ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki abg2da9iabew9aMjabg2da9iaaicdaaaa@37C9@ .

2.      Calculate displacements from the formula

u i = ϕ x i + ijk Ψ k x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadIhadaWg aaWcbaGaamyAaaqabaaaaOGaey4kaSIaeyicI48aaSbaaSqaaiaadM gacaWGQbGaam4AaaqabaGcdaWcaaqaaiabgkGi2kabfI6aznaaBaaa leaacaWGRbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaa qabaaaaOGaaGPaRdaa@48F3@

3.      Calculate stresses from the formula

(1+ν) E σ ij = 2 ϕ x i x j + 1 2 ( ilk 2 Ψ k x l x j + jlk 2 Ψ k x l x i )+ ν δ ij 12ν 2 ϕ x k x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaiikaiaaigdacqGHRaWkcq aH9oGBcaGGPaaabaGaamyraaaacqaHdpWCdaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITdaahaaWcbeqaaiaaik daaaGccqaHvpGzaeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqa aOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGHRaWkda WcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaeyicI48aaSbaaSqa aiaadMgacaWGSbGaam4AaaqabaGcdaWcaaqaaiabgkGi2oaaCaaale qabaGaaGOmaaaakiabfI6aznaaBaaaleaacaWGRbaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamiBaaqabaGccqGHciITcaWG4bWaaS baaSqaaiaadQgaaeqaaaaakiabgUcaRiabgIGiopaaBaaaleaacaWG QbGaamiBaiaadUgaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccqqHOoqwdaWgaaWcbaGaam4AaaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaadYgaaeqaaOGaeyOaIyRaamiEamaaBaaale aacaWGPbaabeaaaaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiab e27aUjabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaakeaacaaIXa GaeyOeI0IaaGOmaiabe27aUbaadaWcaaqaaiabgkGi2oaaCaaaleqa baGaaGOmaaaakiabew9aMbqaaiabgkGi2kaadIhadaWgaaWcbaGaam 4AaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaaaaaaa@8208@

 

You can easily show that this solution satisfies the equations of motion for an elastic solid, by substituting the formula for displacements into the Cauchy-Navier equation

1 12ν 2 u k x k x i + 2 u i x k x k = ρ 0 μ 2 u i t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaaykW6daWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam 4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcaaMcS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaOGaeyyp a0ZaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaaakeaacqaH8o qBaaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWa aSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiDamaaCaaaleqaba GaaGOmaaaaaaaaaa@625B@

The details are left as an exercise.   More importantly, one can also show that the representation is complete, i.e. all dynamic solutions can be derived from some appropriate combination of potentials.

 

 

5.6.2 Pressure suddenly applied to the surface of a spherical cavity in an infinite solid

 

The figure shows a spherical cavity with radius a in an infinite elastic solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3240@ .  The solid is at rest for t<0.   A time t=0, a pressure p is applied to the surface of the hole, and thereafter held fixed.

 

The solution is generated by Love potentials

Ψ i =0ϕ= (1+ν) a 3 p 2ER { 1 2(1ν) e αs sin( βs+γ ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGPbaabeaaki abg2da9iaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqaHvpGzcqGH9aqpcaaMc8UaeyOeI0YaaSaaaeaacaGGOaGaaGymai abgUcaRiabe27aUjaacMcacaWGHbWaaWbaaSqabeaacaaIZaaaaOGa amiCaaqaaiaaikdacaWGfbGaamOuaaaadaGadaqaaiaaigdacqGHsi sldaGcaaqaaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMca aSqabaGccaWGLbWaaWbaaSqabeaacqGHsislcqaHXoqycaWGZbaaaO Gaci4CaiaacMgacaGGUbWaaeWaaeaacqaHYoGycaWGZbGaey4kaSIa eq4SdCgacaGLOaGaayzkaaaacaGL7bGaayzFaaaaaa@6508@

where

α= 12ν 1ν β= 12ν 1ν γ= cot 1 12ν R= x k x k s={ ( c L tR+a)/aRa< c L t 0Ra> c L t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHXoqycqGH9aqpdaWcaaqaai aaigdacqGHsislcaaIYaGaeqyVd4gabaGaaGymaiabgkHiTiabe27a UbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOSdiMaeyypa0Za aSaaaeaadaGcaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4galeqaaa GcbaGaaGymaiabgkHiTiabe27aUbaacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqaHZoWzcqGH9aqpciGGJbGaai4BaiaacshadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaGcaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4 galeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaadkfacqGH9aqpdaGcaaqaaiaadIhada WgaaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaaqa baaakeaacaWGZbGaeyypa0ZaaiqaaeaafaqabeGabaaabaGaaiikai aadogadaWgaaWcbaGaamitaaqabaGccaWG0bGaeyOeI0IaamOuaiab gUcaRiaadggacaGGPaGaai4laiaadggacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam OuaiabgkHiTiaadggacqGH8aapcaWGJbWaaSbaaSqaaiaadYeaaeqa aOGaamiDaaqaaiaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadk facqGHsislcaWGHbGaeyOpa4Jaam4yamaaBaaaleaacaWGmbaabeaa kiaadshaaaaacaGL7baacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7aaaa@24A4@

The displacements and stresses follow as

u i = (1+ν) a 3 p x i 2E R 3 { 1 2(1ν) e αs sin( βs+γ )( βRcot( βs+γ )αR a +1 ) } σ ij = a 3 p 2 R 3 ( 3 x i x j R 2 δ ij ){ 1 2(1ν) e αs sin( βs+γ )( βRcot( βs+γ )αR a +1 ) } + ap 2R ( x i x j R 2 + ν δ ij 12ν ) 2(1ν) e αs sin(βs+γ){ ( α 2 β 2 )2βαcot(βs+γ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaWGPbaabe aakiabg2da9iaaykW7daWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyV d4MaaiykaiaadggadaahaaWcbeqaaiaaiodaaaGccaWGWbGaamiEam aaBaaaleaacaWGPbaabeaaaOqaaiaaikdacaWGfbGaamOuamaaCaaa leqabaGaaG4maaaaaaGcdaGadaqaaiaaigdacqGHsisldaGcaaqaai aaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaSqabaGccaWG LbWaaWbaaSqabeaacqGHsislcqaHXoqycaWGZbaaaOGaci4CaiaacM gacaGGUbWaaeWaaeaacqaHYoGycaWGZbGaey4kaSIaeq4SdCgacaGL OaGaayzkaaWaaeWaaeaadaWcaaqaaiabek7aIjaadkfaciGGJbGaai 4Baiaacshadaqadaqaaiabek7aIjaadohacqGHRaWkcqaHZoWzaiaa wIcacaGLPaaacqGHsislcqaHXoqycaWGsbaabaGaamyyaaaacqGHRa WkcaaIXaaacaGLOaGaayzkaaaacaGL7bGaayzFaaaabaGaeq4Wdm3a aSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabgkHiTmaalaaaba GaamyyamaaCaaaleqabaGaaG4maaaakiaadchaaeaacaaIYaGaamOu amaaCaaaleqabaGaaG4maaaaaaGcdaqadaqaaiaaiodadaWcaaqaai aadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqaaiaadQga aeqaaaGcbaGaamOuamaaCaaaleqabaGaaGOmaaaaaaGccqGHsislcq aH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaWa aiWaaeaacaaIXaGaeyOeI0YaaOaaaeaacaaIYaGaaiikaiaaigdacq GHsislcqaH9oGBcaGGPaaaleqaaOGaamyzamaaCaaaleqabaGaeyOe I0IaeqySdeMaam4CaaaakiGacohacaGGPbGaaiOBamaabmaabaGaeq OSdiMaam4CaiabgUcaRiabeo7aNbGaayjkaiaawMcaamaabmaabaWa aSaaaeaacqaHYoGycaWGsbGaci4yaiaac+gacaGG0bWaaeWaaeaacq aHYoGycaWGZbGaey4kaSIaeq4SdCgacaGLOaGaayzkaaGaeyOeI0Ia eqySdeMaamOuaaqaaiaadggaaaGaey4kaSIaaGymaaGaayjkaiaawM caaaGaay5Eaiaaw2haaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7cqGHRaWkdaWcaaqaaiaadggacaWGWbaabaGaaGOmaiaadkfaaaWa aeWaaeaadaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4b WaaSbaaSqaaiaadQgaaeqaaaGcbaGaamOuamaaCaaaleqabaGaaGOm aaaaaaGccqGHRaWkdaWcaaqaaiabe27aUjabes7aKnaaBaaaleaaca WGPbGaamOAaaqabaaakeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaa aiaawIcacaGLPaaadaGcaaqaaiaaikdacaGGOaGaaGymaiabgkHiTi abe27aUjaacMcaaSqabaGccaWGLbWaaWbaaSqabeaacqGHsislcqaH XoqycaWGZbaaaOGaci4CaiaacMgacaGGUbGaaiikaiabek7aIjaado hacqGHRaWkcqaHZoWzcaGGPaWaaiWaaeaadaqadaqaaiabeg7aHnaa CaaaleqabaGaaGOmaaaakiabgkHiTiabek7aInaaCaaaleqabaGaaG OmaaaaaOGaayjkaiaawMcaaiabgkHiTiaaikdacqaHYoGycqaHXoqy ciGGJbGaai4BaiaacshacaGGOaGaeqOSdiMaam4CaiabgUcaRiabeo 7aNjaacMcaaiaawUhacaGL9baaaaaa@06C0@

The radial and hoop stresses at several time intervals are plotted below:

              

Observe that

  1. A wave front propagates out from the cavity at the longitudinal wave speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadYeaaeqaaa aa@34AB@ ;
  2. Unlike the simple 2D wave problems discussed in Section 4.4, the stress is not constant behind the front.  Instead, each point in the solid experiences a damped oscillation in displacement and stress that eventually decays to the static solution;
  3. Both the radial and hoop stress reverse sign as the wave passes by.  For this reason dynamic loading can cause failures to occur in very unexpected places;
  4. The maximum stress induced by dynamic loading substantially exceeds the static solution. 

 

 

 

5.6.3 Rayleigh waves

 
A Rayleigh wave is a special type of wave which propagates near the surface of an elastic solid.  Assume that

 The solid is an isotropic, linear elastic material with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3132@ , Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@  and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330E@

 The solid has shear wave speed   c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4Caaqabaaaaa@3274@  and longitudinal wave speed c L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamitaaqabaaaaa@324D@

 The surface is free of tractions

 A Rayleigh wave with wavelength λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@321C@  propagates in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction

The displacement and stress due can be derived from Love potentials

ϕ= U 0 ( k 2 + β T 2 ) β L ( k 2 β T 2 ) exp( β L x 2 )exp( ik( x 1 c R t) ) Ψ k = 2ik U 0 δ k3 ( k 2 β T 2 ) exp( β T x 2 )exp( ik( x 1 c R t) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjabg2da9maalaaabaGaamyvam aaBaaaleaacaaIWaaabeaakiaacIcacaWGRbWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaeqOSdi2aa0baaSqaaiaadsfaaeaacaaIYaaaaO Gaaiykaaqaaiabek7aInaaBaaaleaacaWGmbaabeaakiaacIcacaWG RbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaeqOSdi2aa0baaSqaai aadsfaaeaacaaIYaaaaOGaaiykaaaaciGGLbGaaiiEaiaacchadaqa daqaaiabgkHiTiabek7aInaaBaaaleaacaWGmbaabeaakiaadIhada WgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaciGGLbGaaiiEaiaa cchadaqadaqaaiaadMgacaWGRbGaaiikaiaadIhadaWgaaWcbaGaaG ymaaqabaGccqGHsislcaWGJbWaaSbaaSqaaiaadkfaaeqaaOGaamiD aiaacMcaaiaawIcacaGLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqqH OoqwdaWgaaWcbaGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaaikdaca WGPbGaam4AaiaadwfadaWgaaWcbaGaaGimaaqabaGccqaH0oazdaWg aaWcbaGaam4AaiaaiodaaeqaaaGcbaGaaiikaiaadUgadaahaaWcbe qaaiaaikdaaaGccqGHsislcqaHYoGydaqhaaWcbaGaamivaaqaaiaa ikdaaaGccaGGPaaaaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0 IaeqOSdi2aaSbaaSqaaiaadsfaaeqaaOGaamiEamaaBaaaleaacaaI YaaabeaaaOGaayjkaiaawMcaaiGacwgacaGG4bGaaiiCamaabmaaba GaamyAaiaadUgacaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiab gkHiTiaadogadaWgaaWcbaGaamOuaaqabaGccaWG0bGaaiykaaGaay jkaiaawMcaaaaa@9B4A@

where i= 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMgacqGH9aqpdaGcaaqaaiabgkHiTi aaigdaaSqabaaaaa@341F@ , U 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaaGimaaqabaaaaa@3228@  is the amplitude of the vertical displacement at the free surface, k=2π/λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGH9aqpcaaIYaGaeqiWdaNaai 4laiabeU7aSbaa@373E@  is the wavenumber; β L =k 1 c R 2 / c L 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacaWGmbaabeaaki abg2da9iaadUgadaGcaaqaaiaaigdacqGHsislcaWGJbWaa0baaSqa aiaadkfaaeaacaaIYaaaaOGaai4laiaadogadaqhaaWcbaGaamitaa qaaiaaikdaaaaabeaaaaa@3CC5@ , β T =k 1 c R 2 / c s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aInaaBaaaleaacaWGubaabeaaki abg2da9iaadUgadaGcaaqaaiaaigdacqGHsislcaWGJbWaa0baaSqa aiaadkfaaeaacaaIYaaaaOGaai4laiaadogadaqhaaWcbaGaam4Caa qaaiaaikdaaaaabeaaaaa@3CF4@ ; and c R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamOuaaqabaaaaa@3253@  is the Rayleigh wave speed, which is the positive real root of

( 2 c R 2 c s 2 ) 2 4 ( 1 (12ν) 2(1ν) c R 2 c s 2 ) 1/2 ( 1 c R 2 c s 2 ) 1/2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaaGOmaiabgkHiTmaalaaaba Gaam4yamaaDaaaleaacaWGsbaabaGaaGOmaaaaaOqaaiaadogadaqh aaWcbaGaam4CaaqaaiaaikdaaaaaaaGccaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0IaaGinamaabmaabaGaaGymaiabgkHi TmaalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Maaiykaa qaaiaaikdacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaaSaa aeaacaWGJbWaa0baaSqaaiaadkfaaeaacaaIYaaaaaGcbaGaam4yam aaDaaaleaacaWGZbaabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaah aaWcbeqaaiaaigdacaGGVaGaaGOmaaaakmaabmaabaGaaGymaiabgk HiTmaalaaabaGaam4yamaaDaaaleaacaWGsbaabaGaaGOmaaaaaOqa aiaadogadaqhaaWcbaGaam4CaaqaaiaaikdaaaaaaaGccaGLOaGaay zkaaWaaWbaaSqabeaacaaIXaGaai4laiaaikdaaaGccqGH9aqpcaaI Waaaaa@5ED5@

This equation can easily be solved for c R / c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamOuaaqabaGcca GGVaGaam4yamaaBaaaleaacaWGZbaabeaaaaa@351C@  with a symbolic manipulation program, which will most likely return 6 roots.  The root of interest lies in the range 0.65< c R / c s <1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacaGGUaGaaGOnaiaaiwdacqGH8a apcaWGJbWaaSbaaSqaaiaadkfaaeqaaOGaai4laiaadogadaWgaaWc baGaam4CaaqabaGccqGH8aapcaaIXaaaaa@3AD4@  for 1<ν<0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaaigdacqGH8aapcqaH9oGBcq GH8aapcaaIWaGaaiOlaiaaiwdaaaa@37FB@ . The solution can be approximated by c R / c s =0.8750.2ν0.05 (ν+0.25) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamOuaaqabaGcca GGVaGaam4yamaaBaaaleaacaWGZbaabeaakiabg2da9iaaicdacaGG UaGaaGioaiaaiEdacaaI1aGaeyOeI0IaaGimaiaac6cacaaIYaGaeq yVd4MaeyOeI0IaaGimaiaac6cacaaIWaGaaGynaiaacIcacqaH9oGB cqGHRaWkcaaIWaGaaiOlaiaaikdacaaI1aGaaiykamaaCaaaleqaba GaaG4maaaaaaa@4A3D@  with an error of less than 0.6% over the full range of Poisson ratio.

 

The nonzero components of displacement and stress follow as

u 1 = U 0 ik ( k 2 β T 2 ) β L exp( ik( x 1 c R t) ){ ( k 2 + β T 2 )exp( β L x 2 )2 β L β T exp( β T x 2 ) } u 2 = U 0 ( k 2 β T 2 ) exp( ik( x 1 c R t) ){ 2 k 2 exp( β T x 2 )( k 2 + β T 2 )exp( β L x 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaaIXaaabe aakiabg2da9maalaaabaGaamyvamaaBaaaleaacaaIWaaabeaakiaa dMgacaWGRbaabaGaaiikaiaadUgadaahaaWcbeqaaiaaikdaaaGccq GHsislcqaHYoGydaqhaaWcbaGaamivaaqaaiaaikdaaaGccaGGPaGa eqOSdi2aaSbaaSqaaiaadYeaaeqaaaaakiGacwgacaGG4bGaaiiCam aabmaabaGaamyAaiaadUgacaGGOaGaamiEamaaBaaaleaacaaIXaaa beaakiabgkHiTiaadogadaWgaaWcbaGaamOuaaqabaGccaWG0bGaai ykaaGaayjkaiaawMcaaiaaykW7daGadaqaaiaacIcacaWGRbWaaWba aSqabeaacaaIYaaaaOGaey4kaSIaeqOSdi2aa0baaSqaaiaadsfaae aacaaIYaaaaOGaaiykaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOe I0IaeqOSdi2aaSbaaSqaaiaadYeaaeqaaOGaamiEamaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaiabgkHiTiaaikdacqaHYoGydaWg aaWcbaGaamitaaqabaGccqaHYoGydaWgaaWcbaGaamivaaqabaGcci GGLbGaaiiEaiaacchadaqadaqaaiabgkHiTiabek7aInaaBaaaleaa caWGubaabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaaakiaawIcaca GLPaaaaiaawUhacaGL9baaaeaacaWG1bWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0ZaaSaaaeaacaWGvbWaaSbaaSqaaiaaicdaaeqaaaGcba GaaiikaiaadUgadaahaaWcbeqaaiaaikdaaaGccqGHsislcqaHYoGy daqhaaWcbaGaamivaaqaaiaaikdaaaGccaGGPaaaaiGacwgacaGG4b GaaiiCamaabmaabaGaamyAaiaadUgacaGGOaGaamiEamaaBaaaleaa caaIXaaabeaakiabgkHiTiaadogadaWgaaWcbaGaamOuaaqabaGcca WG0bGaaiykaaGaayjkaiaawMcaaiaaykW7daGadaqaaiaaikdacaWG RbWaaWbaaSqabeaacaaIYaaaaOGaciyzaiaacIhacaGGWbWaaeWaae aacqGHsislcqaHYoGydaWgaaWcbaGaamivaaqabaGccaWG4bWaaSba aSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaaiikaiaadU gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaHYoGydaqhaaWcbaGa amivaaqaaiaaikdaaaGccaGGPaGaciyzaiaacIhacaGGWbWaaeWaae aacqGHsislcqaHYoGydaWgaaWcbaGaamitaaqabaGccaWG4bWaaSba aSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaaaaa a@B14B@

σ 11 = U 0 Eexp( ik( x 1 c R t) ) ( k 2 β T 2 )(1+ν)(12ν) β L { k 2 [ ν( β L 2 + β T 2 )(1ν)( k 2 + β T 2 ) ]exp( β L x 2 ) +2 k 2 β T β L (12ν)exp( β T x 2 ) } σ 22 = U 0 Eexp( ik( x 1 c R t) ) ( k 2 β T 2 )(1+ν)(12ν) β L { ( k 2 + β T 2 )[ (1ν) β L 2 ν k 2 ]exp( β L x 2 ) 2 k 2 β T β L (12ν)exp( β T x 2 ) } σ 12 = i U 0 kE( k 2 + β T 2 ) ( k 2 β T 2 )(1+ν) exp( ik( x 1 c R t) ){ exp( β T x 2 )exp( β L x 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaaigdaca aIXaaabeaakiabg2da9maalaaabaGaamyvamaaBaaaleaacaaIWaaa beaakiaadweaciGGLbGaaiiEaiaacchadaqadaqaaiaadMgacaWGRb GaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGJbWa aSbaaSqaaiaadkfaaeqaaOGaamiDaiaacMcaaiaawIcacaGLPaaaae aacaGGOaGaam4AamaaCaaaleqabaGaaGOmaaaakiabgkHiTiabek7a 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ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7daGacaqaaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaeyOeI0IaaGOmaiaadUgadaahaaWcbeqaaiaaikdaaaGccqaHYoGy daWgaaWcbaGaamivaaqabaGccqaHYoGydaWgaaWcbaGaamitaaqaba GccaGGOaGaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaGaciyzaiaa cIhacaGGWbWaaeWaaeaacqGHsislcqaHYoGydaWgaaWcbaGaamivaa qabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaa caGL9baaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaey ypa0ZaaSaaaeaacaWGPbGaamyvamaaBaaaleaacaaIWaaabeaakiaa dUgacaWGfbGaaiikaiaadUgadaahaaWcbeqaaiaaikdaaaGccqGHRa WkcqaHYoGydaqhaaWcbaGaamivaaqaaiaaikdaaaGccaGGPaaabaGa aiikaiaadUgadaahaaWcbeqaaiaaikdaaaGccqGHsislcqaHYoGyda qhaaWcbaGaamivaaqaaiaaikdaaaGccaGGPaGaaiikaiaaigdacqGH RaWkcqaH9oGBcaGGPaaaaiGacwgacaGG4bGaaiiCamaabmaabaGaam yAaiaadUgacaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiabgkHi TiaadogadaWgaaWcbaGaamOuaaqabaGccaWG0bGaaiykaaGaayjkai aawMcaamaacmaabaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisl cqaHYoGydaWgaaWcbaGaamivaaqabaGccaWG4bWaaSbaaSqaaiaaik daaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaciyzaiaacIhacaGGWbWa aeWaaeaacqGHsislcqaHYoGydaWgaaWcbaGaamitaaqabaGccaWG4b WaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzF aaaaaaa@C2D7@

You can use either the real or imaginary part of these expressions for the displacement and stress fields (they are identical, except for a phase difference). Of course, if you choose to take the real part of one of the functions, you must take the real part for all the others as well. Note that substituting x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaaIWaaaaa@3417@  in the expression for σ 22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaaaaa@33CF@  and setting σ 22 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaGccqGH9aqpcaaIWaaaaa@3599@  yields the equation for the Rayleigh wave speed, so the boundary condition σ 22 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaGccqGH9aqpcaaIWaaaaa@3599@  is satisfied.  The variations with depth of stress amplitude and displacement amplitude are plotted below.

      

 

Important features of this solution are:

  1. The wave is confined to a layer near the surface with thickness about twice the wavelength. 
  2. The horizontal and vertical components of displacement are 90 degrees out of phase.  Material particles therefore describe elliptical orbits as the wave passes by.
  3. The speed of the wave is independent of its wavelength MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  that is to say, the wave is non-dispersive.
  4. Rayleigh waves are exploited in a range of engineering applications, including surface acoustic wave devices; touch sensors; and miniature linear motors.  They are also observed in earthquakes, although these waves are observed to be dispersive, because of density variations of the earth’s surface.

 

 

 

5.6.4 Love waves

 

Love waves are a second form of surface wave, somewhat similar to Rayleigh waves, which propagate through a thin elastic layer bonded to the surface of an elastic half space.  Love waves involve motion perpendicular to the plane of the figure. Assume that

 The layer has thickness H, shear modulus μ f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaWGMbaabeaaaa a@3335@  and shear wave speed c sf MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaiaadAgaae qaaaaa@335F@

 The substrate has shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@321E@  and shear wave speed c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4Caaqabaaaaa@3274@

 The wave speeds satisfy c sf < c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaiaadAgaae qaaOGaeyipaWJaam4yamaaBaaaleaacaWGZbaabeaaaaa@3679@  

The displacement and stress associated with a harmonic Love wave with wavelength λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@321C@  which propagates in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction can be derived from Love potentials

Ψ k ={ U 0 μ f γ δ k1 exp(β x 2 ) β( μβsinγH+ μ f γcosγH ) exp(ik( x 1 ct)) x 2 >0 U 0 δ k1 ( μβcos(γ x 2 )+ μ f γsin(γ x 2 ) ) γ( μβsinγH+ μ f γcosγH ) exp(ik( x 1 ct)) x 2 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfI6aznaaBaaaleaacaWGRbaabeaaki abg2da9maaceaabaqbaeqabiqaaaqaamaalaaabaGaeyOeI0Iaamyv amaaBaaaleaacaaIWaaabeaakiabeY7aTnaaBaaaleaacaWGMbaabe aakiabeo7aNjabes7aKnaaBaaaleaacaWGRbGaaGymaaqabaGcciGG LbGaaiiEaiaacchacaGGOaGaeyOeI0IaeqOSdiMaamiEamaaBaaale aacaaIYaaabeaakiaacMcaaeaacqaHYoGydaqadaqaaiabeY7aTjab ek7aIjGacohacaGGPbGaaiOBaiabeo7aNjaadIeacqGHRaWkcqaH8o qBdaWgaaWcbaGaamOzaaqabaGccqaHZoWzciGGJbGaai4Baiaacoha cqaHZoWzcaWGibaacaGLOaGaayzkaaaaaiGacwgacaGG4bGaaiiCai aacIcacaWGPbGaam4AaiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqa aOGaeyOeI0Iaam4yaiaadshacaGGPaGaaiykaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH+aGpcaaIWaaabaWaaS aaaeaacaWGvbWaaSbaaSqaaiaaicdaaeqaaOGaeqiTdq2aaSbaaSqa aiaadUgacaaIXaaabeaakmaabmaabaGaeqiVd0MaeqOSdiMaci4yai aac+gacaGGZbGaaiikaiabeo7aNjaadIhadaWgaaWcbaGaaGOmaaqa baGccaGGPaGaey4kaSIaeqiVd02aaSbaaSqaaiaadAgaaeqaaOGaeq 4SdCMaci4CaiaacMgacaGGUbGaaiikaiabeo7aNjaadIhadaWgaaWc baGaaGOmaaqabaGccaGGPaaacaGLOaGaayzkaaaabaGaeq4SdC2aae WaaeaacqaH8oqBcqaHYoGyciGGZbGaaiyAaiaac6gacqaHZoWzcaWG ibGaey4kaSIaeqiVd02aaSbaaSqaaiaadAgaaeqaaOGaeq4SdCMaci 4yaiaac+gacaGGZbGaeq4SdCMaamisaaGaayjkaiaawMcaaaaaciGG LbGaaiiEaiaacchacaGGOaGaamyAaiaadUgacaGGOaGaamiEamaaBa aaleaacaaIXaaabeaakiabgkHiTiaadogacaWG0bGaaiykaiaacMca caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaamiEamaaBaaaleaacaaIYaaabeaakiabgYda8iaa icdaaaaacaGL7baaaaa@EDA0@

where U 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfadaWgaaWcbaGaaGimaaqabaaaaa@3228@  is the amplitude of the vertical displacement at the free surface, k=2π/λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGH9aqpcaaIYaGaeqiWdaNaai 4laiabeU7aSbaa@373E@  is the wavenumber; β=k 1 c 2 / c s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaadUgadaGcaaqaai aaigdacqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaai4laiaa dogadaqhaaWcbaGaam4Caaqaaiaaikdaaaaabeaaaaa@3B0E@ , γ=k c 2 / c sf 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNjabg2da9iaadUgadaGcaaqaai aadogadaahaaWcbeqaaiaaikdaaaGccaGGVaGaam4yamaaDaaaleaa caWGZbGaamOzaaqaaiaaikdaaaGccqGHsislcaaIXaaaleqaaaaa@3C14@ ; and c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogaaaa@3150@  is the wave speed, which is given by the positive real roots of

tan{ kH c 2 c sf 2 1 } μ μ f 1 c 2 / c s 2 c 2 / c sf 2 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacshacaGGHbGaaiOBamaacmaabaGaam 4AaiaadIeadaGcaaqaamaalaaabaGaam4yamaaCaaaleqabaGaaGOm aaaaaOqaaiaadogadaqhaaWcbaGaam4CaiaadAgaaeaacaaIYaaaaa aakiabgkHiTiaaigdaaSqabaaakiaawUhacaGL9baacqGHsisldaWc aaqaaiabeY7aTbqaaiabeY7aTnaaBaaaleaacaWGMbaabeaaaaGcda WcaaqaamaakaaabaGaaGymaiabgkHiTiaadogadaahaaWcbeqaaiaa ikdaaaGccaGGVaGaam4yamaaDaaaleaacaWGZbaabaGaaGOmaaaaae qaaaGcbaWaaOaaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaOGaai4l aiaadogadaqhaaWcbaGaam4CaiaadAgaaeaacaaIYaaaaOGaeyOeI0 IaaGymaaWcbeaaaaGccqGH9aqpcaaIWaaaaa@5529@

 

This relationship is very unlike the equations for wave speeds in unbounded or semi-infinite solids, and leads to a number of counter-intuitive results.   Note that

  1. The wave speed depends on its wavelength.  A wave with these properties is said to be dispersive, because a pulse consisting of a spectrum of harmonic waves tends to spread out;
  2. The wave speed is always faster than the shear wave speed of the layer, but less than the wave speed in the substrate;
  3. If a wave with wavenumber k 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaaGimaaqabaaaaa@323E@  propagates at speed c, then waves with wavenumber k n = k 0 +nπ/H c 2 / c sf 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgadaWgaaWcbaGaamOBaaqabaGccq GH9aqpcaWGRbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaamOBaiab ec8aWjaac+cacaWGibWaaOaaaeaacaWGJbWaaWbaaSqabeaacaaIYa aaaOGaai4laiaadogadaqhaaWcbaGaam4CaiaadAgaaeaacaaIYaaa aOGaeyOeI0IaaGymaaWcbeaaaaa@4288@  where n is any integer, also propagate at the same speed.   These waves are associated with different propagation modes for the wave.  Each propagation mode has a characteristic displacement distribution through the thickness of the layer, as discussed below.
  4. A wave with a particular wave number can propagate at several different speeds, depending on the mode.  The number of modes that can exist at a particular wave number increases with the wave number.  You can see this in the plot of wave speed MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@ v- wave number below.
  5. Dispersive wave motion is often characterized by relating the frequency of the wave to its wave number, rather than by relating wave speed to wave number.   The (angular) frequency is related to wave number and wave speed by the usual formula ω=ck MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabg2da9iaadogacaWGRbaaaa@3513@ .  Substituting this result into the equation for wave speed yields the Dispersion Relation for the wave

tan{ (ωH/ c sf ) 2 (kH) 2 } μ μ f (kH) 2 (ωH/ c s ) 2 (ωH/ c sf ) 2 (kH) 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacshacaGGHbGaaiOBamaacmaabaWaaO aaaeaacaGGOaGaeqyYdCNaamisaiaac+cacaWGJbWaaSbaaSqaaiaa dohacaWGMbaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaGGOaGaam4AaiaadIeacaGGPaWaaWbaaSqabeaacaaIYaaaaaqa baaakiaawUhacaGL9baacqGHsisldaWcaaqaaiabeY7aTbqaaiabeY 7aTnaaBaaaleaacaWGMbaabeaaaaGcdaWcaaqaamaakaaabaGaaiik aiaadUgacaWGibGaaiykamaaCaaaleqabaGaaGOmaaaakiabgkHiTi aacIcacqaHjpWDcaWGibGaai4laiaadogadaWgaaWcbaGaam4Caaqa baGccaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaaakeaadaGcaaqaai aacIcacqaHjpWDcaWGibGaai4laiaadogadaWgaaWcbaGaam4Caiaa dAgaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaacI cacaWGRbGaamisaiaacMcadaahaaWcbeqaaiaaikdaaaaabeaaaaGc cqGH9aqpcaaIWaaaaa@64B9@

 

The nonzero displacement component is

u 3 ={ U 0 ( μβsin(γ x 2 ) μ f γcos(γ x 2 ) ) ( μβsinγH+ μ f γcosγH ) exp(ik( x 1 ct)) x 2 <0 U 0 μ f γexp(β x 2 ) ( μβsinγH+ μ f γcosγH ) exp(ik( x 1 ct)) x 2 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpdaGabaqaauaabeqaceaaaeaadaWcaaqaaiabgkHiTiaadwfa daWgaaWcbaGaaGimaaqabaGcdaqadaqaaiabeY7aTjabek7aIjGaco hacaGGPbGaaiOBaiaacIcacqaHZoWzcaWG4bWaaSbaaSqaaiaaikda aeqaaOGaaiykaiabgkHiTiabeY7aTnaaBaaaleaacaWGMbaabeaaki abeo7aNjGacogacaGGVbGaai4CaiaacIcacqaHZoWzcaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiykaaGaayjkaiaawMcaaaqaamaabmaaba GaeqiVd0MaeqOSdiMaci4CaiaacMgacaGGUbGaeq4SdCMaamisaiab gUcaRiabeY7aTnaaBaaaleaacaWGMbaabeaakiabeo7aNjGacogaca GGVbGaai4Caiabeo7aNjaadIeaaiaawIcacaGLPaaaaaGaciyzaiaa cIhacaGGWbGaaiikaiaadMgacaWGRbGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccqGHsislcaWGJbGaamiDaiaacMcacaGGPaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH8aapcaaIWaaa baWaaSaaaeaacaWGvbWaaSbaaSqaaiaaicdaaeqaaOGaeqiVd02aaS baaSqaaiaadAgaaeqaaOGaeq4SdCMaciyzaiaacIhacaGGWbGaaiik aiabgkHiTiabek7aIjaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPa aabaWaaeWaaeaacqaH8oqBcqaHYoGyciGGZbGaaiyAaiaac6gacqaH ZoWzcaWGibGaey4kaSIaeqiVd02aaSbaaSqaaiaadAgaaeqaaOGaeq 4SdCMaci4yaiaac+gacaGGZbGaeq4SdCMaamisaaGaayjkaiaawMca aaaaciGGLbGaaiiEaiaacchacaGGOaGaamyAaiaadUgacaGGOaGaam iEamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadogacaWG0bGaaiyk aiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWG4bWaaSbaaSqaaiaaikdaaeqa aOGaeyOpa4JaaGimaaaaaiaawUhaaaaa@E28F@

The nonzero stresses in the layer can be determined from σ 3i = μ f u 3 / x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaamyAaa qabaGccqGH9aqpcqaH8oqBdaWgaaWcbaGaamOzaaqabaGccqGHciIT caWG1bWaaSbaaSqaaiaaiodaaeqaaOGaai4laiabgkGi2kaadIhada WgaaWcbaGaamyAaaqabaaaaa@3F6C@ , but the calculation is so trivial the result will not be written out here.  The wave speed is plotted as a function of wave number below, for the particular case μ= μ f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9iabeY7aTnaaBaaale aacaWGMbaabeaaaaa@35F1@ , c s / c sf =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGcca GGVaGaam4yamaaBaaaleaacaWGZbGaamOzaaqabaGccqGH9aqpcaaI Yaaaaa@37F4@ .   The displacement amplitude a function of depth is also shown for several modes. 

           

 

 

 

5.6.5 Elastic waves in waveguides

 

The surface layer discussed in the preceding section is an example of a waveguide: it is a structure which causes waves to propagate in a particular direction, as a result of the confining effect of its geometry.

 

The picture shows a much simpler example of a waveguide: it is a thin sheet of material, with thickness H and infinite length in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  and x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@324E@  directions.  The strip can guide three types of wave:

  1. Transverse waves, which propagate in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction with particle motion in the x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@324E@  direction;
  2. Flexural waves, which propagate in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction with particle motion in the x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa@324D@  direction;
  3. Longitudinal waves, which propagate in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction with particle motion in the x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa@324D@  direction.

The solutions for cases 2 and 3 are lengthy, but the solution for case 1 is simple, and can be used to illustrate the general features of waves in waveguides.  For transverse waves:

  1. The wave can be any member of the following family of possible displacement distributions

u 3 = U 0 { sin 2 (πn/2)sin(nπ x 2 /2H)+ cos 2 (πn/2)cos(nπ x 2 /2H) }exp(ik( x 1 ct)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGvbWaaSbaaSqaaiaaicdaaeqaaOWaaiWaaeaaciGGZbGa aiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccaGGOaGaeqiWdaNaam OBaiaac+cacaaIYaGaaiykaiGacohacaGGPbGaaiOBaiaacIcacaWG UbGaeqiWdaNaamiEamaaBaaaleaacaaIYaaabeaakiaac+cacaaIYa GaamisaiaacMcacqGHRaWkciGGJbGaai4BaiaacohadaahaaWcbeqa aiaaikdaaaGccaGGOaGaeqiWdaNaamOBaiaac+cacaaIYaGaaiykai GacogacaGGVbGaai4CaiaacIcacaWGUbGaeqiWdaNaamiEamaaBaaa leaacaaIYaaabeaakiaac+cacaaIYaGaamisaiaacMcaaiaawUhaca GL9baaciGGLbGaaiiEaiaacchacaGGOaGaamyAaiaadUgacaGGOaGa amiEamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadogacaWG0bGaai ykaiaacMcaaaa@6CF3@

where n=0,1,2…, and you can use either the real or imaginary part as the solution. This displacement represents a harmonic wave that has wavenumber k=2π/λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacqGH9aqpcaaIYaGaeqiWdaNaai 4laiabeU7aSbaa@373E@  where λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSbaa@321C@  is the wavelength in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction, which propagates in the x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  direction with speed c.  The variation of displacement with x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa@324D@  at any fixed value of x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@  is a standing wave with wavelength H/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIeacaGGVaGaamOBaaaa@32DB@  and angular frequency ω=kc MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabg2da9iaadUgacaWGJbaaaa@3513@ . Each value of n corresponds to a different propagation mode.

  1. The speed of wave propagation (usually referred to as the phase velocity of the wave)  satisfies

c 2 c s 2 = ( nπ 2kH ) 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaam4yamaaCaaaleqabaGaaG OmaaaaaOqaaiaadogadaqhaaWcbaGaam4CaaqaaiaaikdaaaaaaOGa eyypa0ZaaeWaaeaadaWcaaqaaiaad6gacqaHapaCaeaacaaIYaGaam 4AaiaadIeaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSIaaGymaaaa@3F7E@

  1. The wave speeds for modes with n>0 depend on the wave number: i.e. the waves are dispersive;
  2. There are an infinite number of possible wave speeds for each wave number.  Each wave speed is associated with a particular propagation mode n.
  3. The formula for wave speed can be re-written as an equation relating the angular frequency ω=kc MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabg2da9iaadUgacaWGJbaaaa@3513@  to the wave number k

( ωH c s ) 2 = ( nπ 2 ) 2 + ( kH ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaWaaSaaaeaacqaHjpWDcaWGib aabaGaam4yamaaDaaaleaacaWGZbaabaaaaaaaaOGaayjkaiaawMca amaaCaaaleqabaGaaGOmaaaakiabg2da9maabmaabaWaaSaaaeaaca WGUbGaeqiWdahabaGaaGOmaaaaaiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaGccqGHRaWkdaqadaqaaiaadUgacaWGibaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaaaa@43B4@

This is called the Dispersion relation for the wave.

  1. Dispersive waves have a second wave speed associated with them called the Group Velocity.  This wave speed is defined as the slope of the dispersion relation c g =dω/dk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4zaaqabaGccq GH9aqpcaWGKbGaeqyYdCNaai4laiaadsgacaWGRbaaaa@38BA@  (in contrast, the phase velocity is c=ω/k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH9aqpcqaHjpWDcaGGVaGaam 4Aaaaa@35C6@  ).  For the waveguide considered here

c g = dω dk = c s kH (nπ/2) 2 + k 2 H 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4zaaqabaGccq GH9aqpdaWcaaqaaiaadsgacqaHjpWDaeaacaWGKbGaam4AaaaacqGH 9aqpdaWcaaqaaiaadogadaWgaaWcbaGaam4CaaqabaGccaWGRbGaam isaaqaamaakaaabaGaaiikaiaad6gacqaHapaCcaGGVaGaaGOmaiaa cMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGRbWaaWbaaSqabe aacaaIYaaaaOGaamisamaaCaaaleqabaGaaGOmaaaaaeqaaaaaaaa@47F6@

The group velocity, like the phase velocity, depends on the propagation mode and the wave number.  The group velocity has two physical interpretations (i) it is the speed at which the energy in a harmonic wave propagates along the waveguide; and (ii) it is the propagation speed of an amplitude modulated wave of the form

u 3 = U 0 cos(Δk x 1 Δωt)sin(k x 1 ωt)= U 0 { sin[ (k+Δk) x 1 (ω+Δω)t ]+sin[ (kΔk) x 1 (ωΔω)t ] }/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGvbWaaSbaaSqaaiaaicdaaeqaaOGaci4yaiaac+gacaGG ZbGaaiikaiabfs5aejaadUgacaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaeyOeI0IaeuiLdqKaeqyYdCNaamiDaiaacMcaciGGZbGaaiyAaiaa c6gacaGGOaGaam4AaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHsi slcqaHjpWDcaWG0bGaaiykaiabg2da9iaadwfadaWgaaWcbaGaaGim aaqabaGcdaGadaqaaiGacohacaGGPbGaaiOBamaadmaabaGaaiikai aadUgacqGHRaWkcqqHuoarcaWGRbGaaiykaiaadIhadaWgaaWcbaGa aGymaaqabaGccqGHsislcaGGOaGaeqyYdCNaey4kaSIaeuiLdqKaeq yYdCNaaiykaiaadshaaiaawUfacaGLDbaacqGHRaWkciGGZbGaaiyA aiaac6gadaWadaqaaiaacIcacaWGRbGaeyOeI0IaeuiLdqKaam4Aai aacMcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaaiikaiab eM8a3jabgkHiTiabfs5aejabeM8a3jaacMcacaWG0baacaGLBbGaay zxaaaacaGL7bGaayzFaaGaai4laiaaikdaaaa@7F8C@

where Δk<<k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadUgacqGH8aapcqGH8aapca WGRbaaaa@35B6@  and Δω<<ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeM8a3jabgYda8iabgYda8i abeM8a3baa@3770@  are the wave number and frequency of the modulation, and k,ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUgacaGGSaGaeqyYdChaaa@33D5@  are the wavenumber and frequency of the carrier wave.   The carrier wave propagates with speed c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogaaaa@3150@ , while the modulation (which can be regarded as a `group’ of wavelets) propagates at speed c g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4zaaqabaaaaa@3268@ .  Note that for a non-dispersive wave, the group and phase velocities are the same.