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Navier's equations of motion in seismic have 3 solutions, according to how it is resolved (boundary conditions):

  • for Vp (compression waves)
  • for Vs (shear waves)
  • for Vr (Rayleigh waves)

I know the boundary conditions for Vr, that is the half-space (null stress at the surface and null displacements and stress at infinite depth).

What would be the boundary conditions for Vp and Vs solutions and why do they derivate for infinite medium and not half-space? What does an infinite medium imply?

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    $\begingroup$ Could you update your question with the equation you're referring to? It's not clear if you just mean the elastic wave equation or not; I'm not used to finding this from the Navier(-Stokes?) equations. Even then, it isn't entirely clear to me what you're asking for. The compressional and shear waves are body waves, they need a "body" to propagate through -- not a surface with which they'll interact (which will transform them into, e.g., Rayleigh waves). So a typical assumption is vanishing stress and displacement at infinity; look up the "Sommerfeld radiation condition" maybe? $\endgroup$
    – Erik
    Oct 12, 2021 at 14:09
  • $\begingroup$ I support what @Erik is presenting and asking about. I'll supplement the content currently here with just referencing to some classic books that I believe and hope can certainly answer your question(s). Here are two that do an outstanding job of literally deriving various elastic wave equations for half-spaces and other more realistic scenarios: 1] Petroleum Seismology (2005) and 2] Seismic Data Analysis. Also - and I know this is quite hard sometimes - try to isolate one "thing" you aren't sure of and figure that out. Then, move onto other topics. $\endgroup$
    – nate
    Mar 5, 2022 at 1:44

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