1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simplifying generalised Wave Equation

  1. Dec 7, 2014 #1
    1. The problem statement, all variables and given/known data
    Please find question attached. The context is a lecture course on seismic theory for exploration geophysics.

    2. Relevant equations
    The substitution for the elasticity tensor made in my solution is given in the lecture notes.

    Please find my attempt at the solution attached (hopefully readable). For one, I have no idea how to get a negative term in my solution. I wonder if it could be to do with the direction of motion (the sound waves travelling down into the Earth) or perhaps an identity I'm missing.

    Thanks in advance for help.
     

    Attached Files:

  2. jcsd
  3. Dec 8, 2014 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    I know this sounds like a lot of work, but do you think you could transcribe the equations using LaTex or using the provided symbols? It's really helpful for communication in this forum.
     
  4. Dec 8, 2014 #3
    I'll give it a go...

    The given Wave Equation is:

    [itex] \frac{\partial }{\partial x_j}(C_{ijkl}\frac{\partial u_l}{\partial x_k}) - \frac{\partial^2 (\rho u_i)}{\partial t^2} = 0 .[/itex]​

    But we have the simplifications that density is constant in time and that the material is homogeneous. So we can pull out [itex]C_{ijkl} [/itex] and [itex]\rho[/itex] to give:

    [itex] C_{ijkl}(\frac{\partial^2 u_l}{\partial x_j \partial x_k}) - \rho \frac{\partial^2 u_i} {\partial t^2} = 0 .[/itex]​

    We are then told that the material is isotropic. The lecture material gives an expression for the elastic tensor which is invariant under rotations, i.e. isotropic:

    [itex] C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ij} \delta_{kl} + \delta_{il} \delta_{jk}) [/itex]
    Thus, the new Wave Equation is given by:

    [itex] \lambda \delta_{ij} \delta_{kl} (\frac{\partial^2 u_l}{\partial x_j \partial x_k}) + \mu (\delta_{ij} \delta_{kl}(\frac{\partial^2 u_l}{\partial x_j \partial x_k})+ \delta_{il} \delta_{jk}(\frac{\partial^2 u_l}{\partial x_j \partial x_k})) =\rho \frac{\partial^2 u_i} {\partial t^2}[/itex]
    This simplifies to:

    [itex] \lambda \frac{\partial^2 u_k}{\partial x_i \partial x_k} + \mu \frac{\partial^2 u_i}{\partial^2 x_j }+ \mu \frac{\partial^2 u_j}{\partial x_j \partial x_i} =\rho \frac{\partial^2 u_i} {\partial t^2}.[/itex]
    This is not the same as the given solution:

    [itex] (\lambda + \mu) \frac{\partial^2 u_k}{\partial x_i \partial x_k} - \mu \frac{\partial^2 u_i}{\partial^2 x_i } =\rho \frac{\partial^2 u_i} {\partial t^2}.[/itex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Simplifying generalised Wave Equation
  1. Wave equation (Replies: 2)

  2. Wave equation (Replies: 4)

  3. Wave Equation (Replies: 1)

  4. Equation of a Wave (Replies: 4)

  5. Wave equation (Replies: 1)

Loading...