Simplifying generalised Wave Equation

[BOYLANATOR]

Homework Statement

Please find question attached. The context is a lecture course on seismic theory for exploration geophysics.

Homework 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 traveling down into the Earth) or perhaps an identity I'm missing.

Thanks in advance for help.

 

[stevendaryl]

Homework Statement

Please find question attached. The context is a lecture course on seismic theory for exploration geophysics.

Homework 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 traveling down into the Earth) or perhaps an identity I'm missing.

Thanks in advance for help.

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.

 

[BOYLANATOR]

I'll give it a go...

The given Wave Equation is:

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

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

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

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:

$C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ij} \delta_{kl} + \delta_{il} \delta_{jk})$
​

Thus, the new Wave Equation is given by:

$\lambda \delta_{ij} \delta_{kl} (\frac{\partial^2 u_l}{\partial x_j \partial x_k}) + \mu (\delta_{ij} \delta_{kl}(\frac{\partial^2 u_l}{\partial x_j \partial x_k})+ \delta_{il} \delta_{jk}(\frac{\partial^2 u_l}{\partial x_j \partial x_k})) =\rho \frac{\partial^2 u_i} {\partial t^2}$
​

This simplifies to:

$\lambda \frac{\partial^2 u_k}{\partial x_i \partial x_k} + \mu \frac{\partial^2 u_i}{\partial^2 x_j }+ \mu \frac{\partial^2 u_j}{\partial x_j \partial x_i} =\rho \frac{\partial^2 u_i} {\partial t^2}.$
​

This is not the same as the given solution:

$(\lambda + \mu) \frac{\partial^2 u_k}{\partial x_i \partial x_k} - \mu \frac{\partial^2 u_i}{\partial^2 x_i } =\rho \frac{\partial^2 u_i} {\partial t^2}.$
​