# Quasi two dimensional elastic waves

The elastodynamic equation for an isotropic and homogeneous compound is given by \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \left( \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z} \right) \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 w}{\partial y \partial z} \right) \\ \rho \frac{\partial^2 w}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 w}{\partial z^2} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial z} + \frac{\partial^2 v}{\partial y \partial z} \right) \end{align} with $$\mu$$ and $$\lambda$$ the lame constants and $$[u,v,w]^T$$ the displacment vector.

If no variation along the z-direction is assumed ($$\partial / \partial z =0$$), this becomes \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + (\lambda+\mu) \frac{\partial^2 v}{\partial x \partial y} \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial x^2} + (\lambda+\mu)\frac{\partial^2 u}{\partial x \partial y} \\ \rho \frac{\partial^2 w}{\partial t^2} &= \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) \end{align}

Question: is this case with no variation along one direction realistic? Do the equations make sense for a real system (if yes, what kind of system?)? Does this state have a name, for example, a combination of plane stress and antishear strain?

Unfortunately, "realistic" and "makes sense for a real system" are entirely heuristic. For example, it seems you consider the first "full" equations to be realistic and that they make sense for a real system -- but, they are actually approximations made to get the system into a form that can be solved. The approximations are based on some assumptions -- for example, that the system is a continuum.

Is that "realistic?" Does it work for "real systems?"

It does, provided the assumptions are valid. It gives pretty good answers under many conditions that do represent real things, so that's a great start. But it's still just an approximation.

So to your actual question -- for a quasi-2D problem, you drop the terms in the third direction. That means you've made some approximations. Typically you aren't assuming there is no variation in the third direction ($$\partial/\partial z = 0$$), instead you assume that variations in the third direction are much smaller than the other two:

$$\frac{\partial}{\partial z} \ll \frac{\partial}{\partial x}\\ \frac{\partial}{\partial z} \ll \frac{\partial}{\partial y}\\$$

and therefore the terms with $$\partial/\partial z$$ can be neglected with respect to the changes in the other directions.

Is this realistic? Well sure, it's a good model so long as the problem you are interested in has small variations in the third direction relative to the first two. Are there real systems that show this? Again, sure, any time the variations in the third direction are small relative to the other two.

A quick example could be a flat plate where perturbations are introduced on one edge. That would generate motion that is quasi-2D -- in reality, the motion is probably 3D, but the variations in the third direction are really small, and so they can be neglected and you'd still get a good answer for the motion in the other two directions.