Waves in very thin layer

Question

Elastic waves in the earth are described by the elastodynamic equations \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}

Is it possible to have a very thin and stiff layer in a surrounding soft material? If yes, is it possible to approximate the thin layer by a thin film with free boundaries and a uniform displacement over the thickness direction ($\partial / \partial z = 0$)? In this case, the equations of motion become \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} Do these equations describe a realistic case? Does this state have a specific name (like plane stress or antiplane shear ...)?

Answer 1

The problem with the cancelling you suggested is that it is imposed globally for the PDE and not locally (just on the thin layer).

Usually when we talk about vertically layered media we use a plane wave $\exp(i(\mathbf{k}\cdot\mathbf{x}-wt))$ trial solution (you can plug this in to confirm that it satisfies the PDE) and match boundary conditions at the interface of two layers to determine reflection and transmission coefficients. If you are not familiar with this, Aki and Richards (1980) is a good reference. See also the Zoeppritz equations https://en.wikipedia.org/wiki/Zoeppritz_equations

Introducing a very thin, stiff layer is equivalent to placing a 0 displacement (Dirichlet) boundary condition at the location of the layer. Physically, this models an infinite impedance reflector and will generate a total reflection. Waves on a string with both ends fixed are modeled in the same way.

Setting vertical derivatives in displacement to 0 is imposing a Neumann B.C. on the location of the thin layer. This is exactly how the free surface is modeled. Physically, this models vanishing vertical tractions at the boundary location.

So, if you want to model seismic wave propagation with an extremely stiff layer, then use the Dirichelet 0 displacement boundary condition. If the thin layer is only 'very stiff', use the reflection and transmission coefficients provided by the Zoeppritz equations. If you want tractions to disappear, use the Neumann boundary conditions. But, beware; this doesn't make physical sense if the thin layer is sandwiched between two other layers.

Hope this helps. -D